# John Horton Conway

Jump to navigation
Jump to search

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

This article about a mathematician is a stub. You can help Wikiquote by expanding it. |

## Quotes

[edit]- 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.
- Mark Ronan (18 May 2006).
*Symmetry and the Monster: One of the greatest quests of mathematics*. Oxford University Press, UK. pp. 163. ISBN 978-0-19-157938-7.

- Mark Ronan (18 May 2006).

- ... 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."
- Life, Death and the Monster - Numberphile. YouTube (9 May 2014).

*Sphere Packings, Lattices and Groups* (1988)

[edit]- 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 ...book 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]

- Ref: John H. Conway,

#### 1 Sphere Packing and Kissing Numbers

[edit]- 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.

- 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. ...

- [T]he best packings in up to eight dimensions belong to families and , and the corresponding Coxeter–Dynkin diagrams turn up in apparently unrelated areas... [I]n 24 dimensions the Leech lattice has... connections with hyperbolic geometry, Lie algebras, and the Monster simple group... [O]ne day someone will write an article on "The Ubiquity of the Leech lattice." ...There are applications of... packings to
*number theory*... [e.g.,] solving Diophantine equations, and to "the geometry of numbers"... There are... applications of sphere packings... in*digital communications*... a typical question from... spread-spectrum communications for mobile radio... how many spheres of radius 0.25 can be packed in a sphere of radius 1 in 100-dimensional space? ...Two and three-d... packings... circles in a two-d... packing may represent optical fibers... in... a cable. Three-d... packings have applications in*chemistry*and*physics*... biology... antenna design... choosing directions for X-ray*tomography*... and... statistical analysis on spheres... n-dimensional packings may be used in... numerical evaluation of integrals... on the surface of a sphere in or in its interior. ...A related application ...n-dimensional*search*or*approximation*problems ...[I]n physics... dual theory and superstring theory... have involved the and lattices and the related Lorentzian lattices in dimensions 10 and 26...

- The densest possible packings are known for . ...... represents a laminated lattice... the densest lattices in up to 8 dimensions...

## Quotes about John Horton Conway

[edit]- He is Archimedes, Mick Jagger, Salvador Dalí, and Richard Feynman, all rolled into one. He is one of the greatest living mathematicians, with a sly sense of humour, a polymath’s promiscuous curiosity, and a compulsion to explain everything about the world to everyone in it.
- John Horton Conway: the world’s most charismatic mathematician. The Guardian (23 July 2015).