# Questions tagged [symmetry]

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### Classification of maximal point groups

Have the maximal (without finite proper overgroups of the same dimensionality) finite point groups been fully classified in any dimensionality of Euclidean space greater than 4?
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### Can $E_8$ be enlarged?

Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
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### What is the symmetry group of this compound of two polytopes?

The geometric shape in question is a compound of two polytopes: an 11-hypercube with edge length $2$ and an 11-simplex with edge length $\sqrt6$ whose vertices are a subset of the hypercube’s. What is ...
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### A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties? All facets are congruent, it has an ...
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### Balanced Gray codes for powers of 2

All of the binary 4-bit cyclic balanced Gray code sequences can be formed from simple reversals, bit-permutations, and circular shifts of the one Wikipedia example: ...
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### Number of solutions and minimal clues in Sixy Sudoku

Sixy Sudoku is a variation on Latin squares and traditional sudoku played on a $6 \times 6$ grid with an initial clue of several cells filled in with a subset of the digits $1$–$6$. The task is to ...
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### Where can I learn about the discrete symmetries of the complex projective plane (or space)?

I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
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### Matroids which are transitive on minimal basis exchanges

I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that. Consider a finite matroid $M$. Define a ...
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### Symmetric subgraph configurations

Let $G,H$ be two simple graphs. Let's call a subgraph of $H$ that is isomorphic to $G$ a $G$-subgraph. Consider the following construction: Construction: Let $\mathcal G=\mathcal G(G,H)$ be a graph ...
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### Are spherical maps with low distortion locally expanding?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\Hom}[1]{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\S}{\mathbb{S}}$ The question in a nutshell: Are the "best" spherical maps ...
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### Why does $\sqrt 5$ occur in manageable situations of these scenarios?

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
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### Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set \begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
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### Is a vertex- and edge-transitive polytope already a uniform polytope?

I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive). Question: Is every such ...
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### Do cocycles “break” symmetry?

In an article by A. Borovik, “Is mathematics special?”, he talks about the fact that carrying is a cocycle. He then says [A student] discovered that carry is doing what cocycles frequently do: they ...
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### Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
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Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\... 0answers 119 views ### Cencov's “categories of figures” In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-... 2answers 143 views ### Maximal edge length of symmetric polytopes For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ... 0answers 93 views ### Is every “higher-order” harmonic morphism conformal?$\newcommand{\TM}{\operatorname{TM}}\newcommand{\M}{\mathcal{M}}\newcommand{\N}{\mathcal{N}}\newcommand{\TM}{\operatorname{T\M}}\newcommand{\TN}{\operatorname{T\N}}\newcommand{\TstarM}{...
$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth. Let $1 \le k \le d-1$ be fixed....