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] The sides and span of a regular octagon, or truncated square, are in 1 : 1 + √2 silver ratio, and its circumscribing square has a side and diagonal length ratio of 1 : √2; 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.^[4][5] With a central angle of 45 degrees and an internal angle angle of 135 degrees, regular octagonas 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.^[6] 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.^[7] In number theory, figurate numbers representing octagons are called octagonal numbers.^[8]

A cube is a regular polyhedron with 8 vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space.^[9] Through various truncation operations, the cubic honeycomb generates 8 other convex uniform honeycombs under the group .^[10] The octahedron, with 8 equilateral triangles as faces, is the dual polyhedron to the cube and one of 8 convex deltahedra.^[11][12] The cuboctahedron${\displaystyle {\tilde {C}}_{3}}$ is one of only two convex quasiregular polyhedra, with 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.^[13][14]

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.^[16] 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.