An odd mapping of this construct to Euclidean n-space maps a pair of points to the same point by the Borsuk-Ulam theorem. Even-dimensional ones have no non-vanishing continuous tangent vector field. The Möbius transformation is a stereographic projection from the plane to this construct and back. This shape is packed optimally in twenty-four dimensions by the Leech lattice and in three dimensions 
An odd mapping of this construct to Euclidean n-space maps a pair of points to the same point by the Borsuk-Ulam theorem. Even-dimensional ones have no non-vanishing continuous tangent vector field. The Möbius transformation is a stereographic projection from the plane to this construct and back. This shape is packed optimally in twenty-four dimensions by the Leech lattice and in three dimensions by a (*) hexagonal-close packed lattice. The sum of the angles of a triangle on one of these is greater than 180 degrees, and the variables theta, phi, and r represent the coordinate system named for them. For 10 points, name these figures whose volume is four thirds pi times radius cubed.
An odd mapping of this construct to Euclidean n-space maps a pair of points to the same point by the Borsuk-Ulam theorem.
n-spheres (and word forms)