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

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Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
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Determine the formulas that represent the decomposition of vertices/edges/faces into orbits.

I've taken a basic stab at this problem. I feel like I am missing something big. Please help. Thanks! Q: Let $G$ be the group of rotational symmetries of a cube, let $G_v, G_e, G_f$ be the ...
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Which wallpaper patterns can coordinates be chosen so that the group $G$ operates on the lattice $L$?

For which of the seventeen wallpaper patterns can coordinates be chosen so that the group $G$ operates on the lattice $L$? FYI, this is an Artin textbook problem. Please help. Thanks!
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Describe the orbits of poles for the group of rotations of an octahedron.

Can anyone review my work on this problem and tell me if I'm missing anything major? Thanks! Q: Describe the orbits of poles for the group of rotations of an octahedron. There are $|G|=N=24$ ...
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388 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 ...
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1answer
25 views

Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...
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467 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 ...
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38 views

Spontaneous or not spontaneous symmetry breaking? That is the question.

I have the following system of ODEs: \begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ ...
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1answer
375 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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29 views

Show that the point group $\overline{G}$ can not be a subgroup of $G$ generated by a single glide reflection.

I have having trouble with the very last part of this problem: Q: Consider the group $G$ generated by a single glide reflection. Show that the point group is $D_1 = \{\text{id}, r\} \cong ...
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1answer
21 views

basis for symmetric square

If we have the symmetric square Sym$^{2}V$ and $V=\mathbb{C}^{2}$, why is it that $\{x^{2}, xy, y^{2}\}$ form a basis for it? So symmetric square matrices are when the main diagonal acts as a ...
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73 views

What is the value of $\angle BMN$?

Here $\triangle ABC$ is a isosceles triangle with $AC=BC,\angle C=20^\circ$, $\angle ABM=60^\circ$ and $\angle BAN=50^\circ$. What is the value of $\angle BMN$? please help me with this problem.
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40 views

Which mathematical theory investigates distortions?

For transformations like rotations, translations or boosts Lie theory is the appropriate theory. Which theory talks in similar, systematic terms about distortions $$ M \vec{v} = m \vec{v} \quad ...
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48 views

Proving that a group is simple

I want to solve the following: Let $I$ be the group of isometries that preserve the orientation of the icosahedron. Use the class formula to show that $I$ is a simple group (i.e. It does not have ...
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1answer
16 views

Symmetry properties conserved after integration?

I have an integrand consisting of the variables a,b,w,x,y,z. Now I integrate over the variables w,x,y,z (w and x from 0 to 1 and y and z from 0 to infinity). I know that the resulting expression ...
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1answer
61 views

Simple question on group theory

Suppose we have the following system of differential equations: \begin{cases} \frac{dx_{i}}{dt}=f_{i}\left(\boldsymbol{x},\boldsymbol{y}\right), & i=1,\ldots M\\ \\ ...
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Basic questions regarding group theory and symmetry in practice

I have for some months been interested in group theory. I was very fascinated by the level of abstraction I first met when working with groups. Another aspect that has fascinated me lately is ...
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13 views

Formally correct “generator expression” for parameters of a function

I'm trying to express formally correct that a class of functions exists that have a certain property that applies to all concrete "instances" of this class. In that I try to write a "generator ...
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1answer
47 views

If $n\times n$ matrix A is symmetric, is $A^{10}$ symmetric also?

If $n\times n$ matrix A is symmetric, is $A^{10}$ symmetric also? All I can deduce is that $A^{T}=A$, but that doesn't give me much in trying to show that $A^{10}$ is symmetric. Any hints/suggestions? ...
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32 views

Idempotents and symmetries of Zn

[Soft question out of curiosity] While reading about idempotents of a ring, I used $\mathbb Z_n$ as a convenient example. In visualizing the ring's structure, I was intrigued by strange symmetries ...
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1answer
19 views

How does a Lie derivative generate a $U(1)$ isometry?

Consider a $2l$-dimensional Riemannian manifold $(M,g)$ without a boundary and let $V=V^{\mu}\frac{\partial}{\partial^{\mu}}$ be a Killing vector field, i.e. $$ \mathcal{L}_Vg_{\mu \nu} = 0 ...
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17 views

Motions that Leave a Prism Invariant

Identify the group of all motions that leave a right prism invariant. It seems like the normal dihedral group, $D_n$, with respect to the base would be a subgroup of this group. That is to say that ...
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2answers
25 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 ...
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1answer
34 views

Show that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero

I need some help on showing that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero. $A$ is rectangular and can have dependent columns in general. I can show that it cannot have ...
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378 views

I have a 2x2 positive-semidefinite matrix. I am trying to find the equation of its elements.

So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it ...
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132 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 ...
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1answer
31 views

Probability question in group-theoretic context

I'd like to know the probability that four randomly chosen symmetries of the cube generate the whole octahedral group $C_{2}\times S_{4}$... is there some quick way of working this out, i.e. avoiding ...
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739 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 ...
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1answer
43 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
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12 views

Prove that the Gaussian quadrature points are symmetric if the interval is symmetric wrt the origin.

I need to find the proof for the symmetric Gaussian quadrature points in a symmetric interval. I thought it was a popular fact but I couldn't find it on the net. I also want to prove that the ...
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68 views

Programmatically recognizing symmetries of a polyhedron

I'm programming something, but I'm stuck at something which more math-oriented people probably can help me with. I am giving a polyhedron in the following form: for each vertex I get the cyclic order ...
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1answer
26 views

Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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1answer
54 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 \\ ...
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1answer
31 views

Doubt in Peter Olver “Applications of Lie groups to differential equations”

Book: Applications of Lie groups to differential equations. Second edition (1993). Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups. Example: 2.41. The heat equation. Question ...
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37 views

Product of symmetric matrices

Let $A \in \mathbb{R}^{n \times n}$ be symmetric. I am trying to understand under which conditions on $B \in \mathbb{R}^{n \times n}$ the product $AB$ is also symmetric. It is clear that if $B$ is ...
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Symmetry of PDEs

I am trying to solve the following problem: Show that $$(u,t,x) \rightarrow (u, \frac{t}{c^2t^2 - x^2}, \frac{x}{c^2t^2 - x^2})$$ is a symmetry of the Wave equation $u_{tt} - c^2u_{xx} = 0$. Some ...
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online notes on symmetric spaces

Can anyone suggest some good online lecture notes on symmetric spaces? I am interested in reading from Helgason, which is a very tough book to read. So I am searching for some places where the ...
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Integral of symmetric function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be such that $f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for every $n$-permutation $\sigma$, and suppose that ...
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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 ...
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127 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 $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? ...
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216 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 ...
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1answer
29 views

Commutativity of matrix square root

Let $A, B \in \mathbb{R}^{n \times n}$ and let us assume that $A^{1/2}$ exists. I have often seen people write something like $$ AB = A^{1/2}\, B\; A^{1/2} $$ when both $A$ and $B$ are symmetric, in ...
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66 views

What is the smallest cubic bipartite asymmetric graph?

"According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs." If there are any, what is the smallest cubic bipartite asymmetric graph? Kind of a ...
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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 ...
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213 views

What is a “unique” mirror line of symmetry?

What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
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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 ...
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31 views

If $R_1$ and $R_2$ are symmetric and transitive, prove that also $R_1 \cup R_2$ is symmetric and transitive

I have to prove or confute (with a counter example) that, if $R_1$ and $R_2$ are symmetric and transitive, then $R_1 \cup R_2$ is also symmetric and transitive. Note: To prove the transitive case, ...
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1answer
9 views

Finding the Projection without using projection matrix always but to use some symmetry

I am interested in finding the t2' with the help of t1, t1' and t2. Actually I am using some projection matrix T (this will be used to project point x and y which is clear in the Image attached) on x ...
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173 views

Symmetries of Archimedean Solids

There are five platonic solids, and 13/15 (which is correct?) Archimedean Solids. The finite groups of isometries of Euclidean $3$-space are the finite subgroups of $SO(3,\mathbb{R})$ or ...