Questions tagged [symmetry]

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

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A square on the equator of a sphere is a critical point of the electrostatic potential

$\newcommand{\S}{\mathbb{S}^2}$ This is a self-answered question. I learned something from spelling out the details, and I hope this could be interesting to others. I would welcome alternative ...
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Is a planar square on the equator a locally energy minimizing configuration of electrons on $\mathbb{S}^2$?

$\newcommand{\S}{\mathbb{S}^2}$Let$$M=\{(x_1,x_2,x_3,x_4) \in \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Let $...
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How prove a parabola is symmetric without coordinates?

Some years ago a mathematician challenged a friend of mine to prove that a parabola is symmetric without using any coordinates. My friend got stumped, and challenged me to do this problem. I also got ...
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Structure of group of symmetries of a napkin-ring.

Trying to get a better understanding of dicyclic groups, I wonder what the structure of symmetries of a napkin-ring is. Suppose you paint a pattern on the ring that has $n$-fold rotational symmetry ...
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Are symmetry operations commutative or not?

My book (Bishop, David m. Group Theory and Chemistry) states, that $C_3C_3^{-1} = E$ and $C_3^{-1}C_3 = E$. After that it shows an non-commutative example: Ok. But then they say that $(PQ)R = P(QR) =...
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Space and time inversion of PDEs

I am unsure whether I am reversing the $z$ and $t$ dimensions in the following set of PDEs correctly, \begin{align} &\begin{aligned} \partial_{z} \mathcal{E}(z, t) = i \sqrt{d} P(z, t) ...
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A direct sum of Symmetric and Alternating Bilinearforms

Show that the vector space $\text{Bil}(V)$ of all bilinear forms on $V$ can be decomposed in to the direct sum of $\text{Bil}(V)_{\text{sym}} \bigoplus \text{Bil}(V)_{\text{alt}}$, where $\text{Bil}(V)...
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Is the variance of these two variables the same?

Let $X$ and $Y$ be random variables having joint density function $$ f(x,y) = \begin{cases} x + y & \text{for } 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0 & \text{other }x, 0 \leq y \leq 1 \end{...
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Can every tangent vector be realized as an acceleration of a path with a given velocity?

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v \in T_pM$. Define $$ \mathcal{A}_v:=\{ w \in T_pM\,|\,\exists\alpha:(-\epsilon,\epsilon) \to M, \, \, \alpha(0)=p, \...
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Realizing possible accelerations of paths on a sphere

$\newcommand{\al}{\alpha}$ Let $x,v \in \mathbb{S}^n \subseteq \mathbb{R}^{n+1}$, $w \in \mathbb{R}^{n+1}$ satisfy $\langle x,v \rangle=0, \langle x,w \rangle=-1$. Does there exist a smooth path $\...
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The barycenter of a regular tetrahedron coincides with the center of its circumsphere

This is a self-answered question, after some playing around. I would be happy to see alternative solutions. Let $x_1,x_2,x_3,x_4 \in \mathbb{R}^3$ be the vertices of a regular tetrahedron, i.e. $|x_i-...
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The segment from a vertex of a regular tetrahedron to the center of its circumsphere is orthogonal to the opposing face

This is a self-answered question, after some playing around. I would be happy to see alternative solutions. Let $x_1,x_2,x_3,x_4 \in \mathbb{R}^3$ be the vertices of a regular tetrahedron, which lie ...
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A segment having equal angles with two equal segments is perpendicular to the connecting line

Let $a,b,c \in \mathbb{R}^3$ be unit vectors, and suppose that the angles between $a,b$, and between $a,c$ are equal. Is there an elementary, geometric, computation-free proof that $a$ is ...
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The vertices of a tetrahedron lie on a sphere

I am struggling a bit with the following (elementary) question: How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie. I would ...
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Abstract: symmetry, reflexivity, and transitivity!

I need help with how to approach and start a problem regarding a relation $R$. Define a relation $R$ on sets as follows. $$\small R=\{(A,B):{A\text{ and }B\text{ are sets and there is a function }f:A\...
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Node level measure of "symmetry" in a graph

I looked at some of the other answers on the stack exchange but they didn't seem to answer what I was looking for, so let me elaborate. I have a large un-directed tree/graph where I need to extract ...
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Weak symmetrization inequalities

Let $X$ and $X'$ be iid random variables, $X^s=X-X'$ be the symmetrization of $X$, $mX$ be the median of $X$, i.e., the number satisfying $\Pr(X\geqslant mX)\geqslant \frac{1}{2}\leqslant \Pr(X\...
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When will a quotient space has finite choices

Consider $S^1: x^2+y^2=1$ and two symmetry, inversion and reflection. Inversion let $x\to-x,y\to-y$ and $x\to-x,y\to y$. The equivalence relation imposed by inversion and reflection are denoted as $\...
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Unique binary patterns on a Cube

Given a cube where each face can be individually colored either black or white, I know there should be $2^6$ ways to color the cube. How can I tell which cubes are duplicates of another cube but with ...
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Do $4$ balanced points on a sphere form a tetrahedron or lie on a plane?

Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$, and let $x_1,x_2,x_3,x_4 \in \mathbb{S}^2$. Suppose that $\sum_i x_i=0$, where we sum the vectors in $\mathbb{R}^3$. Question: Does one of the ...
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Must an inscribed quadrilateral whose vertices's centroid is the center of the circumscribed circle a rectangle?

Let $\mathbb{S}^1$ be the unit circle in $\mathbb{R}^2$. Let $x_1,x_2,x_3,x_4 \in \mathbb{S}^1$ and suppose that $\sum_i x_i=0$, where we sum the vectors $x_i$ in $\mathbb{R}^2$. Question: Do the $x_i$...
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Does Tetrahedron maximize the total squared distance between $4$ points on a sphere?

Let $x_1,x_2,x_3,x_4$ be points on the unit sphere $\mathbb{S}^2$, that maximizes the quantity $$ \sum_{i < j}\| x_i - x_j \|^2, $$ where $\| x_i - x_j \|$ denotes the Euclidean distance in $\...
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Probability of outcomes in numerology (history)

I was reading some books about the history of mathematics and there is always a chapter on Pythagoreans and their mysticism. One thing I did not know was that there was and still a process called ...
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Understanding permutations of $S_4$ which correspond to symmetries of a regular tetrahedron

I'm trying to understand values of the character of the representation of $S_4$ corresponding to the symmetries of a regular tetrahedron (whew, that's a mouthful!). One illustrative video is found ...
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Symmetric Functional Equation

I am trying to solve this functional equation: $$g(x)+g(1-x)=1$$ Over the reals. Unfortunately, since it's symmetric, usual tricks like trying to manipulate the expression in one of the functional ...
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Galois group roots symmetry and Parabolas symmetry lconnections?

Everyone knows that: quadratic parabola always has mirror geometric symmetry cubic parabola always has rotation geometric symmetry quartic parabola does not have in general (universal) geometric ...
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Asymmetric graphs where all adjacent vertices have different degree

I conjecture the following : If $G$ is a graph so that every pair of adjacent vertices have different degree, then $G$ is not asymmetric graph. I will remind that a graph is a asymmetric if there are ...
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Order of a non-square rectangle with vertices and edges

QUESTION: Suppose $B$ is a symmetry group of a non-square rectangle $R$ that has vertices $W, X, Y, Z$ as well as edges $h = WX, p = XY, k = YZ$ and $n = ZW$. And let $A$ be the set of all $2$-...
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Symmetry becomes invalid when n approaches infinity?

The question is: Let $f(n)$ and $g(n)$ be asymptotically non-negative functions. Judge whether the following statement is true or not. If it is true, prove it. Otherwise, give a counter-...
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3 votes
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Minimal symmetry group of polygon in $\mathbb{R}^2$

Is there a polygon in $\mathbb{R}^2$ whose symmetry group is isomorphic to $\mathbb{Z}\backslash 3\mathbb{Z}$? I believe I found such a polygon, it’s an equilateral triangle with $3$ smaller ...
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What are these 'partial' reflective invariants and equivariants of a multiary function called?

In the univariable case we say that a function is even if $f(x)=f(-x)$ and odd if $-f(x)=f(-x)$ for all $x$ in some space of interest. In the multiary case we would similarly consider a function to be ...
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From which object do these symmetries come from?

$(\mathbb{R}, +)$ is a group. But it’s not immediately clear where the symmetries are. I can answer this problem in two steps. First I can find a substitute group for $(\mathbb{R}, +)$, that will have ...
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Determine symmetry in a 'mostly' disordered (amorphous) 3D object

Symmetry groups have been the backbone of the success that the field of crystallography has achieved. Thinking about symmetry in similar lines as point or space groups, can we determine possible ...
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Is any closed subgroup of $SO(n)$ a rotation group of some compact subset of $\mathbb{R}^n$?

I proved this in dimensions $n = 1, 2, 3$ from the direct classification of subgroups. For $n \geq 4$ I can't understand anything. Also, I can't find any information about this issue. Is this a solved ...
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Proofing the identity of double integrals over symmetric function

In one of my physics classes we had given a double integral over a function $f$ in two variables: $$G=\int_{x\in D}\int_{y\in D}f(x,y)\ \mathrm{d}y\ \mathrm{d}x$$ The function $f$ is symmetric in its ...
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Symmetric boolean function in pairs

I am reading this paper "Efficient Algorithms for Computing Differential Properties of Addition". In Lemma2, the authors claim that $$\Delta_{c_{i+1}}=(x_i \land y_i) \oplus (x_i \land c_i) \...
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How to turn any group into a set of symmetries equipped with composition?

Let's say you have a group like $(\mathbb{R}, +)$, where it’s not immediately clear where the symmetries are (compared to a group $(S, \circ)$ which is a group of symmetries of some object equipped ...
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How can $(\mathbb{R}, +)$ be turned into a group of symmetries?

I'm exploring the claim that group theory is about symmetry. Given a group $G$ that isn’t a set of structure preserving maps with composition, can you always interpret it as some kind of symmetry? My ...
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Symmetry point of experimental data

I have the experimental data of some measurement as a function of angle. Since this is an experimental data, I have the measurements at every 1.8 degree. From the physics, I do know that the data ...
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Can someone explain why odd moments vanish for centered, symmetric-about-0 random variables?

(as in the title). I'm particularly interested in whether there's an intuitive way to understand this. I've done a handful of calculations with moments, but the concept is still a bit new to me.
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If X has a pdf $f(x)=2xe^{-x^2}$ $x>0$ Show that E(X)=$\frac{\sqrt{\pi}}{2}$

It says that you can use the symmetry of the normal standard distribution and that $ \int_{-\infty}^{\infty} \! \frac{1}{\sqrt{2\pi}}e^{-x^2/2} \, dx = 1$ to show that E(X)=$ \int_{-\infty}^{\infty} ...
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Is this sigmoid function point symmetric?

While the usual sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$ is symmetric around the origin, I'm curious as to whether this generalization of the sigmoid is point symmetric around $(\theta, 0.5)$:...
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Reflection = Rotation about a planar axis

For a Dihedral group $D_{2n}$, the operation of reflection about an axis, that is in the same plane as the polygon, can be considered as a rotation in $\mathbb{R}^{3}$. For e.g. for a square, the ...
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Finding the order of rotational symmetry about a given axis

Here is the only example my book gives about rotational symmetry about an axis: I am then asked to find the order of rotational symmetry for the following solids: I think (a) (b) and (c) all have an ...
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Quantum Graphs and the Gross-Pitaevskii equation

I'm a final year mathematical physics student and I've been tasked with researching into the gross-pitaevskii equation and it's solutions on quantum graphs. I have a pretty good understanding but I'm ...
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It makes sense to consider oddness with respect a point which is not $0$?

Let $F$ be a $C^1$ function and consider $$G(x) =ax- F(x+a),$$ with $a\in\mathbb{R}^*_+$. I need $G$ to be even. Clearly, if $F$ is odd in the "usual" sense, so it is $G$. But, actually, I ...
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Is there a 2D shape other than a circle that has infinite number of lines of symmetry? [duplicate]

A circle has an infinite number of lines of symmetry. Is there a 2d polygon that also meets this condition?
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For what function of $\theta$ is the tangent of the mean equal to the cosine of the half difference of values around $\theta=0$?

I would like to find a function $f(\theta)$ on the domain $\frac{-\pi}{2} \leqslant \theta \leqslant \frac{\pi}{2}$ for which $$ \tan \left( \frac{f(\theta) +f(-\theta))}{2}\right) = \cos \left( \frac{...
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Gaussian complexities with or without absolute value?

I am reading the paper Rademacher and Gaussian Complexities: Risk Bounds and Structural Results and I'm struggling with understanding the difference between the (empirical) Gaussian complexity $$ \hat{...
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Are there alternate geometric interpretations of the Riemann tensor based on its symmetries?

In Riemannian geometry, when we think of the Ricci tensor merely as $Ric(\vec a, \vec b)$, then it's hard to describe it other than saying it's the contraction of the Riemann tensor. But when we ...
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