From: baez@math.ucr.edu (john baez)
Newsgroups: sci.physics,rec.arts.sf.science,sci.math
Subject: Re: The Life of the Cosmos - Lee Smolin
Date: 26 Feb 1998 00:35:49 -0800
In article <6d2tfa$65n@enews2.newsguy.com>,
Tommy the Terrorist wrote:
>I haven't figured out if any dodecahedron-based shapes go into 5 or more
>dimensions - if anyone has an idea, tell me.
No, they don't. Here's a complete list of convex regular
polytopes in all dimensions - these being the simplest
generalization of the concept of regular polygon or Platonic
solid to arbitrary dimensions:
1) n-dimensional hypercubes (generalizing the cube)
2) n-dimensional cross-polytopes (generalizing the octahedron)
2) n-dimensional simplices (generalizing the tetrahedron)
4) in 2 dimensions: the regular n-gons for all n > 4
5) in 3 dimensions: the dodecahedron and icosahedron
6) in 4 dimensions: the 24-cell, the 120-cell and the 600-cell
The 24-cell has 24 octahedral faces. If you cross your
eyes while looking at the stereographic pair of images on
Tony Smith's web page:
http://galaxy.cau.edu/tsmith/24anime.html
you will see a 24-cell rotating in 4 dimensions, with the
4th dimension depicted using color.
Since the 24-cell lives in 4 dimensions, we can think of
its vertices as certain unit quaternions. We can take
them to be precisely the unit "Hurwitz integral quaternions",
which are quaternions of the form
a + bi + cj + dk
where a,b,c,d are either all integers or all integers + 1/2.
One can check that the Hurwitz integral quaternions are closed
under multiplication, so the vertices of the 24-cell form a
subgroup of the unit quaternions. A Platonic solid that's
a symmetry group in its own right - ponder *that* while you
cross your eyes and gaze at it spinning around!
The 120-cell has 120 dodecahedral faces; one may think of it
as a kind of hyperdodecahedron. I don't know if there's
any way to think of its vertices as a subgroup of the
unit quaternions. However, its vertices correspond to
the faces of the 600-cell, and vice versa.
The 600-cell has 600 tetrahedral faces; one may think of it
as a kind of hypericosahedron. Since it lives in 4
dimensions we can think of its vertices as certain unit
quaternions. We can take them to be precisely the unit
"icosians". These are quaternions of the form
a + bi + cj + dk
where a,b,c,d all live in the "golden field" - meaning
that they're of the form x + sqrt(5)y where x and y are
rational. Since the icosians are closed under multiplication
a group under multiplication, the vertices of the 120-cell
also form a group!
You can also see stereographic images of the 120-cell and
the 600-cell on Tony Smith's web page. Tony has a theory
of physics where spacetime is a discrete "hyperdiamond lattice",
each cell of which is a 24-cell. David Finkelstein also
has such a theory. You can think of the hyperdiamond lattice,
technically known as D4, as the 4-dimensional version of the
black squares on a checkerboard.
For more fun, try:
Regular polytopes, H. S. M. Coxeter, 3d ed. New York, Dover
Publications, 1973.
Sphere Packings, Lattices and Groups, J. H. Conway and N. J. A. Sloane,
second edition, Grundlehren der mathematischen Wissenschaften 290,
Springer-Verlag, 1993.
I don't know if Gordon already discussed it, but there's a chapter
in Smolin's book called "The Flower and the Dodecahedron", comparing
the beauty of natural forms arising through evolution, and that of
mathematical forms arising out of logical necessity. He wants to
know: which is a better picture of the laws of nature? Theoretical
physicists have traditionally sided with the dodecahedron; Smolin is
arguing for the flower.
I like flowers *and* dodecahedra! Somehow our universe has room for
both.
==============================================================================
From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Newsgroups: sci.physics,rec.arts.sf.science,sci.math
Subject: Re: Dodecahedra **** SPOILER ****
Date: 16 Mar 1998 05:25:36 GMT
lew@ihgp167e.ih.lucent.com (-Mammel,L.H.) writes:
[mentally constructing a 120-cell 4D-hedron of dodecahedra]
|> Each cell center ( the vertices of a 600-cell ) has one of 9
|> distances from the 120 cell centers, including itself, with the counts:
|>
|> 1 0
|> 12 1
|> 20 2
|> 12 3
|> 30 4
|> 12 5
|> 20 6
|> 12 7
|> 1 8
|>
|> The odd distances comprise 12 unique bridges from one pole to
|> the other. If you picture yourself IN the ( surface of ) a
|> 120-cell ( i.e. in one of the cells ) then you can look at the
|> opposite cell from any one of 12 different directions, straight
|> through the perpendicular faces of the 4 intervening cells.
Very nice!
|> The 2nd nearest neighbors are the 5 nn's of the 12 nn's, counted
|> 3 times each.
Giving 20 of them - very neat. Exactly the sort of thing I was thinking of.
Some similar counting below.
|> Since 8,7,6,5 are the complement of 0,1,2,3, that leaves the 4th nn's.
Yep; that's the 4-equatorial symmetry I alluded to of course.
Here's a similar way to get the middle number of your table. Look at
the equatorial joining, where each dodec in your 3rd layer, faces onto
its image dodec in the 5th layer. These two have 5 edges around their
face join, (you better start sketching this - no ascii, sorry!)
Each edge has two pentagonal faces coming away from it, say one up, one
down, each being a face of a 3rd & 5th layer dodec. But they are also
(necessarily) two adjacent faces of an equatorial dodec (in your 4th layer).
Now these two pentagons are part of a thick-&-thin equatorial band around
this equatorial dodec. They point away from the equator, toward the poles
(of the dodec). The are adjacent to two other more fully equatorial
pentagons; and if you follow around the dodec's equator, (you really
need a dodec model here!), you see these 4 pentagons make up exactly
half of the dodec's equator. The other half is another strip of the same
sort, back-to-backing onto our first one. So there is *one more* pair
of up-&-down pentagons on this.
What have we got then? Each equatorial dodec comes form TWO adjacencies
of 3rd-&-5th layer dodecs. So now we can finally count them. There
are 12 3rd-layer dodecs, each with 5 edges available for equatorial
dodecs. That's 60; but each equatorial dodec has been counted twice,
by the paragraph above, so there's 30 in all!
I hope you were doing the sketches... :)
-------------------------------------------------------------------------------
Bill Taylor W.Taylor@math.canterbury.ac.nz
-------------------------------------------------------------------------------
I do not claim my argument is logical, but simply that I'm right.
-------------------------------------------------------------------------------