Talk:Hanner polytope

Examples

I reworked example cases for 4 and 5 dimensions, and David reverted, so I pasted my removed contents below. I don't care to argue with him. This is true, accurate and helpful information, with Klitzing page as a source, notationally all easy to dimensionally verify the products and sums. The main open question for 5D is names and notations used, duoprism and duopyramid. I picked the simplest names and notations I knew. If someone had better sources, feel free! Obviously they're all well known since they're counted to 10-dimensions Hanner_polytope#Combinatorial_enumeration. Tom Ruen (talk) 02:40, 17 November 2022 (UTC)

In higher dimensions the hypercubes and cross polytopes, analogues of the cube and octahedron, are again Hanner polytopes. However, more are possible.{{citation|url=https://bendwavy.org/klitzing/explain/hanner.htm|title=Hanner polytopes|work=Polytopes|first=Richard|last=Klitzing|access-date=2022-11-16}} In four-dimensions, there are four cases as dual pairs: tesseract, {4,3,3}, and 16-cell, {3,3,4}, octahedral prism, {3,4}×{ }, and cubical bipyramid, {4,3}+{ }. In five-dimensions there are eight cases as dual pairs: 5-cube, {4,3,3,3}, and 5-orthoplex, {3,3,3,4}, tesseractic bipyramid, {4,3,3}+{ } and 16-cell prism, {3,3,4}×{ }, cubic bipyramidal prism, ({4,3}+{ })×{ }, and octahedral prismatic bipyramid, ({3,4}×{ })+{ }, square-octahedral duoprism, {4}×{3,4}, and square-cubic duopyramid, {4}+{4,3}.

This paper lists on p.190 4 cases 4D and p.196 lists 4 5D cases and 9 6D cases, ignoring duals. Tom Ruen (talk) 04:30, 17 November 2022 (UTC)

Name key:

C_n=n-cube, C^Δ
_n=dual polytope=n-orthoplex

bip P := P ⊕ [−1, 1] denotes a bipyramid, and prism P refers to the prism construction

1D

#	Name	Translations		f-vector
				f0
1	C1=CΔ 1	line segment	{ }	2

2D

#	Name	Translations		f-vector
				f0	f1
1	C2=CΔ 2	square	{4} = { }×{ } = { }+{ }	4	4

3D

#	Name	Translations		f-vector
				f0	f1	f2
1	CΔ 3	octahedron	{3,4}	6	12	8
2	C3	cube	{4,3}	8	12	6

4D (duals grouped)

#	Name	Translation (Dual)		Vertices (Dual vertices)	f-vector
					f0	f1	f2	f3
1 2	CΔ 4 C4	16-cell (4-cube)	{3,3,4} {4,3,3}	8 (16)	8	24	32	16
3 4	bip C3 prism CΔ 3	cubic bipyramid (octahedral prism)	{4,3}+{ } {3,4}×{ }	8+2 (6×2)	10	28	30	12

5D (duals grouped)

#	Name	Translation (Dual)		Vertices (Dual vertices)	f-vector
					f0	f1	f2	f3	f4
1 2	CΔ 5 C5	5-orthoplex (5-cube)	{3,3,3,4} {4,3,3,3}	10 (32)	10	40	80	80	32
3 4	bip bip C3 prism prism CΔ 3	cubic-square duopyramid (octahedron-square duoprism)	{4,3}+{4} {3,4}×{4}	8+4 (6×4)	12	48	86	72	24
5 6	bip prism CΔ 3 prism bip C3	octahedral prismatic bipyramid (cubic bipyramidal prism)	({3,4}×{ })+{ } ({4,3}+{ })×{ }	6×2+2 (8+2)×2	14	54	88	66	20
7 8	prism CΔ 4 bip C4	16-cell prism (tesseractic bipyramid)	{3,3,4}×{ } {4,3,3}+{ }	8×2 (16+2)	16	56	88	64	18

6D (duals grouped)

#	Name	Translations (dual)		Vertices (Dual vertices)	f-vector
					f0	f1	f2	f3	f4	f5
1 2	CΔ 6 C6	6-orthoplex (6-cube)	{3,3,3,3,4} {4,3,3,3,3}	12 (64)	12	60	160	240	192	64
3 4	bip bip bip C3 prism prism prism CΔ 3	octahedral-cubic duopyramid (octahedral-cubic duoprism)	{3,4}+{4,3} {3,4}×{4,3}	6+8 (6×8)	14	72	182	244	168	48
5 6	C3 ⊕ C3 CΔ 3 × CΔ 3	cubic-cubic duopyramid (octahedral-octahedral duoprism)	{4,3}+{4,3} {3,4}×{3,4}	8+8 (6×6)	16	88	204	240	144	36
7 8	bip bip prism CΔ 3 prism prism bip C3	(octahedral prismatic)-square duopyramid ((cubic bipyramidal)-square duoprism)	{3,4}×{ }+{4} ({4,3}+{ })×{4}	6×2+4 (8+2)×4	16	82	196	242	152	40
9 10	bip prism CΔ 4 prism bip C4	16-cell prismatic bipyramid (tesseractic bipyramidal prism)	{3,3,4}×{ }+{ } ({4,3,3}+{ })×{ }	8×2+2 (16+2)×2	18	88	200	240	146	36
11 12	bip bip C4 prism prism CΔ 4	tesseractic-square duopyramid (16-cell-square duoprism)	{4,3,3}+{4} {3,3,4}×{4}	16+4 (8×4)	20	100	216	232	128	32
13 14	prism CΔ 5 bip C5	5-orthoplex prism (5-cubic bipyramid)	{3,3,3,4}×{ } {4,3,3,3}+{ }	10×2 (32+2)	20	90	200	240	144	34
15 16	bip prism bip C3 prism bip prism CΔ 3	cubic bipyramidal prismatic bipyramid (octahedral prismatic bipyramidal prism)	({4,3}+{ })×{ }+{ } ({3,4}×{ }+{ })×{ }	(8+2)×2+2 (6×2+2)×2	22	106	220	230	122	28
17 18	prism bip bip C3 bip prism prism CΔ 3	cubic-square duopyramidal prism (octahedral-square duoprismatic bipyramid)	({4,3}+{4})×{ } {3,4}×{4}+{ }	(8+4)×2 6×4+2	24	108	220	230	120	26

This sort of piling on of example farm after example farm, with no published sources for any of the calculations or notations within them, and instead only the vaguest hint that because the total number of examples matches up with OEIS then maybe it is calculated without mistakes, is exactly why I reverted the first edit. Please note that per Wikipedia:Talk page guidelines, talk pages should only be for discussion of how to improve the article according to Wikipedia's standards (which include providing sources for all material); they are not an alternative venue for publishing original research. —David Eppstein (talk) 22:12, 17 November 2022 (UTC)

You ask for a published source, and I provide one, which also listed the 18 6D cases, so I gave them too. You can complain about notations, but there's no reason an encyclopedia can't add varied notations, including notations used elsewhere on wikipedia. The operations are simple, products and sum, already explained in this article. You can perhaps find a dozen different papers listing this, and each might have a slightly different naming scheme. Which should be used? All is fine with me if you think it adds readability. Tom Ruen (talk) 23:06, 17 November 2022 (UTC)

[On Kalai’s conjectures concerning centrally symmetric polytopes

Raman Sanyal Axel Werner Gunter M. Ziegler

Discrete Comput Geom (2009) 41: 183–198

DOI 10.1007/s00454-008-9104-8

That source has some tables with some examples, but not all of them and not all of the information you include about them. To pick a random example among many of a piece of information that is claimed in your tables and appears to exist nowhere in the scientific literature (the first example I tried): there are zero hits in Google Scholar for the phrase "cubic bipyramidal prismatic bipyramid". Where is this terminology sourced from? Are you even trying to base your content on sources or are you just making shit up and then finding sources that sort of match something vaguely related only when pressed? And maybe a bigger question should be: in what way do you think overwhelming an article with huge tables of unfamiliar notation is likely to improve the experience of any readers of the article? —David Eppstein (talk) 23:46, 17 November 2022 (UTC)

That case is EXACTLY the same as bip prism bip C3 written more in english rather than abbreviated prefix notation from the paper table, which happens to only exist in that one paper. Say bip C3 or say cubical bipyramid, take your pick. Tom Ruen (talk) 23:56, 17 November 2022 (UTC)

It is the same polytope but it is far from the same data about the polytope. The name you give it, the curly-bracket notation you give it, the expanded formula for its number of vertices, and the correspondence with its dual are all nowhere to be found in that source. —David Eppstein (talk) 01:26, 18 November 2022 (UTC)

f-vector sums

Has anyone noticed the f-vector sums are constant by rank? This matches easy case of hypercube face count sums: 2D (4+4)=8, 3D (8+12+6)=26, 4D (16+32+24+8)=80, 5D (32+80+80+40+10)=242, 6D (64+192+240+160+60+12)=728, …? Does anyone know why? — Preceding unsigned comment added by 2601:447:CE00:3C0:C18C:47C2:9C07:F571 (talk) 15:52, 6 June 2023 (UTC)

P.s. The series 8, 26, 80, 242, 728… is simply 3ⁿ-1 ! oeis.org — Preceding unsigned comment added by 2601:447:CE00:3C0:94DA:2BC0:96:2A53 (talk) 17:20, 8 June 2023 (UTC)

Consider a hypercube of 3ⁿ little cubes. Exactly one of them is hidden. —Tamfang (talk) 22:23, 8 July 2023 (UTC)

This is in our article already under Hanner polytope § Number of faces. See also Kalai's 3^d conjecture. Another easy way to see that a hypercube, specifically, has $3^{d}$ faces (including the cube itself, but not the empty set, as a face) is to put the faces in bijection with the length-$d$ strings over the alphabet $\{0,1,*\}$ where a $0$ or $1$ in position $i$ indicates that we are looking at the subset of the cube whose $i$th coordinate has that value and a $*$ indicates that we are not restricting the $i$th coordinate. —David Eppstein (talk) 00:07, 9 July 2023 (UTC)