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5 (five) is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
, numeral and digit. It is the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
, and
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
, following 4 and preceding 6, and is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. It has attained significance throughout history in part because typical humans have five digits on each hand.


In mathematics

5 is the third smallest
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, and the second super-prime. It is the first
safe prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +  ...
, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
and the third
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
exponent, as well as the third
Catalan number In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Ca ...
, and the third
Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +  ...
. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
s, ( 3, 5) and (5, 7). It is also a
sexy prime In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If o ...
with the fifth prime number and first prime repunit, 11. Five is the third
factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! +&n ...
, an alternating factorial, and an
Eisenstein prime In mathematics, an Eisenstein prime is an Eisenstein integer : z = a + b\,\omega, \quad \text \quad \omega = e^, that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units , itself ...
with no imaginary part and real part of the form 3p1. In particular, five is the first
congruent number In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) c ...
, since it is the length of the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
of the smallest integer-sided right triangle. Five is the second
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
of the form 2^+ 1, and more generally the second Sierpiński number of the first kind, n^n+ 1. There are a total of five known Fermat primes, which also include 3, 17,
257 __NOTOC__ Year 257 ( CCLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Gallienus (or, less frequently, year 10 ...
, and
65537 65537 is the integer after 65536 and before 65538. In mathematics 65537 is the largest known prime number of the form 2^ +1 (n = 4). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
. The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 or 52, while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
number of sides that can be construncted purely with a compass and straight-edge, which includes the five-sided
regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
. Apropos, 31 is also equal to the sum of the maximum number of
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
s inside a circle that are formed from the sides and
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s of the first five n-sided
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem. The number 5 is the fifth
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
, being 2 plus 3. It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a
Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
and a
Markov number A Markov number or Markoff number is a positive integer ''x'', ''y'' or ''z'' that is part of a solution to the Markov Diophantine equation :x^2 + y^2 + z^2 = 3xyz,\, studied by . The first few Markov numbers are : 1, 2, 5, 13, 29, 34, 89 ...
, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13,
194 Year 194 (Roman numerals, CXCIV) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Septimius and Septimius (or, less frequently, year 9 ...
), (5, 29, 433), ... ( lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers. 5 is the third Mersenne prime exponent of the form 2^n1, which yields 31: the prime index of the third
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
and second
double Mersenne prime In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and ...
127, as well as the third double Mersenne prime exponent for the number
2,147,483,647 The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes. The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel ...
, which is the largest value that a signed 32-bit integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime M_ = 223058...93951 − 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers M_ are the only known prime terms, with a sixth possible candidate in the order of 101037.7094. These prime sequences are conjectured to be prime up to a certain limit. Every odd number greater than 1 is the sum of at most five prime numbers, and every odd number greater than 5 is conjectured to be expressible as the sum of three prime numbers. Helfgtott has provided a proof of the latter, also known as the
odd Goldbach conjecture In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three prime number, prime ...
, that is already widely acknowledged by mathematicians as it still undergoes
peer-review Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work ( peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer revie ...
. The sums of the first five non-primes greater than zero 1 + 4 + 6 + 8 + 9 and the first five prime numbers 2 + 3 + 5 + 7 + 11 both equal 28; the 7th
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
and like 6 a
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...
, which also includes 496, the 31st triangular number and perfect number of the form 2^−1(2^1) with a p of 5, by the
Euclid–Euler theorem The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...
. There are a total of five known
unitary perfect number A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself (a divisor ''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n''/''d'' share no common factors). Some perfect ...
s, which are numbers that are the sums of their positive
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
unitary divisor In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 an ...
s. A sixth unitary number, if discovered, would have at least nine odd prime factors. Five is conjectured to be the only odd
untouchable number An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. ...
, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. In
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...
s, 5 is a
pentagonal number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
, with the
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of pentagonal numbers starting: 1, 5, 12, 22, 35, ... * 5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ... Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5. * 5 is a square pyramidal number: 1, 5, 14, 30, 55, ... The sum of the first four members is 50 while the fifth indexed member in the sequence is 55. * 5 is a
centered square number In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each cen ...
: 1, 5, 13, 25, 41, ... The fifth
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usu ...
or 52 is 25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) ''primitive''
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s. The factorial of five, or 5 ! = 120, is the sum of the first fifteen non-zero positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, and 15th
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
, which in turn is the sum of the first five non-zero positive integers and 5th triangular number. 35, which is the fourth or fifth pentagonal and
tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
, is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15. 5 is the value of the central
cell Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery ...
of the only non-trivial normal magic square, also called the ''Lo Shu'' square. Its 3 x 3 array of squares has a magic constant M of 15, where the sums of its rows, columns, and diagonals are all equal to fifteen. 5 is also the value of the central cell the only non-trivial order-3 normal magic hexagon that is made of nineteen cells.
Polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
equations of degree and below can be solved with radicals, while
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s of degree 5, and higher, cannot generally be so solved. This is the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
. This is related to the fact that the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
\mathrm_ is a
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
for ''n'' ⩽ 4 and not solvable for ''n'' ⩾ 5.
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
, e^+ 1 = 0, contains five essential
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s used widely in mathematics: Archimedes' constant \pi,
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
e, the imaginary number i,
unity Unity may refer to: Buildings * Unity Building, Oregon, Illinois, US; a historic building * Unity Building (Chicago), Illinois, US; a skyscraper * Unity Buildings, Liverpool, UK; two buildings in England * Unity Chapel, Wyoming, Wisconsin, US; ...
1, and
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
0, which makes this formula a renown example of beauty in mathematics.


In geometry

A
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a
regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
as self-intersecting edges that are proportioned in
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, \varphi. Its internal geometry appears prominently in Penrose tilings, and is a
facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol . A similar figure to the pentagram is a five-pointed
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
isotoxal star ☆ without self-intersecting edges. Generally, star polytopes that are regular only exist in
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s 2 ⩽ n < 5. In
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, all graphs with 4 or fewer vertices are
planar Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), ...
, however, there is a graph with 5 vertices that is not: ''K''5, the
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
with 5 vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram. By
Kuratowski's theorem In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdi ...
, a finite graph is planar
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
it does not contain a subgraph that is a subdivision of ''K''5, or the complete bipartite
utility graph As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
''K''3,3. A similar graph is the
Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is n ...
, which is
strongly connected In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that ...
and also nonplanar. It is most easily described as graph of a pentagram ''embedded'' inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5. The Petersen graph, which is also a
distance-regular graph In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . ...
, is one of only 5 known
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
graphs with no
Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...
s.Royle, G
"Cubic Symmetric Graphs (The Foster Census)."
The
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the Petersen graph is the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
\mathrm_ of order
120 120 may refer to: *120 (number), the number * AD 120, a year in the 2nd century AD *120 BC, a year in the 2nd century BC *120 film, a film format for still photography * ''120'' (film), a 2008 film * 120 (MBTA bus) * 120 (New Jersey bus) * 120 (Ken ...
= 5!. The
chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...
of the plane is at least five, depending on the choice of set-theoretical axioms: the minimum number of
colors Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associa ...
required to color the plane such that no pair of points at a distance of 1 has the same color. Whereas the hexagonal
Golomb graph In graph theory, the Golomb graph is a polyhedral graph with 10 vertices and 18 edges. It is named after Solomon W. Golomb, who constructed it (with a non-planar embedding) as a unit distance graph that requires four colors in any graph colori ...
and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple four-coloring
Moser spindle In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William, with seven vertices and eleven edges. It is a unit d ...
s are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure. The
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
contains a total of five
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s, or arrays of points defined by discrete
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
operations:
hexagonal In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
,
oblique Oblique may refer to: * an alternative name for the character usually called a slash (punctuation) ( / ) *Oblique angle, in geometry *Oblique triangle, in geometry * Oblique lattice, in geometry * Oblique leaf base, a characteristic shape of the b ...
,
rectangular In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
, centered rectangular, and
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
lattices. The plane can also be tiled monohedrally with convex
pentagons In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...
in fifteen different ways, three of which have
Laves tiling This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ...
s as special cases. Five points are needed to determine a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
, in the same way that two points are needed to determine a line. A
Veronese surface In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giu ...
in the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
\mathbb^5 of a conic generalizes a
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
condition for a point to be contained inside a conic. There are 5
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
in particular contains
pentagonal In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
faces, while the icosahedron, its
dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
, has a
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
that is a regular pentagon. There are also 5: ☆ Regular polyhedron compounds: the
stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...
, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.
Icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
\mathrm I_ is isomorphic to the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
on 5 letters \mathrm A_ of order
120 120 may refer to: *120 (number), the number * AD 120, a year in the 2nd century AD *120 BC, a year in the 2nd century BC *120 film, a film format for still photography * ''120'' (film), a 2008 film * 120 (MBTA bus) * 120 (New Jersey bus) * 120 (Ken ...
, realized by actions on these uniform polyhedron compounds. ☆ Space-filling convex polyhedra: the triangular prism,
hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...
, cube, truncated octahedron, and
gyrobifastigium In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...
. While the cube is the only Platonic solid that can tessellate space on its own, the truncated octahedron and the gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can also tessellate space with their own copies. ☆
Cell-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...
parallelohedra: any parallelepiped, as well as the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...
and
elongated dodecahedron In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...
, and the hexagonal prism and truncated octahedron. The cube is a special case of a parallelepiped, with the rhombic dodecahedron the only
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan s ...
to tessellate space on its own. ☆ Regular abstract polyhedra, which include the
excavated dodecahedron In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef1g1 stellation of the icosahedron. It appears in Magnus We ...
and the
dodecadodecahedron In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The ed ...
. They have combinatorial symmetries transitive on
flags A flag is a piece of textile, fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic desi ...
of their elements, with
topologies In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
equivalent to that of
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
s and the ability to tile the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
. The
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
, or pentatope, is the self-dual fourth-dimensional analogue of the
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
, with
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
symmetry \mathrm_ of order
120 120 may refer to: *120 (number), the number * AD 120, a year in the 2nd century AD *120 BC, a year in the 2nd century BC *120 film, a film format for still photography * ''120'' (film), a 2008 film * 120 (MBTA bus) * 120 (New Jersey bus) * 120 (Ken ...
= 5 ! and \mathrm_ group structure. Made of five tetrahedra, its
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
is a
regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
and its orthographic projection is equivalent to the
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
''K''5. It is one of six regular 4-polytopes, made of thirty-one elements: five vertices, ten edges, ten
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
, five tetrahedral cells and one 4-face. *A regular 120-cell, the dual ''polychoron'' to the regular
600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...
, can fit one hundred and twenty 5-cells. Also, five
24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...
s fit inside a
small stellated 120-cell In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the great gran ...
, the first stellation of the 120-cell. *A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
nonconvex ''duoantiprismatic'' solution in the fourth dimension, constructed from the
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
cartesian product and made of fifty
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, ten
pentagrammic crossed antiprism In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It differs from the pentagrammic antiprism by having oppo ...
s, ten
pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...
s, and fifty vertices. *The
grand antiprism In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope ...
, which is the only known non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges. *The abstract four-dimensional
57-cell In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry or ...
is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge. The
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen dodecahedra. The skeleton of the hemi-dodecahedron is the
Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is n ...
. Overall, the fourth dimension contains five
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
s that form a finite number of uniform polychora: \mathrm A_, \mathrm B_, \mathrm D_, \mathrm F_, and \mathrm H_, with four of these Coxeter groups capable of generating the same finite forms without \mathrm D_; accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. There are also a total of five
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s that generate non-prismatic Eucledian honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of three-dimensional
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
\mathrm_ or four-dimensional \mathrm_ symmetry do not exist in dimensions ''n'' ⩾ 5; however, there is the
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
prismatic group \mathrm_ × \mathrm_ in the fifth dimension which contains prisms of regular and uniform
4-polytopes In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
that have \mathrm_ symmetry. The
5-simplex In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5 ...
is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of n- simplexes in any n dimensions. The 5-simplex has the Coxeter group \mathrm_ as its symmetry group, of order 720 = 6 !, whose group structure is represented by the symmetric group \mathrm_, the only finite symmetric group which has an
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
. The
5-cube In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, ...
, made of ten
tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...
s and the 5-cell as its vertex figure, is also regular and one of thirty-one
uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been deter ...
s under the Coxeter \mathrm B_ hypercubic group. The
demipenteract In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5 ...
, with one hundred and twenty cells, is the only fifth-dimensional
semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...
, and has the
rectified 5-cell In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In t ...
as its vertex figure, which is one of only three semiregular 4-polytopes alongside the
rectified 600-cell In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icos ...
and the
snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces ...
. In the fifth dimension, there are five regular paracompact honeycombs, all with
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
and
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s. There are exclusively twelve complex aperiotopes in \mathbb^n complex spaces of dimensions n5, with fifteen in \mathbb^4 and sixteen in \mathbb^3; alongside complex polytopes in \mathbb^5 and higher under simplex, hypercubic and
orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
groups, the latter with van Oss polytopes. There are five exceptional Lie groups: \mathfrak_2, \mathfrak_4, \mathfrak_6, \mathfrak_7, and \mathfrak_8. The smallest of these, \mathfrak_2, can be represented in five-dimensional complex space and
projected Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released t ...
in the same number of dimensions as a ball rolling on top of another ball, whose
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
is described in two-dimensional space. \mathfrak_8, the largest of all five exceptional groups, also contains the other four as
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s and is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the
600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...
. There are also five
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
s that are excluded from
finite simple group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s of Lie type. The five
Mathieu groups In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 object ...
constitute the first generation in the happy family of
sporadic groups In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. Th ...
. These are also the first five sporadic groups to have been described, defined as \mathrm_ multiply transitive permutation groups on n
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
, with n
In mathematics, an element (or member) of a Set (mathematics), set is any one of the Equality (mathematics), distinct Mathematical object, objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, ...
. In particular, \mathrm_, the smallest of all sporadic groups, has a
rank 3 action Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
on fifty-five points from an induced action on
unordered pair In mathematics, an unordered pair or pair set is a set of the form , i.e. a set having two elements ''a'' and ''b'' with no particular relation between them, where = . In contrast, an ordered pair (''a'', ''b'') has ''a'' as its first ...
s, as well as two five-dimensional faithful complex irreducible representations over the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with ''n'' elements. Of precisely five different
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es of
maximal subgroup In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...
s of \mathrm_, one is the almost simple symmetric group \mathrm_5 (of order 5 !), and another is \mathrm_, also almost simple, that functions as a point stabilizer which has 5 as its largest prime factor in its
group order In mathematics, the order of a finite group is the number of its elements. If a group (mathematics), group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is t ...
: 24·32·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas \mathrm_ is sharply 4-transitive, \mathrm_ is sharply 5-transitive and \mathrm_ is 5-transitive, and as such they are the only two 5-transitive groups that are not
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
s or
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
s. \mathrm_ has the first five prime numbers as its distinct prime factors in its order of 27· 32·5· 7· 11, and is the smallest of five sporadic groups with five distinct prime factors in their order. All Mathieu groups are subgroups of \mathrm_, which under the
Witt design 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
\mathrm_ of
Steiner system 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
S(5, 8, 24) emerges a construction of the extended binary Golay code \mathrm_ that has \mathrm_ as its
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
. \mathrm_ generates ''octads'' from
code words ''Code Words'' is an online publication about computer programming produced by the Recurse Center retreat community. It began publishing in December 2014, and has a quarterly schedule. The journal features original work by participants at the Cent ...
of
Hamming weight The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string ...
8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional
Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
Λ24, which is the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group \mathrm_. There are five non-supersingular primes: 37, 43, 53, 61, and 67, all smaller than the largest of fifteen supersingular
prime divisor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s of the friendly giant, 71.


List of basic calculations


In decimal

5 is the only prime number to end in the digit 5 in decimal because all other numbers written with a 5 in the ones place are multiples of five, which makes it a 1-
automorphic number In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natura ...
. All multiples of 5 will end in either 5 or , and
vulgar fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
with 5 or in the
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
do not yield infinite decimal expansions because they are prime factors of 10, the base. In the powers of 5, every power ends with the number five, and from 53 onward, if the exponent is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, then the hundreds digit is 1, and if it is even, the hundreds digit is 6. A number n raised to the fifth power always ends in the same digit as n.


Evolution of the Arabic digit

The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as for the digits 1 to 4. The Kushana and
Gupta Gupta () is a common surname or last name of Indian origin. It is based on the Sanskrit word गोप्तृ ''goptṛ'', which means 'guardian' or 'protector'. According to historian R. C. Majumdar, the surname ''Gupta'' was adopted by se ...
empires in what is now
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
had among themselves several different forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several different ways, producing from that were more similar to the digits 4 or 3 than to 5. It was from those digits that Europeans finally came up with the modern 5. While the shape of the character for the digit 5 has an ascender in most modern
typeface A typeface (or font family) is the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font. There are thousands o ...
s, in typefaces with
text figures Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numerals designed with varying heights in a fashion that resembles a typical line of running text, hence the ...
the glyph usually has a
descender In typography and handwriting, a descender is the portion of a letter that extends below the baseline of a font. For example, in the letter ''y'', the descender is the "tail", or that portion of the diagonal line which lies below the ''v'' c ...
, as, for example, in . On the
seven-segment display A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
of a calculator, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa.


Science

*The
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
of boron. *The number of appendages on most starfish, which exhibit
pentamerism Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pla ...
. *The most destructive known
hurricanes A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depend ...
rate as Category 5 on the Saffir–Simpson hurricane wind scale. *The most destructive known
tornado A tornado is a violently rotating column of air that is in contact with both the surface of the Earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. It is often referred to as a twister, whirlwind or cyclone, altho ...
es rate an F-5 on the
Fujita scale The Fujita scale (F-Scale; ), or Fujita–Pearson scale (FPP scale), is a scale for rating tornado intensity, based primarily on the damage tornadoes inflict on human-built structures and vegetation. The official Fujita scale category is deter ...
or EF-5 on the
Enhanced Fujita scale The Enhanced Fujita scale (abbreviated as EF-Scale) rates tornado intensity based on the severity of the damage they cause. It is used in some countries, including the United States, Canada, China, and Mongolia. The Enhanced Fujita scale repl ...
.


Astronomy

*
Messier object The Messier objects are a set of 110 astronomical objects catalogued by the French astronomer Charles Messier in his ''Catalogue des Nébuleuses et des Amas d'Étoiles'' (''Catalogue of Nebulae and Star Clusters''). Because Messier was only in ...
M5, a magnitude 7.0
globular cluster A globular cluster is a spheroidal conglomeration of stars. Globular clusters are bound together by gravity, with a higher concentration of stars towards their centers. They can contain anywhere from tens of thousands to many millions of membe ...
in the constellation
Serpens Serpens ( grc, , , the Serpent) is a constellation in the northern celestial hemisphere. One of the 48 constellations listed by the 2nd-century astronomer Ptolemy, it remains one of the 88 modern constellations designated by the International ...
. *The New General Cataloguebr>object
NGC 5, a
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
13 spiral galaxy in the constellation Andromeda. *The Roman numeral V stands for dwarfs ( main sequence stars) in the Yerkes spectral classification scheme. *The Roman numeral V (usually) stands for the fifth-discovered satellite of a planet or minor planet (e.g. Jupiter V). *There are five Lagrangian points in a two-body system.


Biology

*There are generally considered to be five senses. *The five basic
taste The gustatory system or sense of taste is the sensory system that is partially responsible for the perception of taste (flavor). Taste is the perception produced or stimulated when a substance in the mouth reacts chemically with taste receptor ...
s are
sweet Sweetness is a basic taste most commonly perceived when eating foods rich in sugars. Sweet tastes are generally regarded as pleasurable. In addition to sugars like sucrose, many other chemical compounds are sweet, including aldehydes, ketone ...
, salty,
sour The gustatory system or sense of taste is the sensory system that is partially responsible for the perception of taste (flavor). Taste is the perception produced or stimulated when a substance in the mouth reacts chemically with taste receptor ...
,
bitter Bitter may refer to: Common uses * Resentment, negative emotion or attitude, similar to being jaded, cynical or otherwise negatively affected by experience * Bitter (taste), one of the five basic tastes Books * '' Bitter (novel)'', a 2022 nove ...
, and
umami Umami ( from ja, 旨味 ), or savoriness, is one of the five basic tastes. It has been described as savory and is characteristic of broths and cooked meats. People taste umami through taste receptors that typically respond to glutamates and ...
. *Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.


Computing

*5 is the
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
code of the
Enquiry character In computer communications, enquiry is a transmission-control character that requests a response from the receiving station with which a connection has been set up. It represents a signal intended to trigger a response at the receiving end, to se ...
, which is abbreviated to ENQ.


Religion and culture


Hinduism

*The god
Shiva Shiva (; sa, शिव, lit=The Auspicious One, Śiva ), also known as Mahadeva (; ɐɦaːd̪eːʋɐ, or Hara, is one of the principal deities of Hinduism. He is the Supreme Being in Shaivism, one of the major traditions within Hindu ...
has five faces and his mantra is also called (five-worded) mantra. *The goddess
Saraswati Saraswati ( sa, सरस्वती, ) is the Hindu goddess of knowledge, music, art, speech, wisdom, and learning. She is one of the Tridevi, along with the goddesses Lakshmi and Parvati. The earliest known mention of Saraswati as a g ...
, goddess of knowledge and intellectual is associated with or the number 5. *There are five elements in the universe according to
Hindu cosmology Hindu cosmology is the description of the universe and its states of matter, cycles within time, physical structure, and effects on living entities according to Hindu texts. Hindu cosmology is also intertwined with the idea of a creator who all ...
: (earth, fire, water, air and space respectively). *The most sacred tree in Hinduism has 5 leaves in every leaf stunt. *Most of the flowers have 5 petals in them. *The epic
Mahabharata The ''Mahābhārata'' ( ; sa, महाभारतम्, ', ) is one of the two major Sanskrit literature, Sanskrit Indian epic poetry, epics of ancient India in Hinduism, the other being the ''Ramayana, Rāmāyaṇa''. It narrates the s ...
revolves around the battle between Duryodhana and his 99 other brothers and the 5
pandava The Pandavas (Sanskrit: पाण्डव, IAST: Pāṇḍava) refers to the five legendary brothers— Yudhishthira, Bhima, Arjuna, Nakula and Sahadeva—who are the central characters of the Hindu epic ''Mahabharata''. They are acknowledge ...
princes— Dharma, Arjuna, Bhima,
Nakula In the Hindu epic Mahabharata, ''Nakula'' (Sanskrit: नकुल) was fourth of the five Pandava brothers. Nakula and Sahadeva were twins blessed to Madri, by Ashwini Kumaras, the divine physicians. Their parents Pandu and Madri - died e ...
and
Sahadeva Sahadeva (Sanskrit: सहदेव) was the youngest of the Pandava brothers, the five principal protagonists of the epic ''Mahabharata''. He and his twin brother, Nakula, were blessed to King Pandu and Queen Madri by invoking the twin gods As ...
.


Christianity

*There are traditionally five wounds of
Jesus Christ Jesus, likely from he, יֵשׁוּעַ, translit=Yēšūaʿ, label=Hebrew/Aramaic ( AD 30 or 33), also referred to as Jesus Christ or Jesus of Nazareth (among other names and titles), was a first-century Jewish preacher and religious ...
in
Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...
: the
Scourging at the Pillar The Flagellation of Christ, sometimes known as Christ at the Column or the Scourging at the Pillar, is a scene from the Passion of Christ very frequently shown in Christian art, in cycles of the Passion or the larger subject of the '' Life of C ...
, the Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Side Wound of Christ.


Gnosticism

*The number five was an important symbolic number in
Manichaeism Manichaeism (; in New Persian ; ) is a former major religionR. van den Broek, Wouter J. Hanegraaff ''Gnosis and Hermeticism from Antiquity to Modern Times''SUNY Press, 1998 p. 37 founded in the 3rd century AD by the Parthian Empire, Parthian ...
, with heavenly beings, concepts, and others often grouped in sets of five. *
Five Seals In Sethian Gnostic texts, the Five Seals are typically described as a baptismal rite involving a series of five full immersions in holy running or "living water," symbolizing spiritual ascension to the divine realm. The Five Seals are frequently m ...
in
Sethianism The Sethians were one of the main currents of Gnosticism during the 2nd and 3rd century CE, along with Valentinianism and Basilideanism. According to John D. Turner, it originated in the 2nd century CE as a fusion of two distinct Hellenistic ...
* Five Trees in the
Gospel of Thomas The Gospel of Thomas (also known as the Coptic Gospel of Thomas) is an extra-canonical sayings gospel. It was discovered near Nag Hammadi, Egypt, in December 1945 among a group of books known as the Nag Hammadi library. Scholars speculate ...


Islam

*The
Five Pillars of Islam The Five Pillars of Islam (' ; also ' "pillars of the religion") are fundamental practices in Islam, considered to be obligatory acts of worship for all Muslims. They are summarized in the famous hadith of Gabriel. The Sunni and Shia agree o ...
* Muslims pray to Allah five times a day *According to Shia Muslims, the Panjetan or the Five Holy Purified Ones are the members of
Muhammad Muhammad ( ar, مُحَمَّد;  570 – 8 June 632 CE) was an Arab religious, social, and political leader and the founder of Islam. According to Islamic doctrine, he was a prophet divinely inspired to preach and confirm the mo ...
's family:
Muhammad Muhammad ( ar, مُحَمَّد;  570 – 8 June 632 CE) was an Arab religious, social, and political leader and the founder of Islam. According to Islamic doctrine, he was a prophet divinely inspired to preach and confirm the mo ...
,
Ali ʿAlī ibn Abī Ṭālib ( ar, عَلِيّ بْن أَبِي طَالِب; 600 – 661 CE) was the last of four Rightly Guided Caliphs to rule Islam (r. 656 – 661) immediately after the death of Muhammad, and he was the first Shia Imam ...
,
Fatimah Fāṭima bint Muḥammad ( ar, فَاطِمَة ٱبْنَت مُحَمَّد}, 605/15–632 CE), commonly known as Fāṭima al-Zahrāʾ (), was the daughter of the Islamic prophet Muhammad and his wife Khadija. Fatima's husband was Ali, ...
, Hasan, and
Husayn Hussein, Hussain, Hossein, Hossain, Huseyn, Husayn, Husein or Husain (; ar, حُسَيْن ), coming from the triconsonantal root Ḥ-S-i-N ( ar, ح س ی ن, link=no), is an Arabic name which is the diminutive of Hassan, meaning "good", " ...
and are often symbolically represented by an image of the
Khamsa Khamsa (Arabic, lit. "five") may refer to: * Hamsa, a popular amulet in the Middle East and North Africa, also romanized as ''khamsa'' * Al Khamsa, a bloodline for Arabian horses that traces back to five mares * Al Khamsa (organization), a nonpro ...
.


Judaism

*The
Torah The Torah (; hbo, ''Tōrā'', "Instruction", "Teaching" or "Law") is the compilation of the first five books of the Hebrew Bible, namely the books of Genesis, Exodus, Leviticus, Numbers and Deuteronomy. In that sense, Torah means the ...
contains five books—
Genesis Genesis may refer to: Bible * Book of Genesis, the first book of the biblical scriptures of both Judaism and Christianity, describing the creation of the Earth and of mankind * Genesis creation narrative, the first several chapters of the Book of ...
,
Exodus Exodus or the Exodus may refer to: Religion * Book of Exodus, second book of the Hebrew Torah and the Christian Bible * The Exodus, the biblical story of the migration of the ancient Israelites from Egypt into Canaan Historical events * Ex ...
, Leviticus,
Numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
, and Deuteronomy—which are collectively called the Five Books of Moses, the Pentateuch (
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
for "five containers", referring to the scroll cases in which the books were kept), or
Humash ''Chumash'' (also Ḥumash; he, חומש, or or Yiddish: ; plural Ḥumashim) is a Torah in printed and book bound form (i.e. codex) as opposed to a Sefer Torah, which is a scroll. The word comes from the Hebrew word for five, (). A more f ...
(,
Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
for "fifth"). *The book of
Psalms The Book of Psalms ( or ; he, תְּהִלִּים, , lit. "praises"), also known as the Psalms, or the Psalter, is the first book of the ("Writings"), the third section of the Tanakh, and a book of the Old Testament. The title is derived ...
is arranged into five books, paralleling the
Five Books of Moses The Torah (; hbo, ''Tōrā'', "Instruction", "Teaching" or "Law") is the compilation of the first five books of the Hebrew Bible, namely the books of Genesis, Exodus, Leviticus, Numbers and Deuteronomy. In that sense, Torah means the sa ...
. *The
Khamsa Khamsa (Arabic, lit. "five") may refer to: * Hamsa, a popular amulet in the Middle East and North Africa, also romanized as ''khamsa'' * Al Khamsa, a bloodline for Arabian horses that traces back to five mares * Al Khamsa (organization), a nonpro ...
, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by
Jew Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""T ...
s; that same symbol is also very popular in
Arab The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...
ic culture, known to protect from envy and the
evil eye The Evil Eye ( grc, ὀφθαλμὸς βάσκανος; grc-koi, ὀφθαλμὸς πονηρός; el, (κακό) μάτι; he, עַיִן הָרָע, ; Romanian: ''Deochi''; it, malocchio; es, mal de ojo; pt, mau-olhado, olho gordo; ar ...
.


Sikhism

*The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as or the "
Five Ks In Sikhism, the Five Ks ( pa, ਪੰਜ ਕਕਾਰ ) are five items that Guru Gobind Singh Ji, in 1699, commanded Khalsa Sikhs to wear at all times. They are: ''kesh'' (unshorn hair and beard since the Sikh decided to keep it), '' kangha'' (a ...
" because they start with letter K representing in the
Punjabi language Punjabi (; ; , ), sometimes spelled Panjabi, is an Indo-Aryan language of the Punjab region of Pakistan and India. It has approximately 113 million native speakers. Punjabi is the most widely-spoken first language in Pakistan, with 80.5 ...
's
Gurmukhi script Gurmukhī ( pa, ਗੁਰਮੁਖੀ, , Shahmukhi: ) is an abugida developed from the Laṇḍā scripts, standardized and used by the second Sikh guru, Guru Angad (1504–1552). It is used by Punjabi Sikhs to write the language, commonly re ...
. They are: (unshorn hair), (the comb), (the steel bracelet), (the soldier's shorts), and (the sword) (in Gurmukhi: ). Also, there are five deadly evils: (lust), (anger), (attachment), (greed), and (ego).


Daoism

* 5 Elements * 5 Emperors


Other religions and cultures

*According to ancient Greek philosophers such as
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
, the universe is made up of five classical elements:
water Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
,
earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
,
air The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
,
fire Fire is the rapid oxidation of a material (the fuel) in the exothermic chemical process of combustion, releasing heat, light, and various reaction products. At a certain point in the combustion reaction, called the ignition point, flames a ...
, and
ether In organic chemistry, ethers are a class of compounds that contain an ether group—an oxygen atom connected to two alkyl or aryl groups. They have the general formula , where R and R′ represent the alkyl or aryl groups. Ethers can again be ...
. This concept was later adopted by medieval
alchemists Alchemy (from Arabic: ''al-kīmiyā''; from Ancient Greek: χυμεία, ''khumeía'') is an ancient branch of natural philosophy, a philosophical and protoscientific tradition that was historically practiced in China, India, the Muslim world, ...
and more recently by practitioners of Neo-Pagan religions such as
Wicca Wicca () is a modern Pagan religion. Scholars of religion categorise it as both a new religious movement and as part of the occultist stream of Western esotericism. It was developed in England during the first half of the 20th century and w ...
. *The
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
, or five-pointed star, bears religious significance in various faiths including Baháʼí,
Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...
,
Freemasonry Freemasonry or Masonry refers to fraternal organisations that trace their origins to the local guilds of stonemasons that, from the end of the 13th century, regulated the qualifications of stonemasons and their interaction with authorities ...
, Satanism,
Taoism Taoism (, ) or Daoism () refers to either a school of philosophical thought (道家; ''daojia'') or to a religion (道教; ''daojiao''), both of which share ideas and concepts of Chinese origin and emphasize living in harmony with the '' Ta ...
,
Thelema Thelema () is a Western esoteric and occult social or spiritual philosophy and new religious movement founded in the early 1900s by Aleister Crowley (1875–1947), an English writer, mystic, occultist, and ceremonial magician. The word ' ...
, and
Wicca Wicca () is a modern Pagan religion. Scholars of religion categorise it as both a new religious movement and as part of the occultist stream of Western esotericism. It was developed in England during the first half of the 20th century and w ...
. *In
Cantonese Cantonese ( zh, t=廣東話, s=广东话, first=t, cy=Gwóngdūng wá) is a language within the Chinese (Sinitic) branch of the Sino-Tibetan languages originating from the city of Guangzhou (historically known as Canton) and its surrounding ar ...
, "five" sounds like the word "not" (character: ). When five appears in front of a lucky number, e.g. "58", the result is considered unlucky. *In
East Asia East Asia is the eastern region of Asia, which is defined in both Geography, geographical and culture, ethno-cultural terms. The modern State (polity), states of East Asia include China, Japan, Mongolia, North Korea, South Korea, and Taiwan. ...
n tradition, there are five elements: (
water Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
,
fire Fire is the rapid oxidation of a material (the fuel) in the exothermic chemical process of combustion, releasing heat, light, and various reaction products. At a certain point in the combustion reaction, called the ignition point, flames a ...
,
earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
,
wood Wood is a porous and fibrous structural tissue found in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin ...
, and
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
). The
Japanese Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...
names for the days of the week, Tuesday through
Saturday Saturday is the day of the week between Friday and Sunday. No later than the 2nd century, the Romans named Saturday ("Saturn's Day") for the planet Saturn, which controlled the first hour of that day, according to Vettius Valens. The day ...
, come from these elements via the identification of the elements with the five planets visible with the naked eye. Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. *In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance. *Members of The Nation of Gods and Earths, a primarily African American religious organization, call themselves the "Five-Percenters" because they believe that only 5% of mankind is truly enlightened.


Art, entertainment, and media


Fictional entities

*
James the Red Engine James is a fictional anthropomorphic red tender locomotive from ''The Railway Series'' children's books by the Reverend Awdry and the TV series adaptation ''Thomas & Friends''. He is a mixed-traffic engine, which means he is just as capable of ...
, a fictional character numbered 5. *
Johnny 5 ''Short Circuit'' is a 1986 American comic science fiction, science fiction comedy film directed by John Badham and written by S. S. Wilson and Brent Maddock. The film's plot centers upon an experimental military robot that is struck by lightning ...
is the lead character in the film ''Short Circuit'' (1986) *Number Five is a character in
Lorien Legacies ''Lorien Legacies'' is a series of young adult science fiction books, written by James Frey, Jobie Hughes, and formerly, Greg Boose, under the collective pseudonym Pittacus Lore. Lorien Legacies ''I am Number Four'' The first book of The ...
*Sankara Stones, five magical rocks in '' Indiana Jones and the Temple of Doom'' that are sought by the Thuggees for evil purposes *The Mach Five , the racing car Speed Racer ( in the Japanese version) drives in the anime series of the same name (known as "Mach Go! Go! Go!" in Japan) *In the works of
J. R. R. Tolkien John Ronald Reuel Tolkien (, ; 3 January 1892 – 2 September 1973) was an English writer and philologist. He was the author of the high fantasy works ''The Hobbit'' and ''The Lord of the Rings''. From 1925 to 1945, Tolkien was the Rawlins ...
, five wizards ( Saruman,
Gandalf Gandalf is a protagonist in J. R. R. Tolkien's novels '' The Hobbit'' and ''The Lord of the Rings''. He is a wizard, one of the ''Istari'' order, and the leader of the Fellowship of the Ring. Tolkien took the name "Gandalf" from the Old Nor ...
,
Radagast Radagast the Brown is a fictional character in J. R. R. Tolkien's legendarium. A wizard and associate of Gandalf, he appears briefly in ''The Hobbit'', ''The Lord of the Rings'', ''The Silmarillion'', and ''Unfinished Tales''. His role in To ...
, Alatar and Pallando) are sent to Middle-earth to aid against the threat of the Dark Lord Sauron *In the '' A Song of Ice and Fire'' series, the War of the Five Kings is fought between different claimants to the Iron Throne of Westeros, as well as to the thrones of the individual regions of Westeros (
Joffrey Baratheon Joffrey Baratheon is a fictional character in the ''A Song of Ice and Fire'' series of epic fantasy novels by American author George R. R. Martin, and its HBO television adaptation ''Game of Thrones''. Introduced in 1996's ''A Game of Thrones' ...
,
Stannis Baratheon Stannis Baratheon is a fictional character in the ''A Song of Ice and Fire'' series of epic fantasy novels by American author George R. R. Martin, and its television adaptation ''Game of Thrones''. He is the second son of Steffon Baratheon and C ...
,
Renly Baratheon Renly Baratheon is a fictional character in the ''A Song of Ice and Fire'' series of fantasy novels by American author George R. R. Martin, and its television adaptation ''Game of Thrones''. Introduced in 1996's ''A Game of Thrones'', Renly is ...
, Robb Stark and
Balon Greyjoy George R. R. Martin's ''A Song of Ice and Fire'' saga features a large cast of characters. The series follows three interwoven plotlines: a dynastic war for control of Westeros by several families; the rising threat of the superhuman Others bey ...
) *In ''
The Wheel of Time ''The Wheel of Time'' is a series of high fantasy novels by American author Robert Jordan, with Brandon Sanderson as a co-author for the final three novels. Originally planned as a six-book series, ''The Wheel of Time'' spans 14 volumes, in ad ...
'' series, the "Emond's Field Five" are a group of five of the series' main characters who all come from the village of Emond's Field (
Rand al'Thor This article serves as an index of major characters in the fictional setting of Robert Jordan's ''The Wheel of Time'' series, with a description of their main roles or feats in the series. ''The Wheel of Time'' has 2787 distinct named characters. ...
,
Matrim Cauthon This article serves as an index of major characters in the fictional setting of Robert Jordan's ''The Wheel of Time'' series, with a description of their main roles or feats in the series. ''The Wheel of Time'' has 2787 distinct named characters. ...
,
Perrin Aybara This article serves as an index of major characters in the fictional setting of Robert Jordan's ''The Wheel of Time'' series, with a description of their main roles or feats in the series. ''The Wheel of Time'' has 2787 distinct named characters. ...
,
Egwene al'Vere This article serves as an index of major characters in the fictional setting of Robert Jordan's ''The Wheel of Time'' series, with a description of their main roles or feats in the series. ''The Wheel of Time'' has 2787 distinct named characters ...
and
Nynaeve al'Meara This article serves as an index of major characters in the fictional setting of Robert Jordan's ''The Wheel of Time'' series, with a description of their main roles or feats in the series. ''The Wheel of Time'' has 2787 distinct named characters. ...
) * ''Myst'' uses the number 5 as a unique base counting system. In ''
The Myst Reader ''The Myst Reader'' is a collection of three novels based on the ''Myst'' series of adventure games. The collection was published in September 2004 and combines three works previously published separately: ''The Book of Atrus'' (1995), ''The ...
'' series, it is further explained that the number 5 is considered a holy number in the fictional D'ni society. *Number Five is also a character in The Umbrella Academy comic book and TV series adaptation


Films

*Towards the end of the film '' Monty Python and the Holy Grail'' (1975), the character of King Arthur repeatedly confuses the number five with the number
three 3 is a number, numeral, and glyph. 3, three, or III may also refer to: * AD 3, the third year of the AD era * 3 BC, the third year before the AD era * March, the third month Books * '' Three of Them'' (Russian: ', literally, "three"), a 1901 ...
. *''
Five Go Mad in Dorset ''Five Go Mad in Dorset'' was the first of three ''Five Go Mad'' specials from the long-running series of '' The Comic Strip Presents...'' television comedy films. It first aired on the launch night of Channel 4 (2 November 1982), and was writ ...
'' (1982) was the first of the long-running series of '' The Comic Strip Presents...'' television comedy films *''
The Fifth Element ''The Fifth Element'' is a 1997 English-language French science fiction action film conceived and directed by Luc Besson, as well as co-written by Besson and Robert Mark Kamen. It stars Bruce Willis, Gary Oldman, Chris Tucker, and Milla ...
'' (1997), a science fiction film * ''
Fast Five ''Fast Five'' (also known as ''Fast & Furious 5'' or ''Fast & Furious 5: Rio Heist'') is a 2011 American action film directed by Justin Lin and written by Chris Morgan. It is the sequel to ''Fast & Furious'' (2009) and the fifth i ...
'' (2011), the fifth installment of the ''Fast and Furious'' film series. *''
V for Vendetta ''V for Vendetta'' is a British graphic novel written by Alan Moore and illustrated by David Lloyd (with additional art by Tony Weare). Initially published between 1982 and 1985 in black and white as an ongoing serial in the British anthol ...
'' (2005), produced by
Warner Bros. Warner Bros. Entertainment Inc. (commonly known as Warner Bros. or abbreviated as WB) is an American film and entertainment studio headquartered at the Warner Bros. Studios complex in Burbank, California, and a subsidiary of Warner Bros. D ...
, directed by
James McTeigue James McTeigue (born December 29, 1967) is an Australian film and television director. He has been an assistant director on many films, including '' Dark City'' (1998), the ''Matrix'' trilogy (1999–2003) and '' Star Wars: Episode II – Att ...
, and adapted from
Alan Moore Alan Moore (born 18 November 1953) is an English author known primarily for his work in comic books including '' Watchmen'', ''V for Vendetta'', '' The Ballad of Halo Jones'', ''Swamp Thing'', ''Batman:'' ''The Killing Joke'', and '' From He ...
's graphic novel ''
V for Vendetta ''V for Vendetta'' is a British graphic novel written by Alan Moore and illustrated by David Lloyd (with additional art by Tony Weare). Initially published between 1982 and 1985 in black and white as an ongoing serial in the British anthol ...
'' prominently features number 5 and Roman Numeral V; the story is based on the historical event in which a group of men attempted to destroy Parliament on November 5, 1605


Music


Groups

*
Five (group) Five (stylised as 5ive) are a British boy band from London consisting of members Sean Conlon, Ritchie Neville, and Scott Robinson. They were formed in 1997 by the same team that managed the Spice Girls before they launched their career. The ...
, a UK Boy band *
The Five (composers) The Five ( rus, link=no, Могучая кучка, lit. ''Mighty Bunch''), also known as the Mighty Handful, The Mighty Five, and the New Russian School, were five prominent 19th-century Russian composers who worked together to create a distinct ...
, 19th-century Russian composers * 5 Seconds of Summer, pop band that originated in Sydney, Australia *
Five Americans Five Americans was a 1960s American rock band, most famous for their song, "Western Union", which reached number five in the U.S. ''Billboard'' chart and was their only single to chart in the Top 20. In Canada, they had three in the Top 20. C ...
, American rock band active 1965–1969 *
Five Finger Death Punch Five Finger Death Punch, also abbreviated as 5FDP or FFDP, is an American heavy metal band from Las Vegas, Nevada, formed in 2005. The band originally consisted of vocalist and keyboardist Ivan Moody, rhythm guitarist Zoltan Bathory, lead g ...
, American heavy metal band from Las Vegas, Nevada. Active 2005–present *
Five Man Electrical Band The Five Man Electrical Band (known as The Staccatos from 1963 to 1968) is a Canadian rock band from Ottawa, Ontario. They had many hits in Canada, including the top 10 entries "Half Past Midnight" (1967) (as The Staccatos), "Absolutely Right ...
, Canadian rock group billed (and active) as the Five Man Electrical Band, 1969–1975 * Maroon 5, American pop rock band that originated in Los Angeles, California *
MC5 MC5, also commonly called The MC5, is an American rock band formed in Lincoln Park, Michigan, in 1963. The original line-up consisted of Rob Tyner (vocals) Wayne Kramer (guitar), Fred "Sonic" Smith (guitar), Michael Davis (bass), and Dennis ...
, American punk rock band * Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas *
The 5th Dimension The 5th Dimension is an American popular music vocal group, whose repertoire includes pop, R&B, soul, jazz, light opera, and Broadway. Formed as the Versatiles in late 1965, the group changed its name to "the 5th Dimension" by 1966. Betwe ...
, American pop vocal group, active 1977–present *
The Dave Clark Five The Dave Clark Five, also known as the DC5, were an English rock and roll band formed in 1958 in Tottenham, London. Drummer Dave Clark served as the group's leader, producer and co-songwriter. In January 1964 they had their first UK top ten sin ...
, a.k.a. DC5, an English pop rock group comprising Dave Clark,
Lenny Davidson Leonard Arthur 'Lenny' Davidson (born 30 May 1944 in Enfield, Middlesex, England) is an English musician and songwriter, best known as the guitarist for the Dave Clark Five. Career Davidson was born in Enfield as one of three children. H ...
,
Rick Huxley Richard Huxley (5 August 1940 – 11 February 2013) was an English musician who was the bassist for the Dave Clark Five, a group that was part of the British Invasion. Biography Born at Livingstone Hospital, Dartford, Kent, he joined t ...
,
Denis Payton Denis Archibald West Payton (11 August 1943 – 17 December 2006) was an English musician who played tenor saxophone, baritone saxophone, guitar and harmonica in the rock and roll band the Dave Clark Five. Biography Payton was born in Walthamst ...
, and Mike Smith; active 1958–1970 *
The Jackson 5 The Jackson 5 (sometimes stylized as the Jackson 5ive, also known as the Jacksons) are an American pop band composed of members of the Jackson family. The group was founded in 1964 in Gary, Indiana, and for most o ...
, American pop rock group featuring various members of the Jackson family; they were billed (and active) as The Jackson 5, 1966–1975 * Hi-5, Australian pop kids group, where it has several international adaptations, and several members throughout the history of the band. It was also a TV show. *
We Five We Five was a 1960s folk rock musical group based in San Francisco, California. Their best-known hit was their 1965 remake of Ian & Sylvia's " You Were on My Mind", which reached No. 1 on the Cashbox chart, #3 on the ''Billboard'' Hot 100, and ...
: American folk rock group active 1965–1967 and 1968–1977 *
Grandmaster Flash and the Furious Five Grandmaster Flash and the Furious Five were an American hip hop group formed in the South Bronx of New York City in 1978. The group's members were Grandmaster Flash, Melle Mel, Kidd Creole (not to be confused with Kid Creole), Keef Cowboy, S ...
: American rap group, 1970–80's *
Fifth Harmony Fifth Harmony, often shortened to 5H, was an American girl group based in Miami, composed of Ally Brooke, Normani, Dinah Jane, Lauren Jauregui, and previously Camila Cabello until her departure from the group in December 2016. The group sig ...
, an American girl group. * Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013 *
R5 (band) R5 was an American pop rock band formed in Los Angeles in 2009. The band consisted of Ross Lynch (vocals/rhythm guitar), Riker Lynch (bass guitar/vocals), Rocky Lynch (lead guitar/vocals), Rydel Lynch (vocals), and Ellington Ratliff (drums/vo ...
, an American pop and alternative rock group, 2009–2018


Other uses

*A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems. *Modern musical notation uses a
musical staff In Western musical notation, the staff (US and UK)"staff" in the Collin ...
made of five horizontal lines. *In harmonics – the fifth partial (or 4th
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
) of a
fundamental Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third. *The number of completed, numbered piano concertos of
Ludwig van Beethoven Ludwig van Beethoven (baptised 17 December 177026 March 1827) was a German composer and pianist. Beethoven remains one of the most admired composers in the history of Western music; his works rank amongst the most performed of the classic ...
,
Sergei Prokofiev Sergei Sergeyevich Prokofiev; alternative transliterations of his name include ''Sergey'' or ''Serge'', and ''Prokofief'', ''Prokofieff'', or ''Prokofyev''., group=n (27 April .S. 15 April1891 – 5 March 1953) was a Russian composer, ...
, and Camille Saint-Saëns. *Using the Latin root, five musicians are called a quintet. *A scale with five notes per octave is called a
pentatonic scale A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many an ...
. *Five is the lowest possible number that can be the top number of a
time signature The time signature (also known as meter signature, metre signature, or measure signature) is a notational convention used in Western musical notation to specify how many beats (pulses) are contained in each measure (bar), and which note va ...
with an asymmetric
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
.


Television

;Stations *
Channel 5 (UK) Channel 5 is a British free-to-air public broadcast television channel launched in 1997. It is the fifth national terrestrial channel in the United Kingdom and is owned by Channel 5 Broadcasting Limited, a wholly-owned subsidiary of American ...
, a television channel that broadcasts in the United Kingdom *
5 (TV channel) TV5 (also known as 5 and formerly known as ABC) is a Philippine free-to-air television network based in Mandaluyong, with its alternate studios located in Novaliches, Quezon City. It is the flagship property of TV5 Network, Inc. with Cignal ...
(''formerly known as ABC 5 and TV5'') (
DWET-TV DWET-TV, channel 5 (analog) and channel 18 (digital), is the flagship TV station of Philippine television network TV5. The station is owned and operated by TV5 Network Inc., which is owned by MediaQuest Holdings, the multimedia arm of Phi ...
channel 5 In Metro Manila) a television network in the Philippines. ; ;Series *''
Babylon 5 ''Babylon 5'' is an American space opera television series created by writer and producer J. Michael Straczynski, under the Babylonian Productions label, in association with Straczynski's Synthetic Worlds Ltd. and Warner Bros. Domestic Tele ...
'', a science fiction television series *The number 5 features in the television series ''Battlestar Galactica'' in regards to the
Final Five cylons The Cylons are the main antagonists of the human race in the '' Battlestar Galactica'' science fiction franchise, making appearances in the original 1978 series, the 1980 series, the 2004 re-imagining, and the spin-off prequel series '' C ...
and the Temple of Five * ''Hi-5'' (Australian TV series), a television series from Australia * ''Hi-5'' (UK TV series), a television show from the United Kingdom * ''Hi-5'' Philippines a television show from the Philippines *''
Odyssey 5 ''Odyssey 5'' is a Canadian science fiction series, which was shown in 2002 on Space in Canada and on Showtime in the United States. The premise involves five space travelers who witness the destruction of the Earth; they are given the opportu ...
'', a 2002 science fiction television series *'' Tillbaka till Vintergatan'', a Swedish children's television series featuring a character named "Femman" (meaning five), who can only utter the word 'five'. *'' The Five'' (talk show): Fox News Channel roundtable current events television show, premiered 2011, so-named for its panel of five commentators. *''
Yes! PreCure 5 is a Japanese anime series and the fourth installment in Izumi Todo's ''Pretty Cure'' metaseries produced by Toei Animation, featuring the third generation of Cures. The series aired on TV Asahi between February 2007 and January 2008 and ...
'' is a 2007 anime series which follows the adventures of Nozomi and her friends. It is also followed by the 2008 sequel '' Yes! Pretty Cure 5 GoGo!'' *''
The Quintessential Quintuplets is a Japanese manga series written and illustrated by Negi Haruba. It was serialized in Kodansha's ''Weekly Shōnen Magazine'' from August 2017 to February 2020, with its chapters collected into fourteen ''tankōbon'' volumes. The series ...
'' is a 2019 slice of life romance anime series which follows the everyday life of five identical quintuplets and their interactions with their tutor. It has two seasons, and a final movie is scheduled in summer 2022. * ''Hawaii Five-0'',
CBS CBS Broadcasting Inc., commonly shortened to CBS, the abbreviation of its former legal name Columbia Broadcasting System, is an American commercial broadcast television and radio network serving as the flagship property of the CBS Entertainm ...
American TV series.


Literature

* ''The Famous Five'' is a series of children's books by British writer Enid Blyton *''
The Power of Five ''The Power of Five'' (re-titled as ''The Gatekeepers'' in the US) is a series of five fantasy and suspense novels, written by English author Anthony Horowitz. Published between 2005 and 2012, it is an updated re-imagining of Horowitz's ''Pe ...
'' is a series of children's books by British writer and screenwriter Anthony Horowitz *'' The Fall of Five'' is a book written under the collective pseudonym Pittacus Lore in the series ''Lorien Legacies'' *''
The Book of Five Rings is a text on ''kenjutsu'' and the martial arts in general, written by the Japanese swordsman Miyamoto Musashi around 1645. Many translations have been made, and it enjoys an audience considerably broader than other martial artists and people ac ...
'' is a text on kenjutsu and the martial arts in general, written by the swordsman Miyamoto Musashi circa 1645 *''
Slaughterhouse-Five ''Slaughterhouse-Five, or, The Children's Crusade: A Duty-Dance with Death'' is a 1969 semi-autobiographic science fiction-infused anti-war novel by Kurt Vonnegut. It follows the life and experiences of Billy Pilgrim, from his early years, to h ...
'' is a book by Kurt Vonnegut about World War II


Sports

*The
Olympic Games The modern Olympic Games or Olympics (french: link=no, Jeux olympiques) are the leading international sporting events featuring summer and winter sports competitions in which thousands of athletes from around the world participate in a vari ...
have five interlocked rings as their symbol, representing the number of inhabited
continent A continent is any of several large landmasses. Generally identified by convention rather than any strict criteria, up to seven geographical regions In geography, regions, otherwise referred to as zones, lands or territories, are areas t ...
s represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas). * In
AFL Women's AFL Women's (AFLW) is Australia's national semi-professional Australian rules football in Australia, Australian rules football league for women's Australian rules football, female players. The 2017 AFL Women's season, first season of the league ...
, the top level of women's Australian rules football, each team is allowed 5 "
interchanges Interchange may refer to: Transport * Interchange (road), a collection of ramps, exits, and entrances between two or more highways * Interchange (freight rail), the transfer of freight cars between railroad companies * Interchange station, a rai ...
" (substitute players), who can be freely substituted at any time. *In
baseball scorekeeping Baseball scorekeeping is the practice of recording the details of a baseball game as it unfolds. Professional baseball leagues hire official scorers to keep an official record of each game (from which a box score can be generated), but many fans k ...
, the number 5 represents the
third baseman A third baseman, abbreviated 3B, is the player in baseball or softball whose responsibility is to defend the area nearest to third base — the third of four bases a baserunner must touch in succession to score a run. In the scoring system us ...
's position. *In
basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular court, compete with the primary objective of shooting a basketball (approximately in diameter) through the defender's h ...
: **The number 5 is used to represent the position of
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
. **Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball. **The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover. **Under the
FIBA The International Basketball Federation (FIBA ; French: ) is an association of national organizations which governs the sport of basketball worldwide. Originally known as the (hence FIBA), in 1989 it dropped the word ''amateur'' from its nam ...
(used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter. **Under the FIBA rules, A player fouls out and must leave the game after committing five fouls *
Five-a-side football Five-a-side football is a version of minifootball, in which each team fields five players (four outfield players and a goalkeeper). Other differences from football include a smaller pitch, smaller goals, and a reduced game duration. Matches are ...
is a variation of
association football Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
in which each team fields five players. *In
ice hockey Ice hockey (or simply hockey) is a team sport played on ice skates, usually on an ice skating rink with lines and markings specific to the sport. It belongs to a family of sports called hockey. In ice hockey, two opposing teams use ice h ...
: ** A major penalty lasts five minutes. ** There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net). ** The area between the goaltender's legs is known as the
five-hole The five-hole is an ice hockey term for the space between a goaltender's legs. The name and its first recorded usage was in 1976 by Flyer Reggie Leach The phrases ''through the five-hole'' and ''gone five-hole'' are used when a player scores by sh ...
. *In most
rugby league Rugby league football, commonly known as just rugby league and sometimes football, footy, rugby or league, is a full-contact sport played by two teams of thirteen players on a rectangular field measuring 68 metres (75 yards) wide and 11 ...
competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering. *In
rugby union Rugby union, commonly known simply as rugby, is a close-contact team sport that originated at Rugby School in the first half of the 19th century. One of the two codes of rugby football, it is based on running with the ball in hand. In it ...
: ** A
try Try or TRY may refer to: Music Albums * ''Try!'', an album by the John Mayer Trio * ''Try'' (Bebo Norman album) (2014) Songs * "Try" (Blue Rodeo song) (1987) * "Try" (Colbie Caillat song) (2014) * "Try" (Nelly Furtado song) (2004) * " Try (Ju ...
is worth 5 points. ** One of the two starting lock forwards wears number 5, and usually jumps at number 4 in the line-out. ** In the French variation of the bonus points system, a bonus point in the league standings is awarded to a team that loses by 5 or fewer points.


Technology

*5 is the most common number of gears for automobiles with
manual transmission A manual transmission (MT), also known as manual gearbox, standard transmission (in Canada, the United Kingdom, and the United States), or stick shift (in the United States), is a multi-speed motor vehicle transmission system, where gear change ...
. *In radio communication, the term "
Five by five A signal strength and readability report is a standardized format for reporting the strength of the radio signal and the readability (quality) of the radiotelephone (voice) or radiotelegraph (Morse code) signal transmitted by another station as re ...
" is used to indicate perfect signal strength and clarity. *On almost all devices with a
numeric keypad A numeric keypad, number pad, numpad, or ten key, is the palm-sized, usually-17-key section of a standard computer keyboard, usually on the far right. It provides calculator-style efficiency for entering numbers. The idea of a 10-key nu ...
such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the
numpad A numeric keypad, number pad, numpad, or ten key, is the palm-sized, usually-17-key section of a standard computer keyboard, usually on the far right. It provides calculator-style efficiency for entering numbers. The idea of a 10-key n ...
has the raised dot or bar, but the 5 key that shifts with % does not). *On most
telephone A telephone is a telecommunications device that permits two or more users to conduct a conversation when they are too far apart to be easily heard directly. A telephone converts sound, typically and most efficiently the human voice, into e ...
s, the 5 key is associated with the letters J, K, and L, but on some of the
BlackBerry The blackberry is an edible fruit produced by many species in the genus ''Rubus'' in the family Rosaceae, hybrids among these species within the subgenus ''Rubus'', and hybrids between the subgenera ''Rubus'' and ''Idaeobatus''. The taxonomy ...
phones, it is the key for G and H. *The
Pentium Pentium is a brand used for a series of x86 architecture-compatible microprocessors produced by Intel. The original Pentium processor from which the brand took its name was first released on March 22, 1993. After that, the Pentium II and P ...
, coined by Intel Corporation, is a fifth-generation
x86 x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was intr ...
architecture
microprocessor A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit, or a small number of integrated circuits. The microprocessor contains the arithmetic, logic, and control circ ...
. *The
resin identification code The ASTM International Resin Identification Coding System, often abbreviated RIC, is a set of symbols appearing on plastic products that identify the plastic resin out of which the product is made. It was developed in 1988 by the Society of t ...
used in recycling to identify
polypropylene Polypropylene (PP), also known as polypropene, is a thermoplastic polymer used in a wide variety of applications. It is produced via chain-growth polymerization from the monomer propylene. Polypropylene belongs to the group of polyolefins a ...
.


Miscellaneous fields

Five can refer to: *"Give me five" is a common phrase used preceding a high five. *An informal term for the British Security Service,
MI5 The Security Service, also known as MI5 ( Military Intelligence, Section 5), is the United Kingdom's domestic counter-intelligence and security agency and is part of its intelligence machinery alongside the Secret Intelligence Service (MI6), G ...
. *Five babies born at one time are
quintuplets ''Quintuplets'' is an American sitcom that aired 22 episodes on Fox from June 16, 2004 to January 12, 2005. The program starred Andy Richter and Rebecca Creskoff and shared some of their experiences parenting teenage quintuplets. Synopsis The ...
. The most famous set of quintuplets were the
Dionne quintuplets The Dionne quintuplets (; born May 28, 1934) are the first quintuplets known to have survived their infancy. The identical girls were born just outside Callander, Ontario, near the village of Corbeil. All five survived to adulthood. The Di ...
born in the 1930s. *In the United States legal system, the
Fifth Amendment to the United States Constitution The Fifth Amendment (Amendment V) to the United States Constitution addresses criminal procedure and other aspects of the Constitution. It was ratified, along with nine other articles, in 1791 as part of the Bill of Rights. The Fifth Amend ...
can be referred to in court as "pleading the fifth", absolving the defendant from
self-incrimination In criminal law, self-incrimination is the act of exposing oneself generally, by making a statement, "to an accusation or charge of crime; to involve oneself or another ersonin a criminal prosecution or the danger thereof". (Self-incrimination ...
. *
Pentameter Pentameter ( grc, πεντάμετρος, 'measuring five ( feet)') is a poetic meter. А poem is said to be written in a particular pentameter when the lines of the poem have the length of five feet, where a 'foot' is a combination of a particul ...
is verse with five repeating feet per line;
iambic pentameter Iambic pentameter () is a type of metric line used in traditional English poetry and verse drama. The term describes the rhythm, or meter, established by the words in that line; rhythm is measured in small groups of syllables called " feet". "Iam ...
was the most popular form in
Shakespeare William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...
. * Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth) *The designation of an Interstate Highway (
Interstate 5 Interstate 5 (I-5) is the main north–south Interstate Highway on the West Coast of the United States, running largely parallel to the Pacific coast of the contiguous U.S. from Mexico to Canada. It travels through the states of Californi ...
) that runs from
San Diego San Diego ( , ; ) is a city on the Pacific Ocean coast of Southern California located immediately adjacent to the Mexico–United States border. With a 2020 population of 1,386,932, it is the eighth most populous city in the United State ...
,
California California is a state in the Western United States, located along the Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the most populous U.S. state and the 3rd largest by area. It is also the m ...
to Blaine, Washington. In addition, all major north-south Interstate Highways in the United States end in 5. *In the computer game ''
Riven ''Riven'' is a Adventure game#Puzzle adventure, puzzle adventure video game. It is the sequel to ''Myst'' and second in the Myst (series), ''Myst'' series of games. Developed by Cyan Worlds, it was initially published by Red Orb Entertainment, a ...
'', 5 is considered a holy number, and is a recurring theme throughout the game, appearing in hundreds of places, from the number of islands in the game to the number of bolts on pieces of machinery. *''
The Garden of Cyrus ''The Garden of Cyrus'', or ''The Quincuncial Lozenge, or Network Plantations of the Ancients, naturally, artificially, mystically considered'', is a discourse by Sir Thomas Browne. First published in 1658, along with its diptych companion '' ...
'' (1658) by Sir Thomas Browne is a Pythagorean discourse based upon the number 5. *The holy number of Discordianism, as dictated by the
Law of Fives Discordianism is a religion, philosophy, or paradigm centered on Eris, a.k.a. Discordia, the Goddess of chaos. Discordianism uses archetypes or ideals associated with her. It was founded after the 1963 publication of its "holy book," the ''Pri ...
. *The number of Justices on the Supreme Court of the United States necessary to render a majority decision. *The number of dots in a
quincunx A quincunx () is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. The same pattern has other names, including "in saltire" or "in cross" in heraldry (d ...
. *The number of permanent members with veto power on the
United Nations Security Council The United Nations Security Council (UNSC) is one of the six principal organs of the United Nations (UN) and is charged with ensuring international peace and security, recommending the admission of new UN members to the General Assembly, an ...
. *The number of sides and the number of angles in a pentagon. *The number of points in a
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
. *The number of
Korotkoff sounds Korotkoff sounds are the sounds that medical personnel listen for when they are taking blood pressure using a non-invasive procedure. They are named after Nikolai Korotkov, a Russian physician who discovered them in 1905, when he was working at th ...
when measuring blood pressure *The drink
Five Alive Five Alive (french: Déli-cinq) is a line of fruit juice blends created by Minute Maid, a subsidiary of The Coca-Cola Company. Both the name and the five colors of the logo refer to the five fruit juices each variety contains. The juice line was ...
is named for its five ingredients. The drink
punch Punch commonly refers to: * Punch (combat), a strike made using the hand closed into a fist * Punch (drink), a wide assortment of drinks, non-alcoholic or alcoholic, generally containing fruit or fruit juice Punch may also refer to: Places * Pun ...
derives its name after the Sanskrit पञ्च (pañc) for having five ingredients. *The
Keating Five File:AlanCranston.jpg, Alan Cranston (D-CA) File:Dennis DeConcini.jpg, File:John Glenn Low Res.jpg, John Glenn (D-OH) File:John McCain.jpg, John McCain (R-AZ) File:Riegle2.jpg, Donald Riegle (D-MI) The Keating Five were five United States Sen ...
were five
United States Senators The United States Senate is the upper chamber of the United States Congress, with the House of Representatives being the lower chamber. Together they compose the national bicameral legislature of the United States. The composition and powe ...
accused of corruption in 1989. *The
Inferior Five The Inferior Five (or I5) are a parody superhero team appearing in books by the American publisher DC Comics. Created by writer E. Nelson Bridwell and artist Joe Orlando, the team premiered in the DC Comics title ''Showcase'' #62 (May-June 1966 ...
: Merryman, Awkwardman, The Blimp, White Feather, and Dumb Bunny.
DC Comics DC Comics, Inc. (doing business as DC) is an American comic book publisher and the flagship unit of DC Entertainment, a subsidiary of Warner Bros. Discovery. DC Comics is one of the largest and oldest American comic book companies, with thei ...
parody superhero team. * No. 5 is the name of the iconic fragrance created by
Coco Chanel Gabrielle Bonheur "Coco" Chanel ( , ; 19 August 1883 – 10 January 1971) was a French fashion designer and businesswoman. The founder and namesake of the Chanel brand, she was credited in the post-World War I era with popularizing a sporty, c ...
. *The
Committee of Five '' The Committee of Five of the Second Continental Congress was a group of five members who drafted and presented to the full Congress in Pennsylvania State House what would become the United States Declaration of Independence of July 4, 1776. Thi ...
was delegated to draft the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territori ...
Declaration of Independence A declaration of independence or declaration of statehood or proclamation of independence is an assertion by a polity in a defined territory that it is independent and constitutes a state. Such places are usually declared from part or all of th ...
. *The
five-second rule The five-second rule, sometimes known as the three-second rule,(7 February 2006Getting the dirt of the 5-second rule ''Southeast Missourian'' is a food hygiene myth that states a defined time window where it is safe to pick up food (or sometim ...
is a commonly used rule of thumb for dropped food. *555 95472, usually referred to simply as 5, is a minor male character in the comic strip ''Peanuts''.


See also

*
Five Families The Five Families refers to five major New York City organized crime families of the Italian American Mafia formed in 1931 by Salvatore Maranzano following his victory in the Castellammarese War. Maranzano reorganized the Italian American gangs ...
*
Five Nations (disambiguation) Five Nations may refer to: Art, entertainment, and media * Five Nations (accessory), a book about the fictional Eberron location * Five Nations of the Eberron Campaign Setting, a fictional location * '' The Five Nations'', a 1903 poetry collection ...
* 555 (number) *
List of highways numbered 5 Route 5, or Highway 5, may refer to routes in the following countries: International * Asian Highway 5 * European route E05 * European route E005 Argentina * National Route 5 Australia New South Wales * M5 Motorway (Sydney) * The Detou ...


References

*Wells, D. ''
The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, ...
'' London: Penguin Group. (1987): 58–67


External links

* *
The Number 5The Positive Integer 5
{{DEFAULTSORT:5 (Number) Integers 5 (number)