John Horton Conway

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

"When I was on the train from Liverpool to Cambridge to become a student, it occurred to me that no one at Cambridge knew I was painfully shy, so I could become an extrovert instead of an introvert."

"The classical... problem is... how densely a large number of identical spheres ([e.g.,] ball bearings...) can be packed together. ...[C]onsider an aircraft hangar... [A]bout one quarter of the space will not be used... One... arrangement... the face-centered cubic (or fcc) lattice... spheres occupy \pi / \sqrt{18} = .7405... of the total space.... the lattice packing has density .7405... . [H]pwever, there are partial packings that are denser than the face-centered cubic... over larger regions..."

"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."

"Lucretius... was an atomist, a follower of Epicurus. The original people who invented the atomic theory were and Democritus. ...Lucretius is discussing ...atoms ...he says, "at quite indeterminate times and places they swerve" ...because it allows for human free will... and "if the atoms never swerve... what is the source of the free will possessed by living things throughout the earth?" He says, "Although many men are driven by an external source, and often constrained involuntarily to advance or rush headlong, yet there is in the human breast something that can fight against it and resist it... So also in the atoms you must recognize the same possibility. Besides weight and impact, there must be a third cause of movement, the source of this inborn power... due to the slight swerve of the atoms... since nothing can come out of nothing." And then he goes on to say, "the fact that the mind itself has no internal necessity to determine its every act, this is due to the slight swerve of the atoms at no determinate time and place.""

"Let me phrase the free will theorem that Simon and I proved. ...[I]f we... have free will... then so do elementary particles have their... very small quantity of free will... to mean, our behavior is not a function of the past. ...[I]f some experimenters have free will ...then so do elementary particles... even the ones outside us..."

"Descartes... his type of determinism is only partial. I call it disconnected... that the physical world and... our bodies, but not our minds operate mechanically... Were robots, but the mind is different... that "The will is by its nature so free that it can never be constrained." This is normally called Descartes dualism... mind, soul or spirit, and matter... [M]atter operates according to one set of laws, and mind or spirit... to another."

"Hed been working with... Jim Ax for two decades on a theory that is now contradicted by... [our] . ...I admire Simon tremendously."

"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."