This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||
|
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:
In five-dimensions there are eight cases as dual pairs:
|
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:
# | 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 |
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)
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)