4-D and higher dimensional analogues of A065984
Jonathan Post
jvospost3 at gmail.com
Mon Mar 26 00:41:32 CEST 2007
I'm well into a paper on this, a draft of which I can email as a Word
file to anyone who asks. The key results are summarized in the table
at the beginning, which currently reads as shown below.
Sequence of Hypervolumes of Convex Hulls of
4-dimensional Euclidean Points Determined by a Sum of
Four Squares
Draft 3.2, 25 pages, of 23 March 2007
by Jonathan Vos Post
Computer Futures, Inc.
3225 N. Marengo Ave.
Altadena, CA 91001
jvospost3 at gmail.com
ABSTRACT:
What is the content (hypervolume) of the convex hull
of the vertices of P(n) (i.e. of the polytope), as a
function of n? P(n), for a nonnegative integer n, is
the convex hull of all vertices which have integer
coordinates in the Euclidean space Z^4, defined by all
permutations of: (+-h, +-i, +-j, +-k) such that h^2 +
i^2 + j^2 + k^2 = n.
This problem arises as a 4-dimensional analogue of a
3-dimensional problem which was posed and solved by
Wouter Meeussen in 2001. It touches on known results
from Coxeter, Conway, Sloane, and others. Appendices
detail specific example through
n = 32.
Partial results may be summarized in this table:
n vertices hypervolume comment
0 0 0 null polytope
1 8 8/3 hyperoctahedron {3,3,4}
2 24 8 24-cell {3,4,3}
3 32 < 144 rectified 8-cell?
4 24 32 24-cell {3,4,3}?
5 48 16=2a(2) elongated 24-cell
6 96 2a(3) elongated rectified 8-cell?
7 64 32 < V < 120 truncated 8-cell
8 24 32=4a(2) elongated (2 axes) 24-cell
9 104 > 3a(1)
10 144 > 4a(4) runcinated 24-cell?
11 48 3a(3) elongated rectified 8-cell?
12 144
13 112
14 192 6a(3) elongated rectified 8-cell?
15 48 192=6a(4) elongated (2 axes) 24-cell
16 32 > 128/3 = 42.3333
One key to this is from Conway & Sloane, Sphere-packings, Lattices and
Groups, 1993, p.455 where the hypervolume of the 24-cell {3,4,3} is
given, as well as its normalized doimensionless 2nd moment of inertia
= 13/120sqrt2 = 0.0766032... itself a constant that could be in OEIS.
So the first hypervolume that I do not know is for n=3.
I'd suspect that Sloane and Conway know -- they give this their own
name: the ambotesseract.
So, Neil, can you please help me by telling me the hypervolume of an
ambotesseract a.k.a. Rectified tesseract (Norman W. Johnson)
Rit (Jonathan Bowers: for rectified tesseract)
Rectified [four-dimensional] hypercube
Rectified 8-cell
Rectified octachoron
Rectified [four-dimensional] measure polytope
Rectified [four-dimensional regular] orthotope
Runcic tesseract (Norman W. Johnson)
Runcic [four-dimensional] hypercube
Runcic 8-cell
Runcic octachoron
Runcic [four-dimensional] measure polytope
Runcic [four-dimensional regular] orthotope
Then the next I don't know is n = 7, which I think is a truncated
8-cell; and then n=10 which I think is a runcinated 24-cell.
Thanks!
And thank you, Neil, for editing
A127081 One-sided kissing number for spheres in n-dimensional Euclidean space.
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