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 n ...
, 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. Th ...
, 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
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
Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers.
The subsequence begins
:3, 5, 11, 17, 31, ...
.
It is the first
safe prime, the first
good prime A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes.
That is, good prime satisfies the inequality
:p_n^2 > p_ \cdot p_
for all 1 â ...
, the first
balanced prime In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number p_n, where is it ...
, and the first of three known
Wilson prime
In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century E ...
s. Five is the second
Fermat prime 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 17t ...
exponent, as well as the third
Catalan number, and the third
Sophie Germain prime.
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 numberâfor 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 pr ...
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
Alternating may refer to:
Mathematics
* Alternating algebra, an algebra in which odd-grade elements square to zero
* Alternating form, a function formula in algebra
* Alternating group, the group of even permutations of a finite set
* Alternati ...
, and an
Eisenstein prime with no imaginary part and real part of the form
â
.
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) cong ...
, since it is the length of the
hypotenuse of the smallest
integer-sided right triangle.
Five is the second
Fermat prime of the form
+
, and more generally the second
SierpiĆski number of the first kind,
+
. 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 5
2, 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. 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
A shape or figure is a graphics, graphical representation of an obje ...
s inside a circle that are formed from the
sides and
diagonals of the first five
-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 toge ...
s, and equal to the maximum number of areas formed by a six-sided polygon; per
Moser's circle problem The number of and for first 6 terms of Moser's circle problem
In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges an ...
.
The number 5 is the fifth
Fibonacci number, 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 and a
Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5,
13), (2, 5,
29), (5, 13,
194
Year 194 ( 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 947 '' Ab urbe ...
), (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 number
In mathematics, the Perrin numbers are defined by the recurrence relation
: for ,
with initial values
:.
The sequence of Perrin numbers starts with
: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, ...
The number of different maxima ...
s.
5 is the third
Mersenne prime exponent of the form
â
, which yields
: 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 17t ...
and second
double Mersenne prime 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 â 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
In computer architecture, 32-bit computing refers to computer systems with a processor, memory, and other major system components that operate on data in 32-bit units. Compared to smaller bit widths, 32-bit computers can perform large calculation ...
integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime
= 2
23058...93951 â 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of
CatalanâMersenne numbers are the only known prime terms, with a sixth possible candidate in the order of 10
1037.7094. These prime sequences are conjectured to be prime up to a certain limit.
Every odd number greater than
is the sum of at most five prime numbers, and every odd number greater than
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 review ...
.
The sums of the first five non-primes greater than zero
+
+
+
+
and the first five prime numbers
+
+
+
+
both equal
; 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 in ...
and like
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.
T ...
, which also includes
, the 31st triangular number and perfect number of the form
â1(
â
) with a
of
, by the
EuclidâEuler theorem
The EuclidâEuler 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 unitary divisors. A sixth unitary number, if discovered, would have at least nine odd prime factors.
Five is
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
d 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 numbers, 5 is a
pentagonal number, 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
A centered tetrahedral number is a centered figurate number that represents a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, ...
: 1, 5, 15, 35, 69, ... Every centered tetrahedral number with an index of 2, 3 or 4
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
5 is divisible by 5.
* 5 is a
square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broa ...
: 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: 1, 5, 13, 25, 41, ... The fifth
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
or 5
2 is
25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) ''primitive''
Pythagorean triples.
The
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
of five, or
! =
, 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 (â1, â2, â3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 in ...
, 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, is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.
5 is the value of the central
cell of the only non-trivial
normal magic square, also called the
''Lo Shu'' square. Its
x
array of squares has a
magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' â that is ...
of
, 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 exa ...
equations of degree and below can be solved with radicals, while
quintic equations of degree 5, and higher, cannot generally be so solved. This is the
AbelâRuffini theorem. This is related to the fact that the
symmetric group is a
solvable group 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 circum ...
,
+
=
, 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
The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irratio ...
,
Euler's number , the
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
,
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; a h ...
, and
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the HinduâArabic numeral system (or ...
, which makes this formula a renown example of
beauty in mathematics.
In geometry
A
pentagram, or five-pointed
polygram
PolyGram N.V. was a multinational entertainment company and major music record label formerly based in the Netherlands. It was founded in 1962 as the Grammophon-Philips Group by Dutch corporation Philips and German corporation Siemens, to be a ...
, is the first proper
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
constructed from the diagonals of a
regular pentagon as
self-intersecting edges that are proportioned in
golden ratio,
. Its internal geometry appears prominently in
Penrose tilings
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any fin ...
, and is a
facet inside
Kepler-Poinsot star polyhedra and
SchlĂ€fliâHess star polychora, represented by its
SchlÀfli symbol
In geometry, the SchlÀfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The SchlÀfli symbol is named after the 19th-century Swiss mathematician Ludwig SchlÀfli, who generalized Euclidean geometry to more ...
. 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
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...
star â without self-intersecting edges. Generally,
star polytopes that are
regular only exist in
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s 2 ⩜
< 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 conne ...
, all
graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
with 4 or fewer vertices are
planar, however, there is a graph with 5 vertices that is not: ''K''
5, the
complete graph 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 subdivi ...
, 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 ''K''
3,3. A similar graph is the
Petersen graph, which is
strongly connected and also
nonplanar. It is most easily described as graph of a pentagram ''embedded'' inside a pentagon, with a total of 5
crossings
Crossings may refer to:
* ''Crossings'' (Buffy novel), a 2002 original novel based on the U.S. television series ''Buffy the Vampire Slayer''
* Crossings (game), a two-player abstract strategy board game invented by Robert Abbott
* ''Crossings'' ...
, a
girth
Girth may refer to:
;Mathematics
* Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space
* Girth (geometry), the perimeter of a parallel projection of a shape
* Girth ...
of 5, and a
Thue number
In the mathematical area of graph theory, the Thue number of a graph is a variation of the chromatic index, defined by and named after mathematician Axel Thue, who studied the squarefree words used to define this number.
Alon et al. define a ''no ...
of 5. The Petersen graph, which is also a
distance-regular graph, is one of only 5 known
connected 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 cycles.
[Royle, G]
"Cubic Symmetric Graphs (The Foster Census)."
The
automorphism group of the Petersen graph is the
symmetric group of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
120 = 5!.
The
chromatic number of the plane is at least five, depending on the choice of
set-theoretical axioms: the minimum number of
colors 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 colorin ...
and the regular
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has SchlÀfli symbol of or (as a truncated triangular tiling).
English mathemat ...
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
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
.
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 (gen ...
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
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
defined by discrete
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
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,
rectangular,
centered rectangular, and
square lattices. The plane can also be tiled
monohedrally with convex
pentagons
In geometry, a pentagon (from the Greek ÏÎÎœÏΔ ''pente'' meaning ''five'' and ÎłÏÎœÎŻÎ± ''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 ...
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 specia ...
, 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 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 r ...
condition for a point to be contained inside a conic.
There are
Platonic solids 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 (geometry), position of an element (i.e., Point (m ...
: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The
dodecahedron in particular contains
pentagonal faces, while the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
that is a regular pentagon. There are also
:
â
Regular polyhedron compounds: the
stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.
Icosahedral symmetry is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
alternating group on 5 letters
of order
120, realized by actions on these uniform polyhedron compounds.
â
Space-filling convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
: the triangular prism,
hexagonal prism, 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 solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s, respectively, that can also tessellate space with their own copies.
â
Cell-transitive parallelohedra: any
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
, as well as the
rhombic dodecahedron and
elongated dodecahedron, 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 EugĂšne Catalan, who first described them in 1865.
The Catalan sol ...
to tessellate space on its own.
â
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. They have combinatorial symmetries transitive on
flags of their elements, with
topologies equivalent to that of
toroids and the ability to tile the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyaiâ 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 SchlÀfli 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 the o ...
, with
Coxeter group symmetry
of order
120 = 5
! and
group structure
In the social sciences, a social group can be defined as two or more people who interact with one another, share similar characteristics, and collectively have a sense of unity. Regardless, social groups come in a myriad of sizes and varieties ...
. Made of five tetrahedra, its
Petrie polygon is a
regular pentagon and its
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
is equivalent to the
complete graph ''K''
5. It is one of six
regular 4-polytopes
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star re ...
, made of thirty-one
elements: five
vertices, ten
edges
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
, ten
faces, five
tetrahedral cells and one
4-face
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''.
In more technical treatments of the geometry of polyhedra ...
.
*A
regular 120-cell, the dual ''polychoron'' to the regular
600-cell, 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 SchlÀfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
s fit inside a
small stellated 120-cell, the first
stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
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, se ...
nonconvex
''duoantiprismatic'' solution in the fourth dimension, constructed from the
polytope cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''Ă''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
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 antiprisms, and fifty vertices.
*The
grand antiprism, 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 SchlÀfli symbol , with 3 hemi-icosahedr ...
, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven
hemi-dodecahedral cells each with fifteen dodecahedra. The
skeleton
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of the hemi-dodecahedron is the
Petersen graph.
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 Ί is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
s that form a finite number of
uniform polychora
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
:
,
,
,
, and
, with four of these Coxeter groups capable of generating the same finite forms without
; 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 groups 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, as with the
order-5 5-cell honeycomb
In the geometry of hyperbolic 4-space, the order-5 5-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With SchlÀfli symbol , it has five 5-cells around each face. Its dual is the 120-cell honeycomb, . ...
and the
order-5 120-cell honeycomb
In the geometry of hyperbolic 4-space, the order-5 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With SchlÀfli symbol , it has five 120-cells around each face. It is self- dual. It also has 600 1 ...
, 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 or four-dimensional
symmetry do not exist in dimensions ''n'' â©Ÿ 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, se ...
prismatic group Ă
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
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 cosâ1(), or approximately 78.46°.
The 5-s ...
is the
five-dimensional
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions a ...
analogue of the 5-cell, or 4-simplex; the fifth iteration of
-
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es in any
dimensions. The 5-simplex has the Coxeter group
as its symmetry group, of order 720 = 6
!, whose group structure is represented by the symmetric group
, the only finite symmetric group which has an
outer automorphism. 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 SchlÀfli symbol or , constructed as 3 tesseracts, ...
, made of ten
tesseracts 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 Facet (geometry), facets.
The complete set of convex uniform 5-polytopes ...
s under the Coxeter
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
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 w ...
, is the only fifth-dimensional
semiregular polytope, and has the
rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the
rectified 600-cell 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 face ...
. In the fifth dimension, there are five regular paracompact honeycombs, all with
infinite facets 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s. There are exclusively twelve
complex aperiotopes in
complex spaces of dimensions
â©Ÿ
, with fifteen in
and sixteen in
; alongside
complex polytopes in
and higher under
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
,
hypercubic
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perp ...
and
orthoplex groups, the latter with
van Oss polytopes.
There are five
exceptional Lie groups
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
:
,
,
,
, and
. The
smallest of these,
, 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 thei ...
in the same number of dimensions as a
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
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 mea ...
is described in two-dimensional space.
, the largest of all five exceptional groups, also contains the other four as
subgroups and is constructed with one hundred and twenty quaternionic
unit icosians that make up the vertices of the
600-cell. There are also five
solvable groups 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. The ...
. These are also the first five sporadic groups
to have been described, defined as
multiply transitive permutation groups on
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
â . In particular,
, the smallest of all sporadic groups, has a
rank 3 action 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 ele ...
s, as well as two
five-dimensional
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions a ...
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 classes 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 ...
s of
, one is the
almost simple symmetric group
(of order 5
!), and another is
, also almost simple, that functions as a
point stabilizer
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
which has
as its largest
prime factor
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 ...
in its
group order
In mathematics, the order of a finite group is the number of its elements. If a 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 the order of the subgr ...
: 2
4·3
2·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas
is sharply 4-transitive,
is
sharply 5-transitive and
is 5-transitive, and as such they are the only two 5-transitive groups that are not
symmetric groups or
alternating groups.
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
, which under the
Witt design of
Steiner system S(5, 8, 24) emerges a construction of the
extended binary Golay code that has
as its
automorphism group.
generates ''octads'' from
code words of
Hamming weight 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 Î
24, which is the subject of the
second generation of seven sporadic groups that are
subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
s of the automorphism of the Leech lattice,
Conway group .
There are five
non-supersingular primes:
37,
43, 53 (number), 53, 61 (number), 61, and 67 (number), 67, all smaller than the largest of fifteen supersingular prime divisors of the Monster group, friendly giant, 71 (number), 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 Positional notation, ones place are multiples of five, which makes it a 1-automorphic number.
All multiples of 5 will end in either 5 or , and Fraction (mathematics)#Vulgar, proper, and improper fractions, vulgar fractions with 5 or in the fraction (mathematics), denominator do not yield infinite decimal expansions because they are prime factors of 10, the base.
In the Power (mathematics), powers of 5, every power ends with the number five, and from 5
3 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
raised to the fifth power always ends in the same digit as
.
Evolution of the Arabic digit
The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Brahmi numerals, Indian system, as for the digits 1 to 4. The Kushan Empire, Kushana and Gupta Empire, Gupta empires in what is now India had among themselves several different forms that bear no resemblance to the modern digit. The Devanagari, Nagari and Punjabi language, 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 (typography), ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .
On the seven-segment display 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 of boron.
*The number of appendages on most sea star, starfish, which exhibit symmetry (biology)#Pentamerism, pentamerism.
*The most destructive known tropical cyclone, hurricanes rate as SaffirâSimpson hurricane wind scale#Category 5, Category 5 on the SaffirâSimpson hurricane wind scale.
*The most destructive known tornadoes rate an F-5 on the Fujita scale or EF-5 on the Enhanced Fujita scale.
Astronomy
*Messier object Messier 5, M5, a magnitude 7.0 globular cluster in the constellation Serpens.
*The New General Catalogue]
objectNGC 5, a apparent magnitude, magnitude 13 spiral galaxy in the constellation Andromeda (constellation), Andromeda.
*The Roman numeral V stands for dwarfs (main sequence stars) in the stellar classification, Yerkes spectral classification scheme.
*The Roman numeral V (usually) stands for the fifth-discovered satellite of a planet or minor planet (e.g. Amalthea (moon), Jupiter V).
*There are five Lagrangian points in a two-body system.
Biology
*There are generally considered to be five senses.
*The five basic tastes are sweetness, sweet, taste#Saltiness, salty, taste#Sourness, sour, taste#Bitterness, bitter, and umami.
*Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.
Computing
*5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.
Religion and culture
Hinduism
*The god Shiva has five faces and his mantra is also called (five-worded) mantra.
*The goddess Saraswati, goddess of knowledge and intellectual is associated with or the number 5.
*There are Pancha Bhoota, five elements in the universe according to Hindu cosmology: (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 revolves around the battle between Duryodhana and his 99 other brothers and the 5 pandava princesâYudhisthira, Dharma, Arjuna, Bhima, Nakula and Sahadeva.
Christianity
*There are traditionally Five Wounds, five wounds of Jesus Christ in Christianity: the Flagellation of Christ, Scourging at the Pillar, Crown of Thorns, the Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Holy Lance, Side Wound of Christ.
Gnosticism
*The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
*Five Seals in Sethianism
*Five Trees in the Gospel of Thomas
Islam
*The Five Pillars of Islam
*Muslims pray to Allah five times a day
*According to Shia Muslims, the Panjetan-e-Pak, Panjetan or the Five Holy Purified Ones are the members of Muhammad's family: Muhammad, Ali, Fatimah, Hasan ibn Ali, Hasan, and Husayn ibn Ali, Husayn and are often symbolically represented by an image of the Hamsa, Khamsa.
Judaism
*The Torah contains five booksâBook of Genesis, Genesis, Book of Exodus, Exodus, Book of Leviticus, Leviticus, Book of Numbers, Numbers, and Book of Deuteronomy, Deuteronomyâwhich are collectively called the Five Books of Moses, the Pentateuch (Greek language, Greek for "five containers", referring to the scroll cases in which the books were kept), or Chumash (Judaism), Humash (, Hebrew language, Hebrew for "fifth").
*The book of Psalms is arranged into five books, paralleling the Five Books of Moses.
*The Hamsa, Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.
Sikhism
*The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as or the "The Five Ks, Five Ks" because they start with letter K representing in the Punjabi language's Gurmukhi script. 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
*Wuxing (Chinese philosophy), 5 Elements
*Three Sovereigns and Five Emperors, 5 Emperors
Other religions and cultures
*According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water (classical element), water, earth (classical element), earth, air (classical element), air, fire (classical element), fire, and aether (classical element), ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca.
*The
pentagram, or five-pointed star, bears religious significance in various faiths including BahĂĄÊŒĂ Faith, BahĂĄÊŒĂ, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
*In Cantonese, "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 Asian tradition, there are five elements: (water (Wu Xing), water, fire (Wu Xing), fire, earth (Wu Xing), earth, tree (Wu Xing), wood, and metal (Wu Xing), metal). The Japanese language, Japanese names for the week-day names, days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the Classical planet, 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 (number), 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, a fictional character numbered 5.
*Johnny 5 is the lead character in the film ''Short Circuit'' (1986)
*Number Five is a character in Lorien Legacies
*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, five wizards (Saruman, Gandalf, Radagast, Blue Wizards, 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, Stannis Baratheon, Renly Baratheon, Robb Stark and Balon Greyjoy)
*In ''The Wheel of Time'' 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, Matrim Cauthon, Perrin Aybara, Egwene al'Vere and Nynaeve al'Meara)
*Myst (series), ''Myst'' uses the number 5 as a unique base counting system. In ''The Myst Reader'' 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 3, three.
*''Five Go Mad in Dorset'' (1982) was the first of the long-running series of ''The Comic Strip, The Comic Strip Presents...'' television comedy films
*''The Fifth Element'' (1997), a science fiction film
* ''Fast Five'' (2011), the fifth installment of the The Fast and the Furious (series), ''Fast and Furious'' film series.
*''V for Vendetta (film), V for Vendetta'' (2005), produced by Warner Bros., directed by James McTeigue, and adapted from Alan Moore's graphic novel ''V for Vendetta'' 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), a UK Boy band
*The Five (composers), 19th-century Russian composers
*5 Seconds of Summer, pop band that originated in Sydney, Australia
*Five Americans, American rock band active 1965â1969
*Five Finger Death Punch, American heavy metal band from Las Vegas, Nevada. Active 2005âpresent
*Five Man Electrical Band, Canadian rock group billed (and active) as the Five Man Electrical Band, 1969â1975
*Maroon 5, American pop rock band that originated in Los Angeles, California
*MC5, American punk rock band
*Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas
*The 5th Dimension, American pop vocal group, active 1977âpresent
*The Dave Clark Five, a.k.a. DC5, an English pop rock group comprising Dave Clark (musician), Dave Clark, Lenny Davidson, Rick Huxley, Denis Payton, and Mike Smith (Dave Clark Five), Mike Smith; active 1958â1970
*The Jackson 5, American pop rock group featuring various members of the Jackson family; they were billed (and active) as The Jackson 5, 1966â1975
*Hi-5 (Australian group), 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: American folk rock group active 1965â1967 and 1968â1977
*Grandmaster Flash and the Furious Five: American rap group, 1970â80's
*Fifth Harmony, an American girl group.
*Ben Folds Five, an American alternative rock trio, 1993â2000, 2008 and 2011â2013
*R5 (band), an American pop and alternative rock group, 2009â2018
Other uses
*A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.
*Modern musical notation uses a staff (music), musical staff made of five horizontal lines.
*In harmonics â the fifth harmonic series (music), partial (or 4th overtone) of a fundamental frequency, fundamental 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 and minor, major Triad (music), triad chord (music), 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, Sergei Prokofiev, 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.
*Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter (music), meter.
Television
;Stations
*Channel 5 (UK), a television channel that broadcasts in the United Kingdom
*5 (TV channel) (''formerly known as ABC 5 and TV5'') (DWET-TV channel 5 In Metro Manila) a television network in the Philippines.
;
;Series
*''Babylon 5'', a science fiction television series
*The number 5 features in the television series Battlestar Galactica (2004 TV series), ''Battlestar Galactica'' in regards to the Final Five (Battlestar Galactica), Final Five cylons and the Temple of Five
*Hi-5 (Australian TV series), ''Hi-5'' (Australian TV series), a television series from Australia
*Hi-5 (UK TV series), ''Hi-5'' (UK TV series), a television show from the United Kingdom
*Hi-5 Philippines, ''Hi-5'' Philippines a television show from the Philippines
*''Odyssey 5'', 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), The Five'' The Five (talk show), (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 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 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 (2010 TV series), ''Hawaii Five-0'', CBS American TV series.
Literature
*The Famous Five (novel series), ''The Famous Five'' is a series of children's books by British writer Enid Blyton
*''The Power of Five'' 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 swordsman Miyamoto Musashi circa 1645
*''Slaughterhouse-Five'' is a book by Kurt Vonnegut about World War II
Sports
*The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).
* In AFL Women's, the top level of Women's Australian rules football, women's Australian rules football, each team is allowed 5 "Interchange (Australian rules football), interchanges" (substitute players), who can be freely substituted at any time.
*In Baseball scorekeeping#Defensive positions, baseball scorekeeping, the number 5 represents the third baseman's position.
*In basketball:
**The number 5 is used to represent the position of center (basketball), center.
**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 Five-second rule (basketball), "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 (used for all international play, and most non-US leagues) and College basketball, NCAA women's rule sets, a team begins shooting Bonus (basketball), bonus free throws once its opponent has committed five Personal foul (basketball), 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 is a variation of association football in which each team fields five players.
*In ice hockey:
** 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.
*In most rugby league competitions, the starting Rugby league positions#Wing, left wing wears this number. An exception is the Super League, which uses static squad numbering.
*In rugby union:
** A Try (rugby), try is worth 5 points.
** One of the two starting Lock (rugby union), lock forwards wears number 5, and usually jumps at number 4 in the line-out (rugby union), line-out.
** In the National Rugby League (France), French variation of the Rugby union bonus points system, 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.
*In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.
*On almost all devices with a numeric keypad 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 numeric keypad, numpad has the raised dot or bar, but the 5 key that shifts with % does not).
*On most telephones, the 5 key is associated with the letters J, K, and L, but on some of the BlackBerry phones, it is the key for G and H.
*The Pentium, coined by Intel Corporation, is a fifth-generation x86 architecture microprocessor.
*The resin identification code used in recycling to identify polypropylene.
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.
*Five babies born at one time are multiple birth, quintuplets. The most famous set of quintuplets were the Dionne quintuplets born in the 1930s.
*In the United States legal system, the Fifth Amendment to the United States Constitution can be referred to in court as "pleading the fifth", absolving the defendant from self-incrimination.
*Pentameter is verse with five repeating feet per line; iambic pentameter was the most popular form in William Shakespeare, Shakespeare.
*Aether (classical element), 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 System, Interstate Highway (Interstate 5) that runs from San Diego, California to Blaine, Washington. In addition, all major north-south Interstate Highways in the United States end in 5.
*In the computer game ''Riven'', 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'' (1658) by Sir Thomas Browne is a Pythagorean discourse based upon the number 5.
*The holy number of Discordianism, as dictated by the Discordianism#Law of Fives, Law of Fives.
*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.
*The number of permanent members with veto power on the United Nations Security Council.
*The number of sides and the number of angles in a pentagon.
*The number of points in a
pentagram.
*The number of Korotkoff sounds when measuring blood pressure
*The drink Five Alive is named for its five ingredients. The drink Punch (drink), punch derives its name after the Sanskrit à€Șà€à„à€ (pañc) for having five ingredients.
*The Keating Five were five United States Senate, United States Senators accused of corruption in 1989.
*The Inferior Five: Merryman, Awkwardman, The Blimp, White Feather, and Dumb Bunny. DC Comics parody superhero team.
*Chanel No. 5, No. 5 is the name of the iconic fragrance created by Coco Chanel.
*The Committee of Five was delegated to draft the United States United States Declaration of Independence, Declaration of Independence.
*The five-second rule 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
*Five Nations (disambiguation)
*555 (number)
*List of highways numbered 5
References
*Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 58â67
External links
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The Number 5The Positive Integer 5
{{DEFAULTSORT:5 (Number)
Integers
5 (number)