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


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.


And thank you, Neil, for editing

A127081  One-sided kissing number for spheres in n-dimensional Euclidean space.

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