# Questions tagged [symmetry]

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### Minimal Non-Symmetric Closed Tracks Under 45-Degree Rotations

Given an arbitrary number of clockwise and anti-clockwise 45-degree track turns, what is the smallest closed track, that is neither axisymmetric nor rotationally symmetrical? For example, AAAAAAAA ...
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### Can a spherical simplicial complex have more than one "central" inversion?

Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if $\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and $\phi$ is not ...
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### Is the projective symmetry group of a polytope more general than its linear symmetry group?

Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups: \begin{align} \DeclareMathOperator{\Aut}{Aut} \...
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### Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
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### Projection of cocyclic Gaussian primes on the real axis

I just stumbled upon https://math.stackexchange.com/questions/2372062/4-concylic-points-of-gaussian-primes after a quick Google search about cocyclic Gaussian primes. As I've been investigating about ...
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### How dense can a transitive sets of points be?

How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points? As a measure for density I use ...
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### When is a $k$-distance-transitive graph already distance-transitive?

Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
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### A diameter 2 arc-transitive graph whose complement is not arc-transitive?

A graph $G=(V,E)$ is arc-transitive if its symmetry group acts transitively on ordered pairs of adjacent vertices. In general, the complement of an arc-transitive graph is not arc-transitive. But I ...
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### A characterization of root systems via their intersections with halfspaces

In a recent preprint I obtained a nice characterization of root systems as a side product. I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
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### Equiangular lines with symmetry requirements

Listing all possible arrangements of equiangular lines is non-trivial. Does the problem become any easier when we additionally require that the symmetry group of that line arrangement acts ...
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### Examples of Calabi-Yau manifolds with $\mathbb{T}^2$ symmetry
I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially ...