[S]uppose I had a twin brother... [M]y sisters would have had a much better chance... [T]hey could... interrogate us separately... Then we can't chan… - John Horton Conway

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

... I have said for twenty-five or thirty years that the one thing I would really like to know before I die is why the monster group exists."

[T]he strangest contribution of quantum mechanics to this discussion is the EPR paradox. ...That's an essential contribution to our theorem too. ...Despite the fact that information can't be transmitted faster than the speed of light, ...remotely separated events can be correlated ...and this is the content of our TWIN axiom, you can put two particles into a... singleton state... the angular momentum of the pair of particles is zero... [B]y the conservation of angular momentum... if you measure the angular momentum of this in any direction, then for the angular momentum of the other you get the negative answer, but... we're going to square it, that means... the squared component of spin is the same... [T]hese particles have been sort of hypnotized. If you ask... they will give the same answer... like I and my twin brother... [T]he funny thing is, even though the proves that the answers do not exist ahead of time, the equality of the answers can exist...