Activity for WheatWizardâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| 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. |
— | 3 months ago |
| Edit | Post #293232 | Initial revision | — | 3 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) |
— | 3 months ago |
| Edit | Post #291362 |
Post edited: Found another proof. |
— | 11 months ago |
| Edit | Post #291362 | Initial revision | — | 11 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) |
— | 11 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: |
— | about 1 year ago |
| Edit | Post #290492 |
Post edited: A little about the automorphism group. |
— | about 1 year ago |
| Edit | Post #290492 | Initial revision | — | about 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) |
— | about 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: |
— | over 1 year ago |
| Edit | Post #289046 | Initial revision | — | over 1 year 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) |
— | over 1 year ago |
| Edit | Post #288852 |
Post edited: Added linke to atlas. |
— | over 1 year ago |
| Edit | Post #288852 |
Post edited: |
— | over 1 year ago |
| Edit | Post #288852 |
Post edited: |
— | over 1 year ago |
| Edit | Post #288852 | Initial revision | — | over 1 year 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) |
— | over 1 year ago |