# Tagged Questions

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

36 views

### Express rotation of the plane about a point as the product of a translation and a rotation about the origin

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: ...
45 views

37 views

### 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 ...
19 views

### 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: $$C(b,d) -C(a,d) -C(b,c) + C(a,c) \ge 0$$ ...
41 views

### 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 ...
42 views

### 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 ...
19 views

### 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 ...
546 views

### What is a “unique” mirror line of symmetry?

What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
26 views

### 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 ...
41 views

### 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 ...
8 views

### Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
399 views

### Systematic solution to my soccer ball puzzle

I once received a puzzle that can be described as follows: There are $12$ black pentagonal and $20$ white hexagonal pieces. The goal is to form a soccer ball from these (aka. truncated icosahedron). ...
34 views

### 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 ...
41 views

### 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 ...
28 views

### 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 ...
33 views

### 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 ...
47 views

### 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 ...
38 views

### 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 ...
34 views

25 views

### 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 ...
8 views

### 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), ...
6k views

### Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for ...
111 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 ...
48 views

### 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 ...
36 views

### 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 ...
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 ...