8

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.^[3] A regular octagon (or truncated square) has a central angle of 45 degrees and an internal angle angle of 135 degrees, and is therefore able to tessellate two-dimensional space alongside squares in the truncated square tiling.^[4] The sides and span of a regular octagon are in 1 : 1 + √2silver ratio, and its circumscribingsquare has a side and diagonal length ratio of ${\displaystyle ^{\circ }}$${\displaystyle ^{\circ }}$ 1 : √2; with both the silver ratio and the square root of two intimately interconnected through Pell numbers.^[5] In particular, the quotient of successive Pell numbers generates rational approximations for coordinates of a regular octagon.^[6] The Ammann–Beenker tiling features prominent octagonal silver eightfold symmetry.^[7] In number theory, figurate numbers representing octagons are called octagonal numbers.^[8]

A cube is a regular polyhedron with 8 vertices that forms a cubic honeycomb in three-dimensional space.^[9] Through various truncation operations, the cubic honeycomb generates 7 other convex uniform honeycombs, altogether forming 8 honeycombs under the group .^[10] The dual polyhedron to the cube is the octahedron, with 8 equilateral triangles as faces.^[11] It is one of 8 convex deltahedra.^[12][13] The cuboctahedron${\displaystyle {\tilde {C}}_{3}}$, one of only two convex quasiregular polyhedra, has 8 equilateral triangles as faces (alongside 6 squares); it is equal to a rectified cube or rectified octahedron, and its first stellation is the cube-octahedron compound.^[14][15]

Vertex-transitive semiregular polytopes exist up through the 8^th 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. There are no other finite semiregular polytopes in dimensions n > 8.

The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, if O(∞) is the direct limit of the inclusions of real orthogonal groups

Clifford algebras also display a periodicity of 8.^[17] 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.