Why do we care about finding dense packing in n-dimensional space? ... — John Horton Conway

"We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers..."

"The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space \R^n is... a string of real numbersx = (x_1,x_2,x_3, ...,x_n).A sphere in \R^n with center u = (u_1,u_2,u_3, ...,u_n) and radius \rho consists of all points x... satisfying (x_1-u_1)^2 + (x_2-u_2)^2+ ... +(x_n-u_n)^2 = \rho^2. We can describe a sphere packing in \R^n... by specifying the centers u and the radius."

"Im going to present arguments... to strongly support... that we do... have free will, but not... to prove it at the deductive level."

"[[w:Sphere packing|[L]attice packing]]... has the properties that 0 is a center and... if there are spheres with centers u and v then there are spheres with centers u + v and u - v... [i.e.,] the sets of centers forms an . In crystallography these... are... called s... We can find... in general n centers v_1,v_2, ...,v_n for an n-dimensional lattice... such that the set of all centers consists of the sums \sum k_i v_i where k_i are s."

"There has been a great deal of nonsense written... about the mysterious fourth dimension. ...4-dimensional space just consists of points with four coordinates instead of three (...similarly for any number of dimensions). ...[I]magine a telegraph ...over which numbers are ...sent in sets of four. Each set... is a point in 4-d... space."

"In this chapter we discuss the problem of packing spheres in and of packing points on the surface of a sphere. The problem is an important special case of the latter, and asks how many spheres can just touch another sphere of the same size."