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

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15
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2answers
186 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 ...
14
votes
1answer
407 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 ...
12
votes
5answers
584 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 ...
11
votes
3answers
794 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 ...
11
votes
3answers
392 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
0answers
97 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 ...
9
votes
3answers
164 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 ...
8
votes
4answers
249 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
142 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 ...
7
votes
3answers
169 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
697 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
0answers
158 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 ...
6
votes
2answers
2k 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. ...
6
votes
2answers
124 views

Need help to visualise a set (reading Abelian groups)

I am reading Abstract Algebra. I cannot visualise the following example: Let $n$ be a positive integer, and consider the set $S_n$ of all permutations from the set $n = {1, 2, \ldots , n}$ to ...
6
votes
3answers
547 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 ...
6
votes
1answer
384 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
5
votes
3answers
77 views

Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric

I am stumped on a question and am looking for some guidance on how to get it done. The problem gives you: $x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$ ...
5
votes
3answers
164 views

Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
5
votes
1answer
248 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 ...
5
votes
3answers
304 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
5
votes
2answers
217 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if the Riemann Hypothesis is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
5
votes
2answers
71 views

Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total ...
5
votes
1answer
226 views

Magic theorem for cylinders? Symmetry classes according to Conway's notation?

My teacher Kirsi of Mat-1.3000 in Aalto University stated 17 symmetry classes for planes and 14 for spherical things (some lecture slides here). She used Conway Thurston's notation to classify ...
5
votes
1answer
62 views

Points on a 2D plane spanned by a turtle graphics system

Suppose you have a turtle graphics system with a set "turning angle" $\delta$, in which the turtle can execute three commands: $F$: Go forward, by unit length, in the current direction. (The initial ...
5
votes
1answer
268 views

Walks of Even Length on a Bipartite Graph

Given a random walk on a simple $d$-regular bipartite graph $G$. The adjacency matrix $A'$ of $G$ may be split into blocks $$ A'=\pmatrix{ 0 &A^T\\ A&0 }, $$ The propagation operator $M=A'/d$ ...
5
votes
0answers
69 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
4
votes
2answers
247 views

Graphs whose automorphism group is the cyclic group

I would like a good hint for the following problem that takes into account the position at which I am stuck. The problem is as follows Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Find a ...
4
votes
2answers
97 views

Automorphisms of a structure as a powerful tool for studying the structure

This is just an arbitrary testimony of an often repeated slogan: "The group of automorphisms of a given structure is often a powerful tool for studying this structure." D. Lascar, On the ...
4
votes
2answers
95 views

Realizing $\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ as an isometry group

Every finite group arises as the isometry group of a subspace of an euclidean space (Albertsona, Boutin, Realizing Finite Groups in Euclidean Space). What are natural examples of spaces realizing the ...
4
votes
3answers
1k 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 ...
4
votes
1answer
196 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 ...
4
votes
1answer
332 views

Decomposition formulas for rotational symmetries of a cube

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. Problem statement: Let $G$ be ...
4
votes
1answer
77 views

Symmetry in reduced residue systems

This may be a stupid question, but it looks to me like the reduced residue systems modulo N are symmetrical about N/2; that is to say, that the there is the same number of integers not divisible by a ...
4
votes
2answers
41 views

Determining the symmetry group of an infinite horizontal line.

I believe I have a satisfactory answer to the following question: Imagine we have a infinite horizontal line running through the origin, what is the associated symmetry group? Now thinking ...
4
votes
2answers
226 views

Intersection of two tetrahedra, point reflexion

We are given a regular tetrahedron $ABCD$ ($ABC$ is its` base and $D$ is its vertex) and we reflect it through the middle of its height (point reflexion) - and thus we obtain a congruent regular ...
4
votes
1answer
78 views

What exactly is meant by symmetry?

This is something that has troubled me for long. Artin describes 4 types of symmetry: translational, rotational, reflective and glide. I somehow understand symmetry to be a "condition" in which a ...
4
votes
1answer
91 views

Infinitesimal $SO(N)$ transformations

An infinitesimal $SO(N)$ transformation matrix can be written : $$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$ Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric. I've started ...
4
votes
1answer
67 views

Gliders, static structures in various (dynamic) systems

Structures, i.e. symmetries over time, appear in various systems: gliders in cellular automata, like Game of Life or Rule 110, unmatched string's parts in rewrite systems – unchanged in multiple ...
4
votes
1answer
140 views

Immersions of the Klein Bottle

The famous immersion of the Klein bottle lacks symmetry. (I'm talking about this one. http://en.wikipedia.org/wiki/File:Klein_bottle.svg) One can only see one plane of reflection. However, the ...
4
votes
1answer
254 views

Symmetrically splitting an octagon into quadrilaterals

I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
4
votes
0answers
61 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
4
votes
0answers
37 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
4
votes
0answers
128 views

How do you call functions that fulfill $f(x)=\pm f(\pm 1/x)$?

A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd. How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? ...
3
votes
5answers
415 views

What is the intent of this problem, disguised as an eigenvalue - eigenvector problem?

Let $$ A= \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{bmatrix} $$ $a,b,c >0$. Find eigenvalues and a basis of ...
3
votes
3answers
2k views

Exponents in Odd and Even Functions

I was hoping someone could show or explain why it is that a function of the form $f(x) = ax^d + bx^e + cx^g+\cdots $ going on for some arbitrary length will be an odd function assuming $d, e, g$ and ...
3
votes
2answers
33 views

is a line of symmetry going through the center of an octagon parallel to the ground it is on?

I came across this question recently, It wanted me to find angle EDG assuming the line crossing between the octagon/polygon is it's line of symmetry. My answer was 25 degrees, but A few friends ...
3
votes
1answer
85 views

Inverse of diagonally dominant matrix with equal off-diagonal entries

Is there an explicit expression for the inverse of strictly diagonally dominant matrix with identical off-diagonal elements? For example: $$ \begin{pmatrix} a & -b & -b \\ ...
3
votes
1answer
23 views

Point symmetries around lines in 2D

I am trying to remember how to compute the symmetric point to an $(x=a,y=b)$ point with respect to a line, i.e. $y=mx +b$, without luck. Is there a closed form equation for this type of ...
3
votes
1answer
24 views

3-D Realization of Symmetries of a Tetrahedron

We know that the total symmetry group of tetrahedron is $S_4$. I tried to realize these 24 symmetries, but failed to ''realize'' six of them, which correspond to the $4$-cycles $(a \,b\,c\,d)$ (being ...
3
votes
1answer
77 views

Counting 0-1 matrices up to symmetry

I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small? For example, consider ...