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

learn more… | top users | synonyms

7
votes
0answers
154 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 ...
5
votes
0answers
63 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
0answers
59 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
36 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
0answers
131 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
3
votes
0answers
35 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 ...
3
votes
0answers
25 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n ...
3
votes
0answers
58 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
2
votes
0answers
35 views

Trade-off among symmetries

Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am attempting to design a set that has the following symmetries (continuous or discrete) $1.$ Scale symmetry $2.$ Rotation ...
2
votes
0answers
23 views

Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
2
votes
0answers
43 views

What is the symmetry of the Penrose tiling?

What is the symmetry of the Penrose tiling? Simply C5 or bigger? Any simple proof that the tiling is a complete cover of the plane?
2
votes
0answers
27 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of DEs. I'm presently doing research on identifying financial indicators/phenomena that can be ...
2
votes
0answers
30 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
2
votes
0answers
34 views

Ordering binary matrices for reflection/rotation

I have a collection of $n\times n$ binary matrices and I would like to reduce it for symmetry ($D_4$ -- reflections and rotations). The naive method of testing each pair is very slow because the ...
2
votes
0answers
163 views

What to call this kind of symmetry in a sphere?

Geometrically, if the two hemispheres of a spherical distribution of some kind (let's say a spherical gas cloud) are similar such that the properties of the gas as seen by a person standing on a ...
2
votes
0answers
102 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
2
votes
0answers
212 views

An isomorphism between the full tetrahedral symmetry group and the cubic rotation group?

I know that the full symmetry group of the tetrahedron and the rotation group of the cube are both isomorphic to $S_4$. But I want to show a direct, visual isomorphism. I tried looking at a ...
2
votes
0answers
165 views

Make one cube out of 8 little cubes

As part of a puzzle, you have to stack 8 little $1\times 1\times 1$-cubes so that they form one big $2\times 2\times 2$-cube. Now I want to check all possible solution to the puzzle and therefor I'm ...
1
vote
0answers
24 views

Domain enclosed by a simple closed curve with infinity many symmetry axes must be disk?

Let $D\subset\mathbb{R}^2$ be a domain. Suppose that (1) the boundary $\partial D$ is a simple closed curve; (2) the domain $D$ has infinity many symmetry axes. The domain $D$ must be a disk? If ...
1
vote
0answers
33 views

Classification of Differential Equations

I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain. I also know ...
1
vote
0answers
81 views

Is this Fractal New?

I developed some equations relating to symmetry. When used recursively, they produce what I believe is a fractal of symmetries. The fractal is procedurally generated like a snowflake or a gasket, ...
1
vote
0answers
47 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\\ \\ ...
1
vote
0answers
56 views

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 ...
1
vote
0answers
31 views

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 ...
1
vote
0answers
28 views

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 ...
1
vote
0answers
179 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 ...
1
vote
0answers
30 views

Point symmetry group, identification.

Good day. Please, tell me information about the algorithm of identification point symmetry groups for two-dimensional data (timeseries)? Maybee some book ? I have a data file like this: ...
1
vote
0answers
43 views

How to find the center of of rotational symmetry in a Penrose tiling?

The center of Helsinki sports a gorgeous Penrose tiled pavement: Could anybody give me an algorithm that will permit me to find its 'center' - that is, the point around which the five-fold ...
1
vote
0answers
23 views

Characterizing a function regarding symmetry

Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists ...
1
vote
0answers
23 views

One dimensional binary string with periodic boundaries and reflection

I have a binary string $l=(l_1,l_2,\ldots,l_{2n})$ with $l\in\{0,1\}$ and the conditions $l_i \cdot l_{i+n}=0$ for all $i$ and $\sum l_i=n$. Now, I was wondering how many distinct string exist, when a ...
1
vote
0answers
44 views

Constraints on eigenvalues due to the form (symmetry) of Matrix

So I have to deal with a square matrix with complex elements $A$ given by - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
1
vote
0answers
56 views

Types of symmetry for combinatorial graphs

Let $G$ be an undirected, connected graph without loops. Let's call $G$ symmetric iff it has a non-trivial automorphism (that is a permutation $\pi : V(G) \rightarrow V(G) $ – which is not the ...
1
vote
0answers
380 views

Help with Hermitian Symmetry and its inverse Fourier transform in MATLAB.

I have tried to impose Hermitian symmetry on the complex number $z$ which is varies with $x$. I need to take its inverse Fourier transform. A hermitian symmetry should give a real valued inverse FT. ...
1
vote
0answers
59 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
1
vote
0answers
77 views

Dilation Invariance

Given the formula $$F(x)= \sum_{n=-\infty}^{\infty}f(x+n) $$ We know that is invariant under translations of the form $y=x+n$ for any integer $n$. However can we find a similar formula for dilations ...
0
votes
0answers
27 views

Graph, Relation $xRy \Leftrightarrow$ There is a path between $x$ and $y$ - symmetry

I have the relation $xRy$ is equivalent to "There is a path between $x$ and $y$" If I now want to check symmetry, $xRy$ is equivalent to $yRx$, I read that this is true. I thought this is only the ...
0
votes
0answers
7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
0
votes
0answers
19 views

Behavior of eigenvalues of certain matrices

I am trying to analyze the behavior of the 2 highest eigenvalues of matrices of this form : Symmetric $n*n$ matrices that contains only : $1/k$ (for fixed k), -1,1 and 0. My hope is to find some ...
0
votes
0answers
50 views

Is this a spontaneous symmetry-breaking?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f,g$ are some functions, ...
0
votes
0answers
8 views

Modular arithmetic of rotation a triange

A triangle is an n-gon with 3 rotational symmetries and 3 reflection symmetries. Each rotation is 120 degrees. Suppose the triangle begins with initial angle 240 degrees and rotates through 240 ...
0
votes
0answers
17 views

Proving a theorem on a rotation about a line followed by the inversion to show that it is a reflection

A theorem in my textbook is : A rotation about a line followed by the inversion about a point on that line is a reflection or a rotary reflection. I can picture this theorem in my head on a 3D space ...
0
votes
0answers
12 views

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 ...
0
votes
0answers
22 views

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!
0
votes
0answers
33 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 ...
0
votes
0answers
51 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 ...
0
votes
0answers
18 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 ...
0
votes
0answers
16 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 ...
0
votes
0answers
21 views

Several symmetry formulas?

I have lost the book of my math course one day before the test. I want to review several formulas, but I haven't found them anywhere online. The formulas I am looking for are: Symmetry of a line ...
0
votes
0answers
56 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...