# Tagged Questions

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

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### Let $s$ denote reflection of the plane about the vertical axis $x=1$. Find an isometry $g$ such that $grg^{-1}=s$

I need help with the following problem: a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta$ Edit: ...
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### Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...
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0answers
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### Difference of general symmetric function that is non decreasing in its arguments

For a symmetric function $C(x,y)$ and $a,b,c,d \in [0,1]$ with $b\ge a$, $d \ge c$ and further, C(x,y) is non-decreasing in $x,y$. Then, does it hold that: $$C(b,d) -C(a,d) -C(b,c) + C(a,c) \ge 0$$ ...
1answer
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### Is it possible to represent a set of generic things coherently without implicitly creating an order on them?

Sorry if this question seems a little incoherent, I'm not certain of the proper terminology here. I've seen unordered sets represented in various ways, but it recently occurred to me that most of ...
2answers
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### Mathematics of chemistry with focus on particular symmetries

A student of mine came with a somewhat unusual request: My question about group theory and if you have a book you could recommend for some foundation: specifically chemical applications of group ...
2answers
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### Is there a bi-invariant metric (not Riemannian) on $GL_n$?

Is there a bi-invariant* metric $d$ on $GL_n$ (the group of invertible marices) which generates the standard topology on it? (the subspace topology from $\mathbb{R}^{n^2}$) Note: I mean any metric ...
0answers
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### Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
1answer
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### Is a shape formed of two tangents and radii symmetrical?

Is the kite formed by the two tangents and radii in this image symmetrical? Is there a law or reason why? I am assuming that the two tangents are of equal length, but I can't see why. Are any two ...
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### Understanding Cauchy-Schwarz and Holder's inequalities.

Although these inequalities occur in various settings, and I have used them to complete a number of proofs, I can not say that I intuitively understand what their significance is. Holder's ...
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### Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
2answers
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### A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
1answer
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### ODEs are invariant under the given Lie groups?

$\frac{dy}{dx} = \frac{x^{2}y}{x^{3}+xy+y^2}$ is invariant under $(x,y) \mapsto (\frac{x}{1+\varepsilon y},\frac{y}{1+\varepsilon y})$ I can't make both sides equal when I have a variable depends on ...
1answer
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### Efficient search for equilateral triangles, squares and regular pentagons in a set of 3D points

For an algorithm to identify cubic point groups from a set of atom positions $r_i$ forming a molecule, I would need an efficient and fast algorithm to identify equilateral triangles, squares and ...
1answer
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### Eigenvalues of a rotationally symmetric matrix

I have a rotationally symmetric matrix of arbitrary size, for example, A = \begin{pmatrix} a & b & c & b & a \\ b & d & e & d & b \\ c & e ...
1answer
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### Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
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### Prove that if $A$ is a symmetric matric then $A^3$ and $A^2-2A+I$ are symmetric matrices.

I am uncertain on how to approach this proof. For most everything I've encountered concerning symmetry, it has involved taking the transpose in order to show some property. Here, I'm not certain if ...
0answers
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### Symmetry of the multiplier of a l.c.s.c. abelian group

Let $G$ be a l.c.s.c. (locally compact, second countable) Abelian group, and let $\hat{G}$ be its (well-defined) dual. Consider the group $G\oplus\hat{G}$ (which is a l.c.s.c. Abelian group itself), ...
0answers
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### Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
1answer
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### Interesting $\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i$

I found that for m $\in N$ $$\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i.$$ I found it after doing an exercise. For example: $$5^{2}-4^{2}+3^{2}-2^{2}+1^{2} = 1 + 2 + 3 + 4 + 5 = 15.$$ For ...
1answer
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### How can you algebraically determine if a curve is symmetric about $y=-x$?

If I have a curve implicitly defined by say $x^2+xy+y^2=1$, then it is clear that it is symmetric about $y=x$ because if I interchange x's with y's, then I have the exact same equation. However, how ...
1answer
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### How to measure asymmetry of a function?

Let $f(x) = x^{2}$, so $f(x)$ is an upward symmetric parabola. It is a perfectly symmetric function since $f(x) = f(-x)$ for any value of $x$. Now, suppose $f$ is just some function. How would one ...
0answers
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### Can I find the dissections of a figure based on symmetry?

Our teacher gave us a figure, and challenged us to dissect into exactly 4 shapes, that were congruent, in as many ways as possible. I won't reveal details of the specific shape. I am wondering if ...
0answers
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### How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank (k,l) tensor as something that transforms like a tensor as follows: T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
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### How to measure the symmetry of the curve?

In statistics, we can measure the symmetry by skewness, but if we have a curve, in other words, if we have a list of x and y values, how to measure the symmetry of its plot.
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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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### 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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### 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 ...
1answer
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### 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 ...