of unit octonions. Principal -bundle over . Parallelizable. . The 7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered. 8-sphere Equivalent to the octonionic projective line . 23-sphere A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice. Octahedral sphere The octahedral -sphere is defined similarly to the -sphere but using the 1-norm In general, it takes the shape of a cross-polytope. The octahedral 1-sphere is a square (without its interior). The octahedral 2-sphere is a regular octahedron; hence the name. The octahedral -sphere is the topological join of pairs of isolated points. Intuitively, the topological join of two pairs is generated by drawing a segment between each point in one pair and each point in the other pair; this yields a square. To join this with a third pair, draw a segment between each point on the square and each point in the third pair; this gives a octahedron. See also Affine sphere Conformal geometry Exotic sphere Homology sphere Homotopy groups of spheres Homotopy sphere Hyperbolic group Hypercube Inversive geometry Loop (topology) Manifold Möbius transformation Orthogonal group Spherical cap Volume of an -ball Wigner semicircle distribution Notes References External links Multi-dimensional geometry Spheres