Talk:Hanner polytope
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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)
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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:
In five-dimensions there are eight cases as dual pairs:
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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:
- Cn=n-cube, CΔ
n=dual polytope=n-orthoplex - bip P := P ⊕ [−1, 1] denotes a bipyramid, and prism P refers to the prism construction
| # | Name | Translations | f-vector | |
|---|---|---|---|---|
| f0 | ||||
| 1 | C1=CΔ 1 |
line segment | { } | 2 |
| # | Name | Translations | f-vector | ||
|---|---|---|---|---|---|
| f0 | f1 | ||||
| 1 | C2=CΔ 2 |
square | {4} = { }×{ } = { }+{ } | 4 | 4 |
| # | Name | Translations | f-vector | |||
|---|---|---|---|---|---|---|
| f0 | f1 | f2 | ||||
| 1 | CΔ 3 |
octahedron | {3,4} | 6 | 12 | 8 |
| 2 | C3 | cube | {4,3} | 8 | 12 | 6 |
| # | 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 |
| # | 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 |
| # | 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 !