# 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.
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.

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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