# Questions tagged [symmetry]

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

889 questions
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### Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for ...
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### Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
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### Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
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### Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
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### Stuck on a Geometry Problem

$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
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### Why does $\int_1^\sqrt2 \frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$ equal to $0$?

In this question, the OP poses the following definite integral, which just happens to vanish: $$\int_1^\sqrt2 \frac{1}{x}\ln\bigg(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\bigg)dx=0$$ As noticed by one commenter ...
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### What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
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### Why is this definite integral antisymmetric in $s\mapsto s^{-1}$?

I recently happened into the following integral identity, valid for positive $s>0$: $$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$ The ...
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### Definition of group action

I'm currently taking a class in abstract algebra, and the textbook we are using is Ted Shifrin's Abstract Algebra: A Geometric Approach. In the chapter on group actions and symmetry, he defines a ...
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### slick way of transforming an integral?

The function $$(\alpha,\beta) \mapsto \int_0^\beta \frac{\sin\alpha\,d\zeta}{1+\cos\alpha\cos\zeta}$$ is a symmetric function of $\alpha$ and $\beta$. But I don't know a simpler way to see that ...
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### Do Symmetric problems have Symmetric solutions?

There is a general notion of Symmetry in mathematics, that of an object being constant under some transformation. If we think of our object as being a "mathematical problem" we can see certain ...
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### Why is a solution to $y'' + y = 0$ periodic

Solutions to $y'' + y = 0$ are 2$\pi$ periodic. Is this accidental or does this ODE have some symmetries associated with it that force the solutions to be periodic?
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### A volume form on the sphere which gives equal areas to all hemispheres is invariant under the antipodal map?

Let $\omega$ be a volume form on $\mathbb{S}^2$ with the property that the induced area (w.r.t $\omega$) of all the hemispheres is the same. Is it true that $\omega$ is invariant under the antipodal ...
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### Symmetries of Mandelbrot sets with integer exponents

I have been experimenting with recursive formulas of the form: $$\forall c \in \mathbb{C} , z_{n+1} = z_{n}^\alpha + c \tag{1}$$ as well as: \forall c \...
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### Has this connection between groups and the symmetries of relations been studied?

Given four distinct points $a,b,c,d$ on a circle, we can (speaking informally) choose in a unique way two pairs separating each other. For example the pairs $a,c$ and $b,d$ separate each other, if ...
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### A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: \... 0answers 311 views ### ODE system and Lie symmetries The ODE system (see below), where$F$is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some$k$constant. The result that$y$is a translation ... 3answers 970 views ### Is zero the center of the numeric sequence? The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side. Can numbers be said to have different centers of symmetry than zero? Is it possible ... 3answers 4k views ### Why isn't an odd improper integral equal to zero My calculus book says that the integral of$\frac1x$cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and ... 3answers 209 views ### Realizing groups as symmetry groups We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ... 2answers 3k views ### Do symmetric 3d solid figures always have a plane of symmetry? Chapter 3 of Martin Gardner's The New Ambidextrous Universe begins as shown below. As you can see (highlighted), on page 13 he writes that not all solid symmetric objects have a plane of symmetry, and ... 1answer 691 views ### Misconception about geodesics and Killing fields on Lie groups Given a Lie group$G$and a positive-definite inner product on the Lie algebra$\mathfrak{g}$, we can turn$G$into a Riemannian manifold by equipping each tangent space with the metric induced by ... 1answer 733 views ### Is every finite group of isometries a subgroup of a finite reflection group? Is every finite group of isometries in$d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ... 2answers 78 views ### Does the action of a linear map on$k$-dimensional subspaces determine it up to scaling? Let$V$be a real$d$-dimensional vector space, and let$1 \le k \le d-1$be a fixed integer. Let$A,B \in \text{Hom}(V,V)$, and suppose that$AW=BW$for every$k$-dimensional subspace$W \le V$. Is ... 1answer 457 views ### Differences between symmetries and isometries throughout the following question, whenever I'm wrong please correct me! Recently I came across the notions of symmetry and isometry. Though, there is something obscure (definitely in my head) ... 2answers 132 views ### Minimal specification of isometry in terms of norm preservation Let$V,W$be$n$-dimensional (real) inner product spaces, and let$T:V \to W$be a linear map. Let$v_1,...,v_n$be a basis of$V$. It is easy to see that if$|T(v)|_W=|v|_V$for every$v \in \{v_1,....
$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ Let $V$ and $W$ be $d$-dimensional, ...