John Horton Conway

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John Horton Conway in 2005

John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician, and Professor Emeritus of Mathematics at Princeton University in New Jersey.



Sphere Packings, Lattices and Groups (1988)

by J.H. Conway & N.J.A. Sloane (3rd edition, 1993)
  • [I]n two dimensions the... [19 point] hexagonal lattice solves the packing, kissing, covering and quantizing problems. ...[T]his is ...a search for similar nice patterns in higher dimensions.
    • Preface to 1st edition
  • We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers...
    • Ref: John H. Conway, Complex and integral laminated lattices, TAMS 280 (1983) 463-490 [2,6,22]; John H. Conway, The Coxeter–Todd lattice, The Mitchell group, and related sphere packings PCPS 93 (1983) 421-440 [2,4,7,8,22]

1 Sphere Packing and Kissing Numbers

  • In this chapter we discuss the problem of packing spheres in Euclidean space and of packing points on the surface of a sphere. The kissing number problem is an important special case of the latter, and asks how many spheres can just touch another sphere of the same size.
Close-packed spheres
  • 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 of the total space.... the lattice packing has density . [H]pwever, there are partial packings that are denser than the face-centered cubic... over larger regions...
  • The classical... problem... asks: is this the greatest density..? an unsolved problem, one of the most famous...
  • The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space is... a string of real numbers
    A sphere in with center and radius consists of all points ... satisfying
    We can describe a sphere packing in ... by specifying the centers and the radius.
  • 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.
  • [L]attice packing... has the properties that 0 is a center and... if there are spheres with centers and then there are spheres with centers and ... [i.e.,] the sets of centers forms an additive group. In crystallography these... are... called Bravais lattices... We can find... in general centers for an n-dimensional lattice... such that the set of all centers consists of the sums where are integers.
  • Why do we care about finding dense packing in n-dimensional space? ...This is an interesting problem in pure geometry. Hilbert mentioned it in 1900 in his list open problems... [T[he best packings... have connections... with other branches of mathematics. ...
  • The densest possible packings are known for . ...... represents a laminated lattice... the densest lattices in up to 8 dimensions...

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