From: Uncle Al <UncleAl0@hate.spam.net>
Subject: Re: World Crystal Point Groups?
Date: Tue, 20 Nov 2001 16:23:18 GMT
Newsgroups: sci.physics,sci.physics.relativity,sci.physics.particle,sci.math,sci.skeptic
Summary: Enumerating crystallographic groups in  R^4

Jack Sarfatti wrote:
> 
> What are the 4d world crystal classes analogous to the 3d crystal
> space groups? i.e. 32 crystallographic point groups in 3D generalize
> to what in 4d Minkowski spacetime with lattice spacing Lp?
> 
> 5 cubic
> 7 tetragonal
> 7 hexagonal
> 5 trigonal
> 3 orthorhombic
> 3 monoclinic
> 3 triclinic
> 
> what in 4d?
[snip]

Your list adds up to 33 point groups.  It is wrong.  And you forgot
Bravais lattices (Acta Cryst. A47 611 (1991)).

  32 three-dimensional crystallographic point groups,
 230 three-dimensional space groups;
 227 four-dimensional crystallographic point groups,
4895 four-dimensional space groups.
Acta Cryst. A43 294 (1987)

Acta Cryst. A51 211 (1995)

H Brown, R Bülow, J Neubüser, H Wondratschek, H Zassenhaus,
"Crystallographic Groups of Four-Dimensional Space"
Wiley-Interscience, (1978) XIV + 443 p. 

Gött. Nachr. 75 (1910)
Math. Ann. 70 297 (1911)
Math. Ann. 72 400 (1912)
Z. Kristallogr. 73 325 (1930)
Comment. Math. Helv. 21 117 (1948)
"Mathematical Developments Arising from Hilbert Problems"  Felix
Browder Ed., American Mathematical Society, 491-506 (1976) 
R.L.E. Schwarzenberger, "N-Dimensional Crystallography" Pitman,
London, 1980

Hey, Jackoff, how many of those 4895 4-D space groups are chiral?

-- 
Uncle Al
http://www.mazepath.com/uncleal/
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[reformatted --djr]