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

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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 ...
0
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1answer
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, ...
0
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1answer
11 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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0answers
176 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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1answer
65 views

How to find where the magnitude of the gradient of a function is maximized?

How to find where the magnitude of the gradient of ${{e}^{-({{x}^{2}}+{{y}^{2}})}}$ maximizes? I managed to calculate the magnitude - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
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1answer
24 views

Finding all operators that preserve a function

I'm not even sure what field of math this would be, and Googling "symmetry" and "functions" doesn't reveal what I'm looking for. Basically I want to find all $\{\hat{A}\}$ other than the identity ...
3
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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 ...
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1answer
64 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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1answer
53 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
0
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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
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1answer
39 views

are isometries on euclidean plane translations

Let $m$ be an isometry in euclidean plane that changes orientation. Prove that $m \circ m$ is a translation.I do not have an idea how to start the proof of this exercise.
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189 views

Symmetric Properties of Roots (Quadratic Roots)

What is the proof that - $α^2$ + $β^2$ = $(α+β)^2$ - 2αβ $α^3$ + $β^3$ = $(α+β)^3$ - 3αβ(α+β) $α^4+β^4$ = ($α^3+β^3$)(α+β) - αβ($α^2+β^2$) (α+β)4 = α4 + 6α3β + $4α^2β^2$ + $6αβ^3$ + $β^4$ ...
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50 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 ...
1
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1answer
23 views

Fourier series Even vs. Odd and effect of integral bounds?

I understand that when you express a function in fourier series there are 3 coefficients you need to calculate ( a0, an, bn) and I have in the past made use of the symmetry of the function in my ...
0
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1answer
40 views

Combinatorial puzzle concerning labelled equilateral triangles

Consider equilateral triangles $\Delta$ of fixed size and in a fixed position with each side labelled by a label $l \in \{1,\dots,k\}$. Obviously there are $k^3$ such labelled triangles. Let ...
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1answer
47 views

What does it mean for vectors to be symmetrical?

Say we have three $n$-dimensional vectors $A$, $B$ and $C$. Now let's say that $A$ and $B$ are symmetrical about $C$. In two dimensions, this seems to have a fairly obvious meaning, but for higher ...
0
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1answer
27 views

Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
2
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1answer
57 views

Why is $D_5$ a subgroup of the icosahedral group

According to Wikipedia $D_5$ is a subgroup of the group of rotational symmetries of an icosahedron: http://en.wikipedia.org/wiki/Icosahedral_symmetry. I know this isn't very rigorous, but intuitively ...
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1answer
96 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 RH 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 ...
0
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1answer
32 views

Definition of symmetry

Let $X \subseteq R^2$. A symmetry of $X$ is isometry $f: R^2 \to R^2$ such that $f(X) = X$. For example, square has $8$ symmetries one of which is $R_{90}(a, b) = (-b, a)$. Is an element of $X$ ...
2
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1answer
41 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
2
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1answer
104 views

Maximum of a generalized Rayleigh quotient

Given two symmetric positive definite matrices $A,B\in\mathbb{R}^{n\times n}$ and $x\in\mathbb{R}^n$. How do I prove that the generalized Rayleigh quotient $R(A,B,x):=\dfrac{x\cdot A\cdot x}{x\cdot ...
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1answer
30 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
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1answer
148 views

Symmetric matrix if symmetric linear transformation

I want to proof the following theorem: With respect to any orthonormal basis, if the 2 $\times$ 2 matrix $\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ represents a ...
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122 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 ...
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0answers
24 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 ...
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1answer
185 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
3
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1answer
95 views

Fitting a grid inside a circle, is the solution always symmetric?

Let us construct a grid consisting of rectangles of height $h$ and width $w$. When we place a circle of radius $r$ over this grid, there is a certain amount of rectangles that completely lie within ...
2
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1answer
112 views

Hungarian Algorithm on Symmetric Matrix

I have a complete and weighted graph with an even number of vertices. I would like to separate all the vertices into pairs such that the sum of all the edge weights for each edge connecting the ...
0
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1answer
57 views

How can I compute angles & lengths of the following tiling shape

a while back I created a tile made of arrows: I did it using a vector graphics software, without really understanding the properties of this shape. Now, let's say I want to write a program to ...
2
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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 ...
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2answers
43 views

Proving a matrix is always symmetric [duplicate]

$B$ is a square matrix of real numbers. Show that the matrix $BB^T$ is always symmetric.
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91 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
2
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1answer
149 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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1answer
39 views

Symmetry of polar equations

In your opinion how to show symmetry in polar equations without graphing. i thought of these methods :- converting to cartesian then test . check the period of the function . please help me any ...
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29 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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0answers
41 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 ...
0
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2answers
70 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
3
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1answer
53 views

Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = ...
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2answers
50 views

How to make an expression manifestly symmetric

Believe it or not, the following expression is symmetric under the exchange of the indices $j$ and $k$, i.e. $R_{kj}=R_{jk}$: $$ R_{jk}=j s_js_k-\sum_{n=1}^{\min(N-k,j)}(k-j+2n)s_{j-n}s_{k+n} $$ Where ...
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1answer
169 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 ...
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55 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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1answer
31 views

Reading the sphere diagrams in point groups on wikipedia

How do you read/make sense of the sphere diagrams shown here: http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups What do the yellow shaded areas represent? What are the red ...
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1answer
71 views

Visualizing Platonic Solid group symmetries

How do you visualize the rotation symmetries, to classify a icosahedron for example as Ih, H3, [5,3], (*532)
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17 views

For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
5
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3answers
289 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 ...
3
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1answer
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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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?
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201 views

Equivalence Graphs

On the basis of this definition: Two graphs are equivalent if they have the same set of edges (ex. (A,B),(A,C)) how would you determine equivalence for graphs that are not labelled: ex.
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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 ...