# Talk:16-cell honeycomb

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Field:  Geometry

## Densest packing of spheres in 4-space???

The first paragraph of the section D4 lattice reads as follows:

"Its vertex arrangement is called the D4 lattice or F4 lattice. The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.'

But I do not believe that anyone has ever determined the densest possible packing of equal spheres in 4-space!

(Maybe D4 represents the densest possible packing of equal spheres among lattice packings. That is usually much easier to determine than the densest sphere packing among all packings whatsoever.)

I hope someone knowledgeable on this subject will correct this, if in fact it needs correction.Daqu (talk) 20:19, 24 August 2016 (UTC)

I agree that your interpretation is almost certainly correct. --JBL (talk) 21:18, 24 August 2016 (UTC)
Daqu, I have gone ahead and added the word "regular" to make the claim correct. Having looked over that section, though, I'm a bit perplexed: after the first paragraph, none of the statements deal with the 16-cell honeycomb; if the objects they mention are connected to it, they don't say how. I am tempted just to remove that material, but thought I would give others a chance to weigh in. --JBL (talk) 21:02, 25 August 2016 (UTC)
Conway and Sloane call it "the densest sphere-packing [known] in four dimensions." (Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9) Tom Ruen (talk)
Tomruen, now that you've removed "regular", this raises the same question about the kissing number claim in the second half of the sentence. Largest possible, largest known, largest among lattice packings? --JBL (talk) 22:16, 25 August 2016 (UTC)
I checked more carefully and added sources. Its the highest among lattices only. Tom Ruen (talk) 22:58, 25 August 2016 (UTC)
Just to be exceedingly precise: I believe it is not currently known whether the D4 sphere packing is the densest among all sphere packings in 4-space.Daqu (talk) 03:39, 1 September 2016 (UTC)
JBL, Oleg Musin proved in 2003 that 24 (and not 25, the only other possibility at the time) is indeed the kissing number in 4 dimensions.Daqu (talk) 19:17, 17 September 2016 (UTC)

## What is a "demicubic lattice" or "demicubic honeycomb" ???

The article refers to "4-demicubic lattices" and to "5-demicubic honeycombs", but there is no reference to the definition of such things.

The article Alternated hypercubic honeycomb seems to be the only place in Wikipedia where such things are defined, but the definition could hardly be more vague for non-experts:

"In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation."

Whatever "with an alternation operation" means!

Both that article and this one need to have a clear definition in English or reference to what a "demicubic lattice" and a "demicubic honeycomb" mean!

Sure, the Coxeter diagrams are a useful shorthand, but that's no reason to omit a definition in English.Daqu (talk) 17:38, 20 September 2016 (UTC)

I removed the confusing demicubic lattice terms, changing both to D4 lattices. Basically the lattices are the vertex arrangement of a honeycomb, and an alternation remove half the vertices, which create two types of cells in the honeycomb. Tom Ruen (talk) 19:11, 20 September 2016 (UTC)