Activity for WheatWizardâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #293749 | Initial revision | — | 15 days ago |
| Answer | — |
A: What are the known chiral polytopes of full rank? There are more known. Pellicer describes 3 additional chiral 4-polytopes of full rank in Chiral 4-polytopes in ordinary space (2017). These are all apeirotopes in 3 dimensions. (more) |
— | 15 days ago |
| Edit | Post #293740 | Initial revision | — | 16 days ago |
| Question | — |
What are the known chiral polytopes of full rank? The chiral polytopes of full rank are very interesting. It was once erroneously proven by McMullen that there were none, however it has since been revealed to not be the case. I am aware of two chiral polytopes of full rank: (I'm also aware of chiral polytopes of full rank in projective space, but I'... (more) |
— | 16 days ago |
| Edit | Post #293232 |
Post edited: Clarified two things. First that a linear kei need not contain all the lines in n-space, next that I am looking for an equational law. |
— | 4 months ago |
| Edit | Post #293232 | Initial revision | — | 4 months ago |
| Question | — |
Is there an equational law satisfied by "linear kei" but not free kei? A kei is an involutionary quandle, alternatively a magma satisfying three equations: $a \rhd a = a$ $(a \rhd b) \rhd b = a$ $(a \rhd b) \rhd c = (a \rhd c) \rhd (b \rhd c)$ This definition is taken from Kamada, S., 2002. I will also define a "linear kei" as structure generated by a set ... (more) |
— | 4 months ago |
| Edit | Post #291362 |
Post edited: Found another proof. |
— | 12 months ago |
| Edit | Post #291362 | Initial revision | — | 12 months ago |
| Question | — |
Seeking a theorem about lattices I am looking for a reference on the following theorem, or an equivalent statement: > Let $\Lambda$ be an embedding of a free $\mathbb{Z}$-module in $\mathbb{R}^d$. If the rank of $\Lambda$ is greater than $d$ then $\Lambda$ is not discrete. I have proofs here and here, which both seem correct t... (more) |
— | 12 months ago |
| Edit | Post #291124 | Initial revision | — | about 1 year ago |
| Question | — |
Does {7,4|3} have a realization on the Klein quartic? It was pointed out to me recently that the polyhedron {7,4|3} has the same automorphism group as the Klein quartic. Specifically {7,4|3} is: $\langle \rho0,\rho1,\rho2 \mid \rho0^2, \rho1^2, \rho2^2, (\rho0\rho2)^2, (\rho0\rho1)^7, (\rho1\rho2)^4, (\rho0\rho1\rho2\rho1)^3\rangle$ Naturally t... (more) |
— | about 1 year ago |
| Edit | Post #290492 |
Post edited: |
— | over 1 year ago |
| Edit | Post #290492 |
Post edited: A little about the automorphism group. |
— | over 1 year ago |
| Edit | Post #290492 | Initial revision | — | over 1 year ago |
| Question | — |
Is there a $(n_3)$ configuration which is not self-dual? Coxeter points out that for a self-dual configuration $(mc,nd)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $(mc)$. However I'm interested in the other direction of this implication, i.e. is there a configuration $(mc,mc)$ which is not self-dual? For $c=2$ there is none. All polygons ... (more) |
— | over 1 year ago |
| Comment | Post #290196 |
Yes. Thank you. It's easy to forget about connected and compact. :) (more) |
— | over 1 year ago |
| Edit | Post #290196 |
Post edited: Fixed claim per comment. |
— | over 1 year ago |
| Comment | Post #290305 |
> I notice the atlas link lists neither the 11-cell [...] nor the 57-cell [..]
The 11-cell is listed [here](https://www.abstract-polytopes.com/atlas/660/13/1.html). The 57-cell is too large for the atlas to list it. (more) |
— | over 1 year ago |
| Comment | Post #290305 |
What is meant by admits a realization?
In McMullen & Schulte's *Abstract Regular Polytopes* the notion of realization is quite broad, but even their more restrictive notion of a "faithfully symmetric" realization includes a lot. They show **every** abstract regular polytope has a "simplex realizat... (more) |
— | over 1 year ago |
| Edit | Post #290196 |
Post edited: |
— | over 1 year ago |
| Edit | Post #290196 | Initial revision | — | over 1 year ago |
| Question | — |
Is the nth Betti number determined by orientability? I'm interested in a proof of the following claim: > If $M$ is a connected $n$-dimensional compact manifold then the $n$th Betti number, $\betan(M) = 1$ if $M$ is orientable and $\betan(M) = 0$ otherwise. This claim seems true since it basically says that orientable manifolds have some sort of "... (more) |
— | over 1 year ago |
| Edit | Post #289046 |
Post edited: |
— | almost 2 years ago |
| Edit | Post #289046 | Initial revision | — | almost 2 years ago |
| Question | — |
Proving that this relation implies another relation on the Coxeter group [4,3,3,4]. I have a group with five generators $\sigmai$, and the following relations: \begin{split} \sigmai^2 = \varepsilon \\ |i-j| \neq 1 \implies (\sigmai\sigmaj)^2 = \varepsilon \\ (\sigma0\sigma1)^4 = \varepsilon \\ (\sigma1\sigma2)^3 = \varepsilon \\ (\sigma2\sigma3)^3 = \varepsilon \\ (\sigma3\... (more) |
— | almost 2 years ago |
| Edit | Post #288852 |
Post edited: Added linke to atlas. |
— | almost 2 years ago |
| Edit | Post #288852 |
Post edited: |
— | almost 2 years ago |
| Edit | Post #288852 |
Post edited: |
— | almost 2 years ago |
| Edit | Post #288852 | Initial revision | — | almost 2 years ago |
| Question | — |
What is special about the 11-cell and 57-cell? Reading about the 11-cell and 57-cell I find two facts implied often: - They are particularly notable among the abstract regular 4-polytopes. - They are related to each other. I'll establish why I think they are notable: - Both polytopes are notable enough to have their own articles on Wikipedi... (more) |
— | almost 2 years ago |