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Questions tagged [symmetry]

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

162
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5answers
8k views

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 ...
54
votes
9answers
24k views

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 ...
49
votes
1answer
1k views

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 ...
26
votes
1answer
969 views

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 ...
23
votes
12answers
3k views

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 ...
23
votes
1answer
401 views

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 ...
20
votes
3answers
2k views

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 ...
19
votes
2answers
487 views

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 ...
17
votes
3answers
570 views

show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$

let $a,b,c>0$ and such $a+b+c=3abc$, show that $$ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$$ Proposed by wang yong xi since $$\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{...
16
votes
1answer
663 views

Is there an explicit left invariant metric on the general linear group?

Consider $GL_n^+$, the group of (real) invertible matrices with positive determinant. Is it possible to find an explicit formula for a metric on $GL_n^+$ which is left-invariant, i.e $$d(A,B)=d(gA,...
14
votes
5answers
1k views

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 ...
14
votes
2answers
269 views

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 ...
12
votes
3answers
2k views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
11
votes
9answers
4k views

Why $e^x$ is always greater than $x^e$?

I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the ...
11
votes
3answers
3k views

What's the intuition of the transpose of a matrix? [duplicate]

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
10
votes
1answer
705 views

Systematic solution to my soccer ball puzzle

I once received a puzzle that can be described as follows: There are $12$ black pentagonal and $20$ white hexagonal pieces. The goal is to form a soccer ball from these (aka. truncated icosahedron). ...
10
votes
1answer
386 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
10
votes
2answers
263 views

Pedagogical examples of distinguishing “types of symmetry”

When speaking to interested parties lacking formal mathematical background, I've illustrated how different objects can have the same number of symmetries and yet have different types of symmetry with ...
10
votes
1answer
272 views

Interesting tiling with a lot of symmetrical shapes

I have such an interesting observation: if I take a square grid and rotate it over itself by atan(3/4) , it forms a structure which has four axes of reflection symmetry: The resulting structure is ...
10
votes
0answers
111 views

Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
9
votes
2answers
12k views

Odd order moments of a symmetrical distribution

Is it true that for every symmetrical distribution all odd-order moments are equal to zero? If yes, how would I be able to prove such a thing?
9
votes
2answers
6k views

Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?

Define a game with S players to be Symmetric if all players have the same set of options and the payoff of a player depends only on the player's choice and the set of choices of all players. ...
9
votes
3answers
194 views

Breaking symmetries

Back when I was studying electromagnetism and Maxwell's equations, our teacher told us a quote. I can't recall it exactly, but the meaning was roughly the following: Symmetry in a problem is ...
9
votes
2answers
1k views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint $g(x,y,z,\...
9
votes
1answer
206 views

What are the finite subgroups of isometries of a flat triangular torus?

Let $\mathbb{Z}^2$ act on $\mathbb{R}^2$ as follows: $(1,0)$ acts by translating the plane by $(1,0)$, and $(0,1)$ acts by translating plane by $(1/2, \sqrt{3}/2)$. Now consider the torus $\mathbb{R}...
9
votes
0answers
215 views

Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
8
votes
2answers
999 views

Is this contraction of metric tensor derivatives symmetric?

A couple of times when I've tried to prove symmetries of various tensors (for learning), I've ended up with the expression below, and the fact that either a) I made mistake, or b) the expression is ...
8
votes
2answers
824 views

Reflection with respect to a parabola

I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have ...
8
votes
4answers
364 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
8
votes
1answer
623 views

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 ...
8
votes
2answers
87 views

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?
8
votes
1answer
221 views

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 ...
8
votes
1answer
211 views

Symmetries of Mandelbrot sets with integer exponents

I have been experimenting with recursive formulas of the form: \begin{equation} \forall c \in \mathbb{C} , z_{n+1} = z_{n}^\alpha + c \tag{1} \end{equation} as well as: \begin{equation} \forall c \...
8
votes
1answer
76 views

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 ...
8
votes
1answer
101 views

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 ...
8
votes
0answers
158 views

Isometries of the canonical left invariant metric on $GL_n$

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 \,: \...
8
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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 ...
7
votes
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) ...
7
votes
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,....
7
votes
2answers
276 views

A map which commutes with Hodge dual is conformal?

$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ Let $V$ and $W$ be $d$-dimensional, ...
7
votes
1answer
90 views

Visualizing and understanding the roots of $f(z) = z^2 - e^{i\varphi}$

[I've added another animated picture below, showing how the actions of the groups $Q_\alpha$, $R_\beta$ coincide every now and then.] The roots of $f(z) = z^2 - e^{i\varphi}$ are simply $\pm e^{i\...
7
votes
1answer
203 views

Space of arbitrary rotations of a cube

Suppose I have a cube $[-1,1]^3\subset\mathbb{R}^3$. I am allowed to rotate it about any angle/axis through the origin rather than just $90^\circ$ about the coordinate axes, e.g., by applying ...
6
votes
3answers
571 views

Does Young's inequality hold only for conjugate exponents?

Suppose that $ab \leq \frac{1}{p}a^p+\frac{1}{q}b^q$ holds for every real numbers $a,b\ge 0$. (where $p,q>0$ are some fixed numbers). Is it true that $ \frac{1}{p}+\frac{1}{q}=1$? I guess so, and ...