Questions tagged [symmetry]

Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

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Symmetry for irreducible polynomials over the integers

Some investigations make believe that for all polynomials $\Sigma a_kx^k$ of degree $n$, all $a_k\neq 0$, in $\mathbb Z[x]$ it holds that $\Sigma a_kx^k$ is irreducible iff $\Sigma a_{n-k}x^k$ is ...
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Does every convex planar set contain a centrally symmetric subset with at least $2/3$ its area?

Let $S$ be a bounded convex subset of the plane of unit area. Can we guarantee the existence of a centrally-symmetric subset $C⊆ S$ of area $2/3$? If $S$ is any triangle, this bound is tight, attained ...
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prove wallpaper groups are not isomorphic

In the chapter I am reading on wallpaper groups, it outlines proofs that all the wallpaper groups are not isomorphic and hence different. But I do not fully see why what they are saying is true. For ...
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Is there an area-preserving concentric diffeomorphism of the ellipse?

$\newcommand{Vol}{\text{Vol}}$ Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse $$E=\{(x,y) \in \mathbb R^2 \, | \, (\frac{x}{\frac{1}{b}})^...
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Can a heterogeneous binary relations be symmetric

This Wikipedia page lists the properties homogeneous relations may have. I am wondering whether it makes sense or not to consider symmetry of non-homogeneous relations, or if necessarily : $∀x∈X,∀y∈Y,(...
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Why does the cumulative summation of cos(n^2) have a strange symmetry?

I was messing around in Desmos and typed this in: $\displaystyle{\sum_{n=1}^{floor(x)}\cos(n^2)}$ It produces this graph in the $XY$-plane (keep in mind the axis ratios): $XY$-plane" /> It appears to ...
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How to prove the action of the symmetry group of a cube on its pairs of opposite faces defines a surjective homomorphism from $S_4$ to $S_3$?

Show that the action of the symmetry group of a cube on pairs of opposite faces of the cube defines a surjective homomorphism from the symmetric group $S_4$ to $S_3$. My attempt/logic: I know that ...
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wallpaper groups explanation [closed]

I'm reading Chapter 26 Wallpaper Patterns of Armstrong's Groups and Symmetry, and am really struggling to understand the case-by-case analysis of all the different 17 groups. Does anyone know any ...
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References for groups as symmetry elements acting on some structure

The definition that I knew about groups were the abstract definition, that any set $G$ together with an operation $*$ is a group $(G,*)$ iff $G$ is closed under group operation, associative, and has ...
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Averaging over Mahalanobis distance vectors of different clusters

Given vectors from two different clusters (in particular in my example from two different experimental conditions, called "CS" and "US") where the Mahalanobis Distance is ...
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Estimation in uniform distribution

Is there a case when uniform distribution posses unbiased and umvue estimate of $\theta$? Suppose $X_1, X_2, \dots, X_{45}$ ~ uniform on interval $[\theta-1/2\ , \theta+1/2]$ My views: I know $\max(...
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Decomposing a sum of symmetric tensors components

Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}$ for all $1\leq i,j,k\leq N$. By symmetric I mean that if I permute any indices ...
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Is the function $f(x,y)$ describing the given graph is symmetric for any $x\neq y$? Or, only for $x=y$?

We have an infinite periodic graph with the two lengths $x$ and $y$ and we have a function $f(x,y)$ that describes a specific property of this periodic structure. Then, according to the symmetry of ...
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Can one imagine spaces where point reflections are not bijective?

Not a mathematician, and the question probably does not make sense but asking it anyway: In usual euclidian spaces, central inversion through O(0,...,0) of point M(x1,...,xN) gives you one and only ...
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How many axes of symmetry for a skewed cuboid?

How many axes of rotational symmetry are there for the following object? Also, is there an algorithmic procedure for finding the axes of rotational symmetry for a general 3D object? Here are all the ...
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Computational complexity of the Cholesky factorization

According to the Cholesky factorization on Wikipedia, the computational complexity of it is $\frac{n^3}{3}$ FLOPs where $n$ is the size of the considered matrix $\mathbf{A}$. There are various ...
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Mathematics of dewarping a reflection in a generalized cylinder

Because this question involves physics, I've posted a closely related question here. Before I go to the effort of deriving all the equations, I'm wondering if anyone knows of a reference paper for ...
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1answer
24 views

How to classify discrete groups of motions

I'm studying chapter 5 of Artin's Algebra, and it talks about the classification of the discrete groups of motions(in the plane), and that there are only 17 types. I thought about what it means for ...
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Argument by symmetry that $\sum_{k=0}^{n-1} e^{2\pi i\frac{k}{n}}=0$

By summing up the geometric series, it can be shown that, $$\sum_{k=0}^{n-1} e^{2\pi i\frac{k}{n}}=\frac{1- e^{2\pi i\frac{n}{n}}}{1- e^{2\pi i\frac{1}{n}}}=0.$$ However despite having proof of this ...
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Antisymmetric (1 1) tensor

I understand the (2 0) tensor $F^{\nu \mu}$ is antisymmetric $\iff$ $F^{\nu \mu} = - F^{\mu \nu}$, and that the (0 2) tensor $F_{\nu \mu}$ is antisymmetric $\iff$ $F_{\nu \mu} = - F_{\mu \nu}$. But, ...
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Does this tetrahedron have a name?

Consider the tetrahedron with vertices at: $$(0,0,0)$$ $$(2,-\sqrt2,0)$$ $$(2,\sqrt2,0)$$ $$(2,0,2)$$ This tetrahedron is not regular but does it have any notable properties? It appears to have some ...
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Symmetric distributions in $\mathbb{R}$.

Let $X$ be a real random variable central symmetric with respect to $c\in\mathbb{R}$, that is, $X - c$ is equal in distribution to $c - X$, that is, $$ \mathbb{P}(X - c\leq t) = \mathbb{P}(c - X\leq t)...
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What is the meaning of symmetry of equalities?

For the equation: $$a+\frac1b=b+\frac1c=c+\frac1a=t$$ According to @FundThmCalculus's Answer We have symmetry of equalities. I wonder what does that mean. in other word when equations are symmetric ...
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$n$-th order ODE admits a symmetry group if and only if it can be rewritten in terms of the differential invariants.

I am trying to figure out a proof of the following theorem. Let $G$ be a one-parameter Lie group acting on plane $(x,y)$. Assume that on an open subset $W^{(n)} \subset J^n$ (where $J^n$ is n-th jet ...
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How to find a tessellation from a basis of linear transformations?

Sorry if this is a repeat. It seems like something that should be well known, but I have never found a really good way to write computer code to solve this problem, despite having ad hoc solutions for ...
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Isometry group of a prime constellation

Disclaimer: this question was first asked on Mathoverflow and downvoted there as it doesn't seem to suit it. I hope it is nevertheless sufficiently interesting to be posted here. This question is a ...
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1answer
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Twisted ribbons and centre of gravity

NOTE After posting this question, I realised I had not defined the overall shape of the form. The final note at the bottom was intended to correct that omission. I am no longer certain how this ...
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1answer
29 views

$T \cdot W=0$ for every skew symmetric tensor $T$ implies that $T$ is a symmetric tensor

I'm trying to prove by myself the following easy fact. The context is continuum mechanics (in particular I'm using Gurtin's book): Let $T$ a tensor and $W%$ a skew-symmetric tensor. If $T \cdot W =0 $...
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1answer
150 views

Retrying Leray: new solutions to Navier-Stokes

This is an attempt to revisit Leray's self-similar solutions of the 3-dimensional incompressible Navier-Stokes equation (NSE) and to show, that they are an isolated case of the general set of self-...
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Can each edge of a Wythoffian polytope be flipped by a reflection?

A Wythoffian polytope $P\subset\Bbb R^d$ is an orbit polytope of a finite reflection group, that is, $$P:=\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in\Gamma\},$$ where $\Gamma$ is a finite ...
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Half wave, quarter wave symmetry

Is my understanding correct? If a function is anti-symmetric about $T/2$ where $T$ is the period, then it posses half wave symmetry. If a function has half-wave symmetry and symmetry about the ...
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30 views

Symmetries of a graph with unconnected vertex

How do you determine the number of symmetries in the following graph: https://i.stack.imgur.com/gviNB.png I know how to determine the number of symmetries of this graph if the two middle vertices ...
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42 views

Is every subgroup of $\mathrm{Isom}(\mathbb{R}^n)$ a semidirect product $T \rtimes Q$?

It is well known that the group of isometries of Euclidean space $\mathrm{Isom}(\mathbb{R}^n)$ splits as the semidirect product $\mathbb{R}^n \rtimes O(n)$. However, is it also true that every ...
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1answer
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Shapes with $7$ lines of symmetry

I am trying to find shapes with $7$ lines of symmetry. Regular $7$-gon(Heptagon) has this property. but can you give example of other shapes with $7$ lines of symmetry? (I know I can draw an small ...
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1answer
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Divisibility Rule for 21 similar to rule for 12

I already understand the divisibility rule for 12. $$10\equiv -2 \pmod{12} \implies 10^n\equiv (-2)^n \pmod{12}$$ Then for some number $n = abcd = 1000a + 100b + 10c + d,  abcd\equiv1000a + 100b + ...
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Generate Symmetry consistent combinations from equivalent positions

I want to generate all possible substitutions from a simple molecule, for example Naphthalene Here usually we have, position 1 = 8 = 5 = 4; position 7 = 6 = 2 = 3 However if i have three groups, a,b,...
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Fourier Transform Symmetry Misunderstanding

I learned Fourier Transform about 2 years ago, but recently I found that I can't understand a simple property of that. there is a lot of proof that shows Fourier Transform of even/odd signal is even/...
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Rigorous proof that $E(X|X+Z) = E(Y|Y+Z)$ when $X, Y, Z$ are independent and $X\overset{d}{=} Y$

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and let $X, Y, Z:\Omega\rightarrow[-\infty,\infty]$ be independent random variables, such that $X, Y$ are $\mathbb{P}$-integrable and ...
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Representation of the action of $ S^3 $ on $ (1,2,-3) \in M$ On the plane

Let's consider the action of permutation of $S^3$ on the vector space $M =\lbrace (\lambda_1, \lambda_2, \lambda_3)\in \mathbb{R}^3, \lambda_1+\lambda_2+\lambda_3=0\rbrace$. The problem I have is the ...
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Symmetric vs Symmetrical

I was reading an engineering book. It says " A turbomachine with symmetrical velocity triangles...". I personally felt more naturally right away to say symmetric velocity triangles. What is ...
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Is there a 3D shape that is infinitely rotationally symmetrical in exactly 2 axis?

A sphere is completely rotationally symmetrical in all directions. You can apply any combination of roll, pitch and yaw to it and it would be indistinguishable from the sphere you started with. A ...
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An attempt to revisit the criticality of the Navier-Stokes Equation

I am looking for feedback on this attempt to revisit the criticality of the Navier-Stokes equation (NSE) for incompressible fluids in 3 dimensions. It has been said, that the NSE is supercritical (see ...
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Which reflection groups contain central inversion?

Question: Which finite irreducible reflection groups $\Gamma\subseteq\mathrm O(\Bbb R^d)$ contain the central inversion $-\mathrm{Id}$, and how can this be spotted from the Coxeter diagram? The ...
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How to prove that a rotation invariant gradient must be in the direction of the radius

Let $f(x,y)$ be a smooth function on $\mathbb{R}^2$. Suppose that its gradient vector field $$ \nabla f:=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right) $$ is invariant under ...
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1answer
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How to prove uniqueness of approximate “jump” values of an $L^1$ function?

Let $v \in \mathbb{S}^{n-1}$ be a unit vector. Given $r>0$, Let $B_r(x)$ be the ball of radius $r$ around $x \in \mathbb{R}^n$, and define $$B_r^+(x,v)=\{ y \in B_r(x) \, | \, \langle y-x,v\rangle &...
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How to find the order of the group of rigid motions of platonic solids in $\mathbb{R}^3$?

The following appear as exercises in Dummit and Foote's Algebra (Section $1.2$ - Dihedral Groups): Let $G$ be the group of rigid motions in $\mathbb{R}^3$ of a tetrahedron. Show that $|G| = 12$ Let $...
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Why does a convex polyhedron being vertex-, edge-, and face-transitive imply that it is a Platonic solid?

Suppose that we have a convex polyhedron $P$, such that the symmetry group of $P$ acts transitively on its vertices, edges, and faces (that is, it is isogonal, isotoxal, and isohedral). It then ...
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Unambiguous geometry terms for specific kinds of circular symmetry

I'm writing a paper on asymmetries in the human visual system and want to ensure that I am using correct/unambiguous terminology to describe the asymmetries in question. Unfortunately, I've had ...
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1answer
51 views

Symmetries in a dihedral group

How many symmetries does $D_{2n}$ have exactly? I imagine it has rotations written in the form $\frac{x\pi}{n}$ such that $x \in \{0, 1, \dots, n-1\}$, and can also be flipped. But what about ...
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Algorithms for detecting the symmetry group of an object

Given a list of points in 3-dimensional space, what's the state-of-the-art in algorithms for calculating the symmetry group ?

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