8th Symphony Of Mahler, Kölner Philharmonie, 27-6-2009

8 (eight) is the natural number following 7 and preceding 9.

In mathematics

8 is:
* a composite number, its proper divisors being , , and . It is twice 4 or four times 2.
* a power of two, being 2 (two cubed), and is the first number of the form , being an integer greater than 1.
* the first number which is neither prime nor semiprime.
* the base of the octal number system, which is mostly used with computers. In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet.
* a Fibonacci number, being plus . The next Fibonacci number is . 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube.
* the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.
* the order of the smallest non-abelian group all of whose subgroups are normal.
* the dimension of the octonions and is the highest possible dimension of a normed division algebra.
* the first number to be the aliquot sum of two numbers other than itself; the discrete biprime , and the square number .

A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary.

A polygon with eight sides is an octagon. The sides and span of a regular octagon, or truncated square, are in silver ratio, 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. With a central angle of 45 degrees and an internal angle of 135 degrees, regular octagons are able to tessellate two-dimensional space alongside squares in the truncated square tiling, as well as fill a plane-vertex with a regular triangle and a regular icositetragon. The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal ''silver'' eightfold symmetry, and is the two-dimensional orthogonal projection of the 8-8 duoprism. In number theory, figurate numbers representing octagons are called octagonal numbers.

A cube 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 group $_3$. The octahedron, with eight equilateral triangles as faces, is the dual polyhedron to the cube and one of eight convex deltahedra. The stella octangula, or ''eight-pointed star'', is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that form a cubic faceting, composed of two self-dual tetrahedra that makes it the simplest of five regular compound polyhedra. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra. It contains eight equilateral triangular faces alongside six squares, whose first stellation is the cube-octahedron compound. The hexagonal prism, which classifies as an irregular octahedron that is a parallelohedron, like the cube, is able to tessellate space as a three-dimensional analogue of the hexagon. The gyrobifastigium, with four square faces and four triangular faces, is the only Johnson solid that is able to tessellate space, while the truncated octahedron, also a parallelohedron, is the permutohedron of order four, with eight hexagonal faces alongside six squares that is likewise the only Archimedean solid that can generate a honeycomb on its own.

Vertex-transitive semiregular polytopes whose facets are ''finite'' exist up through the 8th dimension. In the third dimension, 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 are semiregular polytopes. For dimensions five through eight, the demipenteract and the k₂₁ polytopes 2₂₁, 3₂₁, and 4₂₁ are the only semiregular (Gosset) polytopes. Collectively, the k₂₁ family of polytopes contains eight figures that are rooted in the triangular prism, 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 figures in the k₂₁ family. There are no other finite semiregular polytopes or honeycombs in dimensions ''n'' > 8.

Sphenic numbers always have exactly eight divisors.

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 algebras 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 are closely related to the properties of the octonions.

The spin group Spin(8) is the unique such group that exhibits the phenomenon of triality.

The lowest-dimensional even unimodular lattice is the 8-dimensional $\mathrm E_$ lattice. Even positive definite unimodular lattices exist only in dimensions divisible by 8.

A figure 8 is the common name of a 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.

List of basic calculations

Etymology

English ''eight'', from Old English ''eahta, æhta'', Proto-Germanic ''*ahto''
is a direct continuation of Proto-Indo-European '' *oḱtṓ(w)-'', and as such cognate with Greek and Latin ''octo-'', 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''.
The adjective ''octuple'' (Latin ''octu-plus'') 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, written (Mandarin: ''bā''; Cantonese: ''baat''), is from Old Chinese ''*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 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 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 other than zero, originates with the Brahmi numerals.
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 ८); 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 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"). 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 typefaces, in typefaces with text figures 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 for "one thousand" CIƆ, or alternatively from the final Greek letter, ω.

In science

Physics

* In nuclear physics, the second magic number.
* In particle physics, the eightfold way is used to classify sub-atomic particles.
* In statistical mechanics, the eight-vertex model has 8 possible configurations of arrows at each vertex.

Astronomy

* Messier object M8, a magnitude 5.0 nebula in the constellation of Sagittarius.
* The New General Cataloguebr>object
NGC 8, a double star in the constellation Pegasus.
* Since the demotion of Pluto to a dwarf planet on 24 August 2006, in our Solar System, eight of the bodies orbiting the Sun are considered to be planets.

Chemistry

* The atomic number of oxygen.
* The most stable allotrope of a sulfur molecule is made of eight sulfur atoms arranged in a rhombic form.
* The maximum number of electrons that can occupy a valence shell.
* The red pigment lycopene consists of eight isoprene units.

Geology

* A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.

Biology

* All spiders, and more generally all arachnids, have eight legs. Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.
* The octopus and its cephalopod relatives in genus ''Argonauta'' have eight arms (tentacles).
* Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.
* Sea anemones of genus '' Edwardsia'' 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 in fungi of the class Ascomycetes, 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.
* There are eight cervical nerves on each side in man and most mammals.

Psychology

* There are eight Jungian cognitive functions, according to the MBTI models by John Beebe and Linda Berens.
* Timothy Leary identified a hierarchy of eight levels of consciousness.

In technology

* A byte is eight bits.
* Many (mostly historic) computer architectures are eight-bit, among them the Nintendo Entertainment System.
* Standard-8 and Super-8 are 8 mm film formats.
* Video8, Hi8 and Digital8 are related 8 mm video formats.
* 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, eight pints in a gallon and eight tablespoonfuls in a gill.
* There are eight furlongs in a mile.
* The clove, an old English unit of weight, was equal to eight pounds when measuring cheese.
* An eight may be an article of clothing of the eighth size.
* Force eight is the first wind strength attributed to a gale

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