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

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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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26 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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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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10 views

Proving that rotation-inversion axis n contains rotation axis n/2. [closed]

How to prove that rotation-inversion axis of n order contains rotation axis of n/2 order when n is even?
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34 views

How can we assume that the all parabolas in the form 'ax^2 + bx +c' are symmetrical?

So I was reading an answer to a question pertaining to the derivation of the line of symmetry. It reads as follows: The vertex occurs on the vertical line of symmetry, which is not affected by ...
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1answer
43 views

What do you call the property of a shape that looks the same for some rotations?

Specifically I mean a fragment of a sphere(e.g. for $x,y,z > 0$). It looks the same if you look at this from $(1,0,0)$ or $(0,1,0)$ or $(0,0,1)$. What do you call the property? I thought it would ...
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100 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 ...
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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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1answer
18 views

If $s$ and $t$ are symmetries of a plane such that they agree on three non collinear points then show that $s=t$

This is a problem based on "Symmetry" of the plane $\mathbb{R^2}$. Suppose $A$, $B$, $C$ are the three points in plane which are after the corresponding actions by $s$ and $t$ are in the places $D$, ...
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170 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 ...
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1answer
27 views

How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
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49 views

A symmetric group question. [closed]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
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Class equation question from Artin's book.

Let G be a group of order n that operates nontrivially on a set of order r. Prove that if n > r!, then G has a proper normal subgroup. Also I am not very clear of the term "operates nontrivially", ...
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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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1answer
32 views

Global Optimization, symmetric solutions

Does anyone have the idea to solve the global multivariate minimization problem as below? $$\text{minimizes}\quad (x_1x_2x_3+x_1x_4x_5+x_1x_6x_7+x_2x_4x_6+x_2x_5x_7+x_3x_4x_7)-(x_1+x_2+x_4+x_7) \\ ...
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2answers
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The class equation of the octahedral group

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$ I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the ...
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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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5answers
418 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 ...
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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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How to describe the degree of symmetry of an object? [closed]

A vast number of objects around us exhibit symmetry, but how to describe the degree of it? For example, is an equilateral triangle as symmetrical as a regular tetrahedron?
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1answer
46 views

Seeking Better (Symmetry Exploiting) Solution and Generalization of An Inequality

Given positive variable $(x,y,a,b)$ where $x+y=1$, how does one "slickly" prove the following inequality? $$f(x,y) := \frac{xa+yb}{\sqrt{xa^2+yb^2}}\ge \frac{2\sqrt{ab}}{a+b}.$$ or simply $$f(x,y) := ...
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1answer
40 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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1answer
36 views

How can point group symmetry operations be used to reduce the number of independent crystal properties?

How can point group lattice symmetry operations be applied to reduce the full second-rank elasticity tensor (in Voigt notation) from: to, for example, in the cubic case, this: A reference would ...
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1answer
20 views

Counting symmetries using elementary method

I am studying group theory using Armstrong's Groups and Symmetry, one of the biggest problem is that there is no solution manual available. Thus I will rely on you guys! Find all the rotational ...
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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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61 views

Why does symmetry happen in reset-based random walks?

Studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it with ...
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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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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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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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1answer
20 views

Finding the pattern in number of (next and up) nearest neighbours of $N$ points on a circle

The question is rather vague (I couldn't phrase it more clearly though) so let me explain. Say you have a circle, and you put four equidistant points on it. You then have two types of neighbours: ...
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65 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 ...
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1answer
26 views

List of two-sided wallpaper groups?

I'm interested in the symmetries of two-dimensional patterns that have two sides. In other words, what discrete groups can be formed from the three-dimensional Euclidean isometries which preserve a ...
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4answers
78 views

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not ...
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1answer
28 views

Symmetry and periodicity of ODEs solution

Consider a set of smooth ODE: $$(1)~~~~~~~~~\dot{x} = f(x)$$ with $x \in \mathbb{R}^n$ and $f : \mathbb{R}^n \to \mathbb{R}^n$. Consider also a linear transformaton $\gamma : \mathbb{R}^n \to ...
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43 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
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1answer
35 views

How are there 21 pairs are symmetric, but not the 36 pairs of a pair of dice..

In the book "Knowing the Odds - An Introduction to Probability" by John B. Walsh, pp.12-13 he states the following: first of all, Symmetry Principle says, Symmetry Principle. If two events are ...
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28 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
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Show $d(x,y) = d(y,x)$

If $$\mu (x,y) = \min\{n\in\mathbb{N} \ | \ x_n \not= y_n \}$$ and $$d(x,y) = \frac{1}{\mu(x,y)}$$ How can I show that $$d(x,y)=d(y,x)$$ For me it's pretty obvious, but I don't know how to ...
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35 views

Geometric proof that (symmetry w/r to $x$ and $y$ axes) $\implies$ (symmetry w/r to origin)

I'm trying to prove that reflecting a point about the x and y axes is equivalent to reflecting it about the origin. Is my proof valid? How could I improve it? Proof: Take a point $a$ in the first ...
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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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1answer
41 views

Is there a name for this kind of space?

Assume a Riemannian symmetric space $G/H$ where the decomposition of the Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$. It is a known fact that if $\mathfrak{h}$ is the Lie ...
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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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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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Problem on symmetric matrices

Let $A$ be square non-singular matrix of order $n \geq 2$. If $A$ is symmetric, then $A^2$ is symmetric positive definite. If $A^2$ is symmetric positive definite, then $A$ is symmetric. I think I ...
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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 ...
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855 views

Points of symmetry of tessellation.

I was given this irregular hexagon: Then I was told to tessellate it: Now, I am being asked to find all the points on the hexagon (first picture) which are points of symmetry of my tessellation ...
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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, ...
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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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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 ...
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73 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...