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

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112 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 ...
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164 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 ...
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74 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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62 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 ...
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38 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
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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
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144 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
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37 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
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26 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
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60 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
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88 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, ...
2
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36 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 ...
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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
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45 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
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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
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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
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42 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
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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
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108 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
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215 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
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173 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 ...
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16 views

Proofs of the multiplication or chain rule for derivatives that invoke symmetry

Introductory calculus texts sometimes include direct proofs of the multiplication and chain rules for derivatives by: Introducing a pair of differences $D_f=\frac{f(x+h)-f(x)}{h}-f'(x)$ and ...
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10 views

Is self-similarity a form of symmetry, is it the other way around, or is it something else?

I'm aware that self-similarity is a form of symmetry, however I'm interested in getting a more in depth explanation of the relationship. Could you consider symmetry to be a form of self-similarity? ...
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30 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 ...
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35 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 ...
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49 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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61 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 ...
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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 ...
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29 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 ...
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181 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 ...
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32 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: ...
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46 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 ...
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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 ...
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24 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 ...
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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 - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
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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 ...
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398 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. ...
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61 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. ...
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82 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 ...
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23 views

Determining the corresponding vector field to a group action.

Im having trouble trying to understand how to determine the corresponding vector field to a group action on a symplectic manifold. I feel this will be easier if I give two examples which are confusing ...
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16 views

Proving that rotation-inversion axis 2 and rotation axis n/2 induce rotation axis n.

How to prove that rotation axis of n/2 order and rotation-inversion axis of 2 order induce rotation-inversion axis of n order?
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21 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
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56 views

The Fixed Point Theorem in Artin's book

Theorem 7.3.2 Let G be a p-group, and let S be a finite set on which G operates. If the order of S is not divisible by p, there is a fixed point for the operation of G on S - an element s whose ...
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37 views

What will be Terms after repeating this step(Differentiation and multiplication) F times.

I was solving a probability problem and got stuck on the following situation, where each x_i is independent of others: $$f=(x_1+x_2+..x_k)^N$$ I'm interested in the expression obtained after ...
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15 views

Tetrahedron symmetries

What are the order of lines of symmetry, plane of symmetry and rotational symmetry in a tetrahedron with its base is an equilateral triangle and other sides are all isosceles triangles?
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29 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 ...
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8 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 ...
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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 ...
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53 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, ...
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11 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 ...