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8 (eight) 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 ...
following 7 and preceding 9.


Etymology

English ''eight'', from Old English '', æhta'',
Proto-Germanic Proto-Germanic (abbreviated PGmc; also called Common Germanic) is the reconstructed proto-language of the Germanic branch of the Indo-European languages. Proto-Germanic eventually developed from pre-Proto-Germanic into three Germanic branc ...
''*ahto'' is a direct continuation of
Proto-Indo-European Proto-Indo-European (PIE) is the reconstructed common ancestor of the Indo-European language family. Its proposed features have been derived by linguistic reconstruction from documented Indo-European languages. No direct record of Proto-Indo-E ...
'' *oḱtṓ(w)-'', and as such cognate with Greek and Latin , both of which stems are reflected by the English prefix oct(o)-, as in the ordinal adjective ''octaval'' or ''octavary'', the distributive adjective is ''
octonary : ''For the base-8 numeral system, see octal.'' An octonary is an eight-line section in a poem, song or psalm. The most notable example is found in Psalm 119Calvin ''Bible Commentaries: Psalms, Part IV'' p287 "Some call this the octonary psalm, bec ...
''. The adjective ''octuple'' (Latin ) may also be used as a noun, meaning "a set of eight items"; the diminutive '' octuplet'' is mostly used to refer to eight siblings delivered in one birth. The Semitic numeral is based on a root ''*θmn-'', whence Akkadian ''smn-'', Arabic ''ṯmn-'', Hebrew ''šmn-'' etc. The
Chinese numeral Chinese numerals are words and characters used to denote numbers in Chinese. Today, speakers of Chinese use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous sy ...
, written (
Mandarin Mandarin or The Mandarin may refer to: Language * Mandarin Chinese, branch of Chinese originally spoken in northern parts of the country ** Standard Chinese or Modern Standard Mandarin, the official language of China ** Taiwanese Mandarin, Stand ...
: ''bā'';
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 are ...
: ''baat''), is from
Old Chinese Old Chinese, also called Archaic Chinese in older works, is the oldest attested stage of Chinese language, Chinese, and the ancestor of all modern varieties of Chinese. The earliest examples of Chinese are divinatory inscriptions on oracle bones ...
''*priāt-'', ultimately from Sino-Tibetan ''b-r-gyat'' or ''b-g-ryat'' which also yielded Tibetan '' brgyat''. It has been argued that, as the
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 ...
is the highest number of items that can universally be cognitively processed as a single set, the etymology of the numeral ''eight'' might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar. The
Turkic Turkic may refer to: * anything related to the country of Turkey * Turkic languages, a language family of at least thirty-five documented languages ** Turkic alphabets (disambiguation) ** Turkish language, the most widely spoken Turkic language * ...
words for "eight" are from a Proto-Turkic stem ''*sekiz'', which has been suggested as originating as a negation of ''eki'' "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not being held up"); this same principle is found in Finnic '' *kakte-ksa'', which conveys a meaning of "two before (ten)". The Proto-Indo-European reconstruction '' *oḱtṓ(w)-'' itself has been argued as representing an old dual, which would correspond to an original meaning of "twice four". Proponents of this "quaternary hypothesis" adduce the numeral ', which might be built on the stem ''new-'', meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).


Evolution of the Arabic digit

The modern digit 8, like all modern
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
other than zero, originates with the
Brahmi numerals The Brahmi numerals are a numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are a non positional decimal system. They are the direct graphic ancestors of the modern Hindu–Arabic numeral s ...
. The Brahmi digit for ''eight'' by the 1st century was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of the left line and the upper half of the right line removed. However, the digit for eight used in India in the early centuries of the Common Era developed considerable graphic variation, and in some cases took the shape of a single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our digit 5. The digits as used in
Al-Andalus Al-Andalus DIN 31635, translit. ; an, al-Andalus; ast, al-Ándalus; eu, al-Andalus; ber, ⴰⵏⴷⴰⵍⵓⵙ, label=Berber languages, Berber, translit=Andalus; ca, al-Àndalus; gl, al-Andalus; oc, Al Andalús; pt, al-Ândalus; es, ...
by the 10th century were a distinctive western variant of the glyphs used in the Arabic-speaking world, known as ''ghubār'' numerals (''ghubār'' translating to "
sand table A sand table uses constrained sand for modelling or educational purposes. The original version of a sand table may be the abax used by early Greek students. In the modern era, one common use for a sand table is to make terrain models for milita ...
"). In these digits, the line of the ''5''-like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the ''8''-shape that became adopted into European use in the 10th century. Just as 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 list of type ...
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 character for the digit 8 usually has an ascender, as, for example, in . The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the digit 8 in origin; it is first used (in the mathematical meaning "infinity") in the 17th century, and it may be derived from the
Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
for "one thousand" CIƆ, or alternatively from the final Greek letter, ω.


Mathematics

Eight is the third
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
, lying between the fourth prime number ( 7) and the fourth composite number ( 9). 8 is the first non-unitary cube
prime 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 ...
of the form ''p''3. With
proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s , , and , it is the third
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
(2). 8 is the first number which is neither
prime 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 ...
nor
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...
and the only nonzero perfect power that is one less than another perfect power, by
Mihăilescu's Theorem Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are ...
. * 8 is the first proper
Leyland number In number theory, a Leyland number is a number of the form :x^y + y^x where ''x'' and ''y'' are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are : 8, 17, 32, 54, 57, 100, 145, 177, ...
of the form , where in its case and both equal 2. * 8 is the sum between the first pair of twin-primes ( 3, 5), and the only twin-prime sum that is not a multiple of 3 or 12. * 8 is the sixth Fibonacci number and the first even, non-prime Fibonacci number. It is also the only positive Fibonacci number aside from 1 that is a
perfect cube In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . ...
. * 8 is the third
refactorable number A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, ...
, as it has exactly four positive divisors, and 4 is one of them. * 8 is the only composite number with a prime aliquot sum of 7 (1 + 2 + 4) that is part of the aliquot sequence (8, 7, 1, 0). * 8 is the first number to be the aliquot sum of two numbers: the discrete
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...
10 = 5 × 2, and
squared A square is a regular quadrilateral with four equal sides and four right angles. Square or Squares may also refer to: Mathematics and science *Square (algebra), multiplying a number or expression by itself *Square (cipher), a cryptographic block ...
prime 49 = 72. * 8 is the number of known
primary pseudoperfect number In mathematics, and particularly in number theory, ''N'' is a primary pseudoperfect number if it satisfies the Egyptian fraction equation :\frac + \sum_\frac = 1, where the sum is over only the prime divisors of ''N''. Properties Equivalently, ...
s, the smallest four (2, 6, 42, 1806) are one less than the first four numbers in Sylvester's sequence. Sphenic numbers always have exactly eight divisors.


Octagon

A
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 ...
with eight sides is an
octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
. The sides and
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan es ...
of a
regular octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
, or truncated square, are in
silver ratio In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
, and its circumscribing square has a side and diagonal length ratio of ; with both the silver ratio and the square root of two intimately interconnected through Pell numbers, where in particular the quotient of successive Pell numbers generates rational approximations for coordinates of a
regular octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
. With a
central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
of 45 degrees and an internal angle of 135 degrees, a regular octagon can fill a plane-vertex with a regular
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
and a regular
icositetragon In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. Regular icositetragon The ''regular polygon, regular icositetragon'' is represented by S ...
, as well as tessellate two-dimensional space alongside squares in the truncated square tiling. This tiling is one of eight
Archimedean tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Eucli ...
s that are semi-regular, or made of more than one type of regular
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 ...
, and the only tiling that can admit a regular octagon. The
Ammann–Beenker tiling In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. Th ...
is a nonperiodic tesselation of prototiles that feature prominent octagonal ''silver'' eightfold symmetry, that is the two-dimensional
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 ...
of the four-dimensional
8-8 duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...
. In number theory, figurate numbers representing octagons are called
octagonal number An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341 ...
s.


Cube and octahedron

A
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
is a regular polyhedron with eight vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space. Through various truncation operations, the cubic honeycomb generates eight other convex uniform honeycombs under the cubic group _3. The octahedron, with eight equilateral triangles as faces, is the
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. ...
to the cube and one of eight convex deltahedra. The stella octangula, or ''eight-pointed star'', is the only
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 ...
with octahedral symmetry. It has eight triangular faces alongside eight vertices that forms a cubic
faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted pol ...
, composed of two self-dual
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 ...
that makes it the simplest of five regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex
quasiregular polyhedra In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular polygon, regular faces, which alternate around each vertex (geometry), vertex. They are vertex-transitive and edge-transitive, hence a step closer ...
. It contains eight equilateral triangular faces, whose 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 ...
is the cube-octahedron compound. There are also eight
uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...
s made purely of octahedra, including the regular compound of five octahedra. ;Truncated figures The
truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...
is the simplest Archimedean solid, made of four triangles and four hexagons, the hexagonal prism, which classifies as an irregular octahedron and parallelohedron, is able to tessellate space as a three-dimensional analogue of the
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...
, and the
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 ...
, with four square faces and four triangular faces, is the only
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 ...
that is able to tessellate space. The
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, also a parallelohedron, is the
permutohedron In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...
of order four, with eight hexagonal faces alongside six squares is likewise the only
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
that can generate a honeycomb on its own.


Polychora

A tesseract or 8-cell is the
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
analogue of the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
. It is one of six regular ''polychora'', with a total of eight cubical
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 ...
, hence its name. Its dual figure is the analogue of the octahedron, with twice the amount of cells and simply termed the
16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...
, that is the
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of vectors in four dimensions. Whereas a
tesseractic honeycomb In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets. Its verte ...
is self-dual, a
16-cell honeycomb In Four-dimensional space, four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycomb (geometry), honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensiona ...
is dual to a
24-cell honeycomb In Four-dimensional space, four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular polytope, regular space-filling tessellation (or honeycomb (geometry), honeycomb) of 4-dimensional Euclidean space by ...
that is made of
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, oct ...
s. The 24-cell is also regular, and made purely of octahedra whose vertex arrangement represents the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of Hurwitz integral quaternions. Both the tesseract and the 16-cell can fit inside a 24-cell, and in a 24-cell honeycomb, eight 24-cells meet at a vertex. Also, the Petrie polygon of the tesseract and the 16-cell is a regular octagon.


The octonions

The
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s are a hypercomplex normed division algebra that are an extension of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. They are realized in eight dimensions, where they have an isotopy group over the real numbers that is
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...
Spin(8), the unique such group that exhibits a phenomenon of
triality In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8) ...
. As a double cover of
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
SO(8), Spin(8) contains the special orthogonal Lie algebra D4 as its Dynkin diagram, whose order-three outer automorphism 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 symmetric group S3, giving rise to its triality. Over finite fields, the eight-dimensional Steinberg group 3D4(''q''3) is
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 ...
, and one of sixteen such groups in the classification of
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. As is Lie algebra E8, whose complex form in 248 dimensions is the largest of five
exceptional Lie algebra In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: \mathfrak_2, \mathfrak_4, \mathfrak_6, \mathfrak_7, \mathfrak_8; their respective ...
s that include E7 and E6, which are
held Held may refer to: Places * Held Glacier People Arts and media * Adolph Held (1885–1969), U.S. newspaper editor, banker, labor activist *Al Held (1928–2005), U.S. abstract expressionist painter. *Alexander Held (born 1958), German television ...
inside E8. The smallest such algebra is G2, that is the automorphism group of the octonions. In
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, special unitary group SO(3) has an eight-dimensional
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
whose colors are ascribed gauge symmetries that represent the vectors of the eight
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
s in the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
. The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. If O(\infty) is the direct limit of the inclusions of real orthogonal groups O(1)\hookrightarrow O(2)\hookrightarrow\ldots\hookrightarrow O(k)\hookrightarrow\ldots, the following holds: :\pi_(O(\infty))\cong\pi_(O(\infty)).
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
s also display a periodicity of 8. For example, the algebra ''Cl''(''p'' + 8,''q'') is isomorphic to the algebra of 16 by 16 matrices with entries in ''Cl''(''p'',''q''). We also see a period of 8 in the K-theory of spheres and in the representation theory of the rotation groups, the latter giving rise to the 8 by 8 spinorial chessboard. All of these properties (that also tie with
Lorentzian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
, and
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...
s) are closely related to the properties of the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s, which occupy the highest possible dimension for a normed division algebra.


E8

The \mathrm E_ lattice Γ8 is the smallest positive even
unimodular lattice In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...
. As a lattice, it holds the optimal structure for the densest packing of
240 __NOTOC__ Year 240 ( CCXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Venustus (or, less frequently, year 993 ''Ab u ...
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s in eight dimensions, whose lattice points also represent the root system of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
E8. This honeycomb arrangement is shared by a unique complex tessellation of
Witting polytope In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3333, and Coxeter diagram . It has 240 vertices, 2160 3 edges, 2160 Möbius–Kantor polygon, 33 faces, and 240 Hessian polyhedron, 333 cells. It is s ...
s, also with 240 vertices. Each complex Witting polytope is made of Hessian polyhedral cells that have
Möbius–Kantor polygon In geometry, the Möbius–Kantor polygon is a regular complex polygon 33, , in \mathbb^2. 33 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a ''Möbius–Kantor polygon'' for sharing th ...
s as faces, each with eight vertices and eight complex equilateral triangles as edges, whose Petrie polygons form regular octagons. In general, positive even unimodular lattices only exist in dimensions proportional to eight. In the 16th dimension, there are two such lattices : Γ8 ⊕ Γ8 and Γ16, while in the 24th dimension there are precisely twenty-four such lattices that are called the
Niemeier lattice In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such lattic ...
s, the most important being the Leech lattice, which can be constructed using the octonions as well as with three copies of the ring of icosians that are isomorphic to the \mathrm E_ lattice. The order of the smallest non-abelian group all of whose subgroups are normal is 8.
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 ...
semiregular polytopes whose facets are ''finite'' exist up through the 8th dimension. In the
third dimension 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 informal ...
, they include the Archimedean solids and the infinite family of uniform prisms and antiprisms, while in the fourth dimension, only the rectified 5-cell, 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 ...
are semiregular polytopes. For dimensions
five 5 is a number, numeral, and glyph. 5, five or number 5 may also refer to: * AD 5, the fifth year of the AD era * 5 BC, the fifth year before the AD era Literature * ''5'' (visual novel), a 2008 visual novel by Ram * ''5'' (comics), an awa ...
through eight, 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- ...
and the k21 polytopes 221, 321, and 421 are the only semiregular (
Gosset Gosset, founded in 1584, is the oldest wine house in Champagne. In 1584, Pierre Gosset, alderman of Aÿ and wine-grower, made still, mostly red, wines from the grapes he harvested from his own vines. In those days, two wines vied for pride of pl ...
) polytopes. Collectively, the k21 family of polytopes contains eight figures that are rooted in the
triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
, which is the simplest semiregular polytope that is made of three cubes and two equilateral triangles. It also includes one of only three semiregular Euclidean honeycombs: the affine 521 honeycomb that represents the arrangement of vertices of the eight-dimensional \mathrm E_ lattice, and made of 421 facets. The culminating figure is the ninth-dimensional 621 honeycomb, which is the only affine semiregular paracompact hyperbolic honeycomb 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 in the k21 family. There are no other finite semiregular polytopes or honeycombs in dimensions ''n'' > 8.


Finite simple groups

In the classification of sporadic groups, the third generation consists of eight groups, four of which are centralizers of \mathrm (itself the largest group of this generation), with another three transpositions of
Fischer group In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...
\mathrm . 8 is the difference between 53 and 61, which are the two smallest prime numbers that do not divide the order of any sporadic group. The largest supersingular prime that divides the
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 ...
of \mathrm is 71, which is the eighth self-convolution of Fibonacci numbers (where
744 __NOTOC__ Year 744 ( DCCXLIV) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 744 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
, which is essential to
Moonshine theory In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
, is the twelfth). While only two sporadic groups have eight prime factors in their order (
Lyons group In the area of modern algebra known as group theory, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of order :    283756711313767 : = 51765179004000000 : ≈ 5. History ''Ly'' is one of the 26 sporad ...
\mathrm and Fischer group \mathrm ),
Mathieu group 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 objec ...
\mathrm holds a semi-presentation whose
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 ...
is equal to o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 8.


List of basic calculations


In other bases

A number is divisible by 8 if its last three digits, when written in
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, are also divisible by 8, or its last three digits are 0 when written in
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
. 8 is the base of the
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
number system, which is mostly used with
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
s. In octal, one digit represents three bits. In modern computers, a
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
is a grouping of eight bits, also called an
octet Octet may refer to: Music * Octet (music), ensemble consisting of eight instruments or voices, or composition written for such an ensemble ** String octet, a piece of music written for eight string instruments *** Octet (Mendelssohn), 1825 compos ...
.


In science


Physics

* In nuclear physics, the second magic number. * In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the eightfold way is used to classify sub-atomic particles. * In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the
eight-vertex model In statistical mechanics, the eight-vertex model is a generalisation of the ice-type (six-vertex) models; it was discussed by Sutherland, and Fan & Wu, and solved by Baxter in the zero-field case. Description As with the ice-type models, the ei ...
has 8 possible configurations of arrows at each vertex.


Astronomy

* Messier object M8, a magnitude 5.0
nebula A nebula ('cloud' or 'fog' in Latin; pl. nebulae, nebulæ or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral or molecular hydrogen and also cosmic dust. Nebulae are often star-forming regio ...
in the
constellation A constellation is an area on the celestial sphere in which a group of visible stars forms Asterism (astronomy), a perceived pattern or outline, typically representing an animal, mythological subject, or inanimate object. The origins of the e ...
of Sagittarius. * The
New General Catalogue The ''New General Catalogue of Nebulae and Clusters of Stars'' (abbreviated NGC) is an astronomical catalogue of deep-sky objects compiled by John Louis Emil Dreyer in 1888. The NGC contains 7,840 objects, including galaxies, star clusters and ...
object NGC 8, a double star in the constellation
Pegasus Pegasus ( grc-gre, Πήγασος, Pḗgasos; la, Pegasus, Pegasos) is one of the best known creatures in Greek mythology. He is a winged divine stallion usually depicted as pure white in color. He was sired by Poseidon, in his role as hor ...
. * Since the demotion of
Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
to a dwarf planet on 24 August 2006, in our
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
, eight of the bodies orbiting the Sun are considered to be
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s.


Chemistry

* 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
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
. * The most stable allotrope of a
sulfur Sulfur (or sulphur in British English) is a chemical element with the symbol S and atomic number 16. It is abundant, multivalent and nonmetallic. Under normal conditions, sulfur atoms form cyclic octatomic molecules with a chemical formula ...
molecule is made of eight sulfur atoms arranged in a rhombic form. * The maximum number of electrons that can occupy a valence shell (except for transitional elements), see the octet rule. * The red pigment lycopene consists of eight
isoprene Isoprene, or 2-methyl-1,3-butadiene, is a common volatile organic compound with the formula CH2=C(CH3)−CH=CH2. In its pure form it is a colorless volatile liquid. Isoprene is an unsaturated hydrocarbon. It is produced by many plants and animals ...
units.


Geology

* A
disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...
crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the
tetragonal crystal system In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square ...
has eight similar faces whose alternate interfacial angles only are equal.


Biology

* All
spider Spiders ( order Araneae) are air-breathing arthropods that have eight legs, chelicerae with fangs generally able to inject venom, and spinnerets that extrude silk. They are the largest order of arachnids and rank seventh in total species ...
s, and more generally all arachnids, have eight legs. Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes. * The
octopus An octopus ( : octopuses or octopodes, see below for variants) is a soft-bodied, eight- limbed mollusc of the order Octopoda (, ). The order consists of some 300 species and is grouped within the class Cephalopoda with squids, cuttle ...
and its cephalopod relatives in genus ''Argonauta'' have eight arms (tentacles). * Compound coelenterates of the subclass or order
Alcyonaria Octocorallia (also known as Alcyonaria) is a class of Anthozoa comprising around 3,000 species of water-based organisms formed of colonial polyps with 8-fold symmetry. It includes the blue coral, soft corals, sea pens, and gorgonians (sea fans ...
have polyps with eight-branched tentacles and eight septa. * Sea anemones of genus ''
Edwardsia ''Edwardsia'' is a genus of sea anemones, the type of the family Edwardsiidae. They have eight mesenteries and live in tubes in the sand. The name, in New Latin, commemorates the French zoologist Henri Milne-Edwards. The genus contains the fo ...
'' have eight mesenteries. * Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia. * The eight-spotted forester (genus ''Alypia'', family Zygaenidae) is a diurnal moth having black wings with brilliant white spots. * The
ascus An ascus (; ) is the sexual spore-bearing cell produced in ascomycete fungi. Each ascus usually contains eight ascospores (or octad), produced by meiosis followed, in most species, by a mitotic cell division. However, asci in some genera or s ...
in fungi of the class
Ascomycetes Ascomycota is a phylum of the kingdom Fungi that, together with the Basidiomycota, forms the subkingdom Dikarya. Its members are commonly known as the sac fungi or ascomycetes. It is the largest phylum of Fungi, with over 64,000 species. The defi ...
, following nuclear fusion, bears within it typically eight ascospores. * Herbs of genus '' Coreopsis'' (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each. * In human adult dentition there are eight teeth in each quadrant. The eighth tooth is the so-called
wisdom tooth A third molar, commonly called wisdom tooth, is one of the three molars per quadrant of the human dentition. It is the most posterior of the three. The age at which wisdom teeth come through ( erupt) is variable, but this generally occurs betwe ...
. * There are eight cervical nerves on each side in man and most mammals.


Psychology

* There are eight
Jungian cognitive functions Cognitive functions, also referred to as psychological functions, as described by Carl Jung in his book ''Psychological Types'', are particular mental processes within a person's psyche that are present regardless of common circumstance. This is a ...
, according to the MBTI models by
John Beebe John Beebe (born June 24, 1939) is an American psychiatrist and Jungian analyst in practice in San Francisco. Beebe was born in Washington, D.C. He received degrees from Harvard College and the University of Chicago medical school. He is a past ...
and
Linda Berens Linda may refer to: As a name * Linda (given name), a female given name (including a list of people and fictional characters so named) * Linda (singer) (born 1977), stage name of Svetlana Geiman, a Russian singer * Anita Linda (born Alice Lake i ...
. * Timothy Leary identified a hierarchy of eight levels of consciousness.


In technology

* A
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
is eight bits. * Many (mostly historic) computer architectures are eight-bit, among them the
Nintendo Entertainment System The Nintendo Entertainment System (NES) is an 8-bit third-generation home video game console produced by Nintendo. It was first released in Japan in 1983 as the commonly known as the The NES, a redesigned version, was released in America ...
. * Standard-8 and
Super-8 Super 8 mm film is a motion-picture film format released in 1965 by Eastman Kodak as an improvement over the older "Double" or "Regular" 8 mm home movie format. The film is nominally 8 mm wide, the same as older formatted 8& ...
are 8 mm
film formats A film format is a technical definition of a set of standard characteristics regarding image capture on photographic film for still images or film stock for filmmaking. It can also apply to projected film, either slides or movies. The primary ch ...
. * Video8, Hi8 and Digital8 are related
8 mm video format The 8mm video format refers informally to three related videocassette formats. These are the original Video8 (analog recording) format and its improved successor Hi8 (analog video and analog audio but with provision for digital audio), as well as ...
s. * On most phones, the 8 key is associated with the letters T, U, and V, but on the BlackBerry Pearl it is the key for B and N. * An eight may refer to an eight-cylinder engine or automobile. A V8 engine is an internal combustion engine with eight cylinders configured in two banks (rows) of four forming a "V" when seen from the end. * A figure-eight knot (so named for its configuration) is a kind of stopper knot. * The number eight written in parentheses is the code for the musical note in Windows Live Messenger. * In a seven-segment display, when an 8 is illuminated, all the display bulbs are on.


In measurement

* The SI prefix for 10008 is yotta (Y), and for its reciprocal, yocto (y). * In liquid measurement (United States customary units), there are eight fluid ounces in a Cup (volume), cup, eight pints in a gallon and eight tablespoonfuls in a gill (volume), gill. * There are eight furlongs in a mile. * The clove, an old English units#Avoirdupois, English unit of weight, was equal to eight pounds when measuring cheese. * An eight may be an article of clothing of the eighth clothing sizes, size. * Force eight is the first wind strength attributed to a gale on the Beaufort scale when announced on a Shipping Forecast.


In culture


Currency

* Sailors and civilians alike from the 1500s onward referred to evenly divided parts of the Spanish dollar as "pieces of eight", or "bits".


Architecture

* Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos and multi-roomed pagodas (descended from stupas; see religion section below). * Eight Glossary of architecture#C, caulicoles rise out of the leafage in a Corinthian order, Corinthian capital, ending in leaves that support the volutes.


In religion, folk belief and divination


Hinduism

* As sourced from the Mahabharata, there are 8 vasus who are given elemental names: *# Anala or Agni (fire) *# Dhara or Prithvi (earth) *# Anila or Vayudeva (wind) *# Apa (water) *# Prabhasa or Dyauh (sky) *# Pratyusha *# Soma *# Dhruva * The goddess of wealth and prosperity, Lakshmi, has eight forms known as Ashta Lakshmi and worshipped as:
"''Maha-lakshmi, Dhana-lakshmi, Dhanya-lakshmi, Gaja-lakshmi,
Santana-lakshmi, Veera-lakshmi, Vijaya-lakshmi and Vidhya-lakshmi''" *There are eight ''nidhi'', or seats of wealth, according to Hinduism. *There are eight guardians of the directions known as ''Astha-dikpalas''. *There are eight Hindu monasteries established by the saint Madhvacharya in Udupi, India popularly known as the ''Ashta Mathas of Udupi''.


Buddhism

* The Dharmacakra, a Buddhism, Buddhist symbol, has eight spokes. The Buddha's principal teaching—the Four Noble Truths—ramifies as the Noble Eightfold Path and the Buddha emphasizes the importance of the eight attainments or jhanas. * In Mahayana Buddhism, the branches of the Eightfold Path are embodied by the Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra (Bodhisattva), Samantabhadra). These are later (controversially) associated with the Eight Consciousnesses according to the Yogacara school of thought: consciousness in the five senses, thought-consciousness, self-consciousness, and unconsciousness-"consciousness" or "store-house consciousness" (alaya-vijñana). The "irreversible" state of enlightenment, at which point a Bodhisattva goes on "autopilot", is the Eight Ground or ''bhūmi''. In general, "eight" seems to be an auspicious number for Buddhists, e.g., the "eight auspicious symbols" (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the self-replenishing amphora; the white ''kamala'' lotus-flower; the white conch; the eternal (Celtic-style, infinitely looping) knot; the banner of imperial victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). Similarly, Buddha's Birthday, Buddha's birthday falls on the 8th day of the 4th month of the Chinese calendar.


Judaism

* The religious rite of brit milah (commonly known as circumcision) is held on a baby boy's eighth day of life. * Hanukkah is an eight-day Jewish holiday that starts on the 25th day of Kislev. * Shemini Atzeret (Hebrew language, Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately following the seven-day holiday of Sukkot.


Christianity

* The spiritual The eighth day (Christian), Eighth Day, because the number 7 refers to the days of the week (which repeat themselves). * The number of Beatitudes. * wikisource:Bible (King James)/1 Peter#3:20, 1 Peter 3:20 states that there were eight people on Noah's Ark. * The Antichrist is the eighth king in the Book of Revelation.


Islam

* In Islam, eight is the number of angels carrying the Throne of God, throne of Allah in heaven. * The number of gates of heaven. * The octagram ''Rub el Hizb'' is often used in Islamic symbology.


Taoism

* ''Bagua'' * ''Ba Xian'' * ''Baduanjin''


Other

* In Wicca, there are eight Sabbats, festivals, seasons, or spokes in the Wheel of the Year. * In Ancient Egyptian mythology, the Ogdoad (Egyptian), Ogdoad represents the Ancient Egyptian creation myths, eight primordial deities of creation. * In Scientology there are Scientology beliefs and practices#The eight dynamics, eight dynamics of existence. * There is also the Ogdoad (Gnosticism), Ogdoad in Gnosticism.


As a lucky number

* The number eight is considered to be a Numbers in Chinese culture, lucky number in Chinese and other Asian cultures. Eight (; Chinese numerals#Numeral characters, accounting ; pinyin ''bā'') is considered a Numbers in Chinese culture#Eight, lucky number in Chinese culture because it sounds like the word meaning to generate wealth (; Pinyin: ''fā''). Property with the number 8 may be valued greatly by Chinese. For example, a Hong Kong Vehicle registration plate, number plate with the number 8 was sold for $640,000. The opening ceremony of the 2008 Summer Olympics, Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008. * In Pythagorean numerology the number 8 represents victory, prosperity and overcoming. * is also considered a lucky number in Japan, but the reason is different from that in Chinese culture. Eight gives an idea of growing prosperous, because the letter () broadens gradually. * The Japanese thought of as a holy number in the ancient times. The reason is less well-understood, but it is thought that it is related to the fact they used eight to express large numbers vaguely such as (literally, eightfold and twentyfold), (literally, eight clouds), (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as , which is also guessed to be a holy number in those times because it indicates the world (north, south, east, and west) might be considered a very holy number. * In numerology, 8 is the number of building, and in some theories, also the number of destruction.


In astrology

* In astrology, Scorpius, Scorpio is the 8th astrological sign of the Zodiac. * In the Middle Ages, 8 was the number of "unmoving" stars in the sky, and symbolized the perfection of incoming planetary energy.


In music and dance

* A note played for one-eighth the duration of a whole note is called an eighth note, or quaver. * An octave, the interval between two musical notes with the same letter name (where one has double the frequency of the other), is so called because there are eight notes between the two on a standard major or minor diatonic scale, including the notes themselves and without chromatic deviation. The ecclesiastical musical mode, modes are ascending diatonic musical scales of eight notes or tones comprising an octave. * There are eight notes in the octatonic scale. * There are eight musicians in a double quartet or an octet (music), octet. Both terms may also refer to a musical composition for eight voices or instruments. * Caledonians is a square dance for eight, resembling the quadrille. * Albums with the number eight in their title include ''8'' by the Swedish band Arvingarna, ''8 (Incubus album), 8'' by the American rock band Incubus (band), Incubus, ''The Meaning of 8'' by Minnesota indie rock band Cloud Cult and ''8ight'' by Anglo-American singer-songwriter Beatie Wolfe. * Dream Theater's eighth album ''Octavarium (album), Octavarium'' contains many different references to the number 8, including the number of songs and various aspects of the music and cover artwork. * "Eight maids a-milking" is the gift on the eighth day of Christmas in the carol "The Twelve Days of Christmas (song), The Twelve Days of Christmas". * The 8-track tape, 8-track cartridge is a musical recording format. * "#8" is the stage name of Slipknot (band), Slipknot vocalist Corey Taylor. * "Too Many Eights" is a song by Athens, Georgia's Supercluster (band), Supercluster. * Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song "Kiss You (When It's Dangerous)". * "Eight Days a Week (song), Eight Days a Week" is a #1 single for the music group the Beatles. * Figure 8 (album), ''Figure 8'' is the fifth studio album by singer-songwriter Elliott Smith, released in the year 2000, an album released by Julia Darling in 1999, and an album released by Outasight in 2011. * Ming Hao from the k-pop group Seventeen (South Korean band)#Performance team, Seventeen goes by the name "The8". * "8 (circle)" is the eighth song on the album ''22, A Million'' by the American band Bon Iver. * "8" is the eighth song on the album ''When We All Fall Asleep, Where Do We Go?'' by Billie Eilish.


In film and television

* ''8 Guys'' is a 2003 short film written and directed by Dane Cook. * ''8 Man'' (or ''Eightman''): 1963 Japanese manga and anime superhero. * 8 Mile (film), ''8 Mile'' is a 2002 film directed by Curtis Hanson. * 8mm (film), ''8 mm'' is a 1999 film directed by Joel Schumacher. * ''8 Women'' (Original French title: ) is a 2001 film directed by François Ozon. * ''Eight Below'' is a 2006 film directed by Frank Marshall (film producer), Frank Marshall. * ''Eight Legged Freaks'' is a 2002 film directed by Ellory Elkayem. * ''Eight Men Out'' is a 1988 film directed by John Sayles. * ''Jennifer Eight'', also known as ''Jennifer 8'', is a 1992 film written and directed by Bruce Robinson. * ''Eight Is Enough'' is an American television comedy-drama series. * In ''Stargate SG-1'' and ''Stargate Atlantis'', dialing an 8-chevron address will open a wormhole to another galaxy. * ''The Hateful Eight'' is a 2015 American western mystery film written and directed by Quentin Tarantino. * ''Kate Plus 8'' is an American reality television show. * ''Ocean's 8'' is an American heist comedy film directed by Gary Ross.


In sports and other games

* Eight-ball pool (cue sports), pool is played with a cue ball and 15 numbered balls, the black ball numbered 8 being the middle and most important one, as the winner is the player or side that legally pockets it after first pocketing its numerical group of 7 object balls (for other meanings see ''Eight ball (disambiguation)''). * In chess, each side has eight pawns and the board is made of 64 squares arranged in an eight by eight lattice. The eight queens puzzle is a challenge to arrange eight queens on the board so that none can capture any of the others. * In the game of eights or Crazy Eights, each successive player must play a card either of the same suit or of the same rank as that played by the preceding player, or may play an eight and call for any suit. The object is to get rid of all one's cards first. * In association football, the number 8 has historically been the number of the Central Midfielder. * In Australian rules football, the top eight teams at the end of the Australian Football League regular season qualify for the AFL finals series, finals series (i.e. playoffs). * In baseball: ** The center fielder is designated as number 8 for scorekeeping purposes. ** The College World Series, Men's College World Series, the final phase of the NCAA Division I Baseball Championship, NCAA Division I tournament, features eight teams. * In rugby union, the only position without a proper name is the Number eight (rugby union), Number 8, a forward position. * In rugby league: ** Most competitions (though not the Super League, which uses static squad numbering) use a position-based player numbering system in which one of the two starting props wears the number 8. ** The Australia-based National Rugby League has its own 8-team finals series, similar but not identical in structure to that of the Australian Football League. * In Rowing (sport), rowing, an "eight" refers to a sweep-oar racing boat with a crew of eight rowers plus a coxswain. * In the 2008 Summer Olympics, 2008 Games of the XXIX Olympiad held in Beijing, the official opening was on 08/08/08 at 8:08:08 p.m. China Standard Time, CST. * In rock climbing, climbers frequently use the figure-eight knot to tie into their harnesses. * The Women's College World Series, the final phase of the NCAA Division I softball tournament, like its men's counterpart in baseball, features eight teams. * In curling an 8-point 'Eight Ender' is a perfect end. Each team delivers 8 Stones per end.


In foods

* Nestlé sells a brand of chocolates filled with peppermint-flavoured cream called After Eight, referring to the time 8 p.m. * There are eight vegetables in V8 (beverage), V8 juice.


In literature

* Eights may refer to octosyllable, octosyllabic, usually Choliamb, iambic, Meter (poetry), lines of verse. * The drott-kvaett, an Old Icelandic verse, consisted of a stanza of eight regular lines. * In Terry Pratchett's ''Discworld'' series, eight is a magical number and is considered taboo. Eight is not safe to be said by wizards on the Discworld (world), Discworld and is the number of Bel-Shamharoth. Also, there are eight days in a Disc week and eight colours in a Disc spectrum, the eighth one being octarine. * Lewis Carroll's poem ''The Hunting of the Snark'' has 8 "fits" (cantos), which is noted in the full name "The Hunting of the Snark – ''An Agony, in Eight Fits''". * Eight apparitions appear to Macbeth (character), Macbeth in Act 4 scene 1 of Shakespeare's ''Macbeth'' as representations of the eight descendants of Banquo.


In slang

* An "eighth" is a common measurement of cannabis (drug), marijuana, meaning an eighth of an ounce. It is also a common unit of sale for psilocybin mushrooms. * Avril Lavigne's song "Sk8er Boi" uses this convention in the title. * The Section 8 (housing), Housing Choice Voucher Program, operated by the United States Department of Housing and Urban Development, is commonly referred to as the Section 8 program, as this was the original section of the Act which instituted the program. * In Colombia and Venezuela, "volverse un ocho" (meaning to tie oneself in a figure 8) refers to getting in trouble or contradicting oneself. * In China, "8" is used in chat speak as a term for parting. This is due to the closeness in pronunciation of "8" (bā) and the English word "bye".


Other uses

* A figure 8 is the common name of a geometry, geometric shape, often used in the context of sports, such as skating. Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.


See also

* The Magical Number Seven, Plus or Minus Two *List of highways numbered 8


References


External links


The Octonions
John C. Baez {{DEFAULTSORT:8 (Number) Integers 8 (number)