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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Find plane of symmetry (knowing reflection positions)
I have a set of 3 distincts points which I know the positions in 3D (say D1, D2, D3).
I also know they are the plane-symmetry of 3 other distinct points in 3D (say B1, B2, B3) which I also know the ...
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Commutativity and symmetry of equalities
Is there any mathematical context, useful or not, in which a=b does not imply b=a? If yes and if it is useful please explain how.
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Generalised what?? CIrcle or line or ???..
What is the term for all objects that share the following property
For any two different points in the shape the perpendicular bisector is an axis of symmetry of the shape.
So for 2 dimensional ...
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Symmetry-invariant conservation laws using Multi-reduction [closed]
I am studying the article "Symmetry Multi-reduction Method for Partial Differential equations with Conservation Laws" by Anco and Gandarias and I am using Maple to code the examples in this ...
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How to perform Symmetrization?
Let $f:\mathbb{R}^n\to\mathbb{R}$ be an asymmetric function with respect to its variables. How can I transform it to become symmetric?
For example, when $n=2$, take $f(x,y)=x+y^2$. Pairwise adding ...
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Linear combination of tensor products of Pauli matrices that produce +/- paired eigenvalues
Eigenvalues of tensor products of Pauli matrices $\sigma_\mu\otimes\sigma_\nu (\mu,\nu\in\{x,y,z\})$ always come in pairs, as a consequence of the fact that eigenvalues of Pauli matrices are $\pm 1$. ...
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How many identical building blocks can make an icosahedrally symmetrical structure?
I'm working on an applied problem that requires designing symmetrical structures. I'd like to form an arrangement with full icosahedral symmetry (point group Ih in 3D) out of building blocks with Cs ...
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Symmetry group of rectangular cuboid
In this Wikipedia page it is said that the symmetry group of the rectangular cuboid (a box with three unequal dimensions) is a dihedral group $D_8$ (well, some people may call it $D_4$...)
However, I ...
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Does convergence along a diagonal grid imply convergence along other paths going to infinity? [closed]
Suppose that for each natural $k$, we have a sequence of real numbers $a_{n,k}$, and that $\lim_{n \to \infty} a_{n,k}=\alpha \in \mathbb{R}$ is independent of $k$.
Suppose furthermore that $\lim_{n \...
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Justification for the definition of relative error, why is it not a metric?
The absolute error and relative error operators are very commonly encountered while reading about topics from the fields of floating-point arithmetics or approximation theory.
Absolute error is
${ae(a,...
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Symmetric mollifier on cube instead of ball
It is known that there exists a symmetric, positive mollifier $\phi\in C^\infty_0(B_1(0))$, with $B_1(0)\subset\mathbb{R}^n$ the open unit ball. As far as I know, there also exists some positive ...
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Is there a finite group which is not the symmetry group of any n-space?
This is a natural follow-up to my previous question, here: A finite group that is not a symmetry group. An $n$-space symmetry is an isometry of $\mathbb{R}^n$. Given a subset $S$ of $\mathbb{R}^n$, ...
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How to generate evenly spaced rotations in SO(3) and/or a regular grid over $\mathbb{S}^3$ (to be used to divide the space of unitary quaternions)?
I am trying to split the space of $\mathrm{SO}(3)$ into spaces $(S_i)_{i\in\{1, \cdots, n\}}$ of rotations providing a regular paving of $\mathrm{SO}(3)$.
At least I would need that $\bigcup_{i=1}^n ...
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Is the MVT true for $b<a$ (instead of $a<b$) ? Is this adapted absolute value version true?
The Mean Value Theorem for $f\in \mathbb R ^{\mathbb R}$, is stated only for $a < b$ as:
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists $x\in (a,b)$ such as: $$f(b)...
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Tifr gs 2021: symmetry of coloring chessboard under rotation [duplicate]
Let $\mathcal{C}$ denote the set of colorings of an $8\times 8$ chessboard, where each square is coloredeither black or white. Let $\sim$ denote the equivalence relation on $\mathcal{C}$ defined as ...
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Could the word "symmetry" represent different things in different contexts? (naive question)
I just wanted to bring up some discussion about an apparently essential concept for some fields in mathematics as so as for some in physics, as already mentioned in the title, I'm referring to the ...
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The actual $384$ matrices corresponding to the symmetries of a Tesseract
A tesseract has $384$ symmetries. How do I get the actual $384$ matrices corresponding to them?
My attempt: The Tesseract has $8$ hyper-faces (which are regular 3-d cubes). One can rotate it so that ...
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Which manifolds admit symmetric tilings?
Let $M$ be a connected, boundaryless manifold. A tiling of $M$ is a regular cellulation of $M$, and a tiling is regular (i.e. symmetric) if the group of self-homeomorphisms of $M$ that map cells to ...
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Can every inscribable 3-polytope be circumscribed while still preserving the maximum symmetry of its canonical form?
According to Steinitz's theorem, it is possible to prove that every 3-connected graph G(P) represents the skeleton of a convex 3-polytope P, which can be realized in R3 from its skeleton G(P). Various ...
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What function $f(x) = f(1/x)$ has limits $\lim_{x \to 0} f(x) = 0, \lim_{x \to \infty} f(x) = 0$, and $\lim_{x \to 1} f(x) = \infty$?
What (simple) function has the following properties:
It maps the positive reals (except for 1) to positive reals: $f: (0,1) \cup (1, +\infty) \to (0, +\infty)$.
It is symmetric about reciprocals: $...
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Do solids retain their symmetry after uniform truncation?
I was looking at platonic solids and Archimedean solids. I observed that the point group of the solids still the same after uniform truncation. Examples of this is that Cube, truncated cube, ...
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3D vector representation of $H_2O$ molecule?
I'm struggling with a group theory question on the symmetries of the water molecule. First off, I'm giving the following image of the molecule in question:
Now, I've constructed the character table ...
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Symmetries of the dihedral groups $D_n$ and $D_{nh}$.
According to my lecture book (Linear Representations of Finite Groups by Serre), the dihedral group $D_n$ consists of $n$ rotations and $n$ reflections in the plane that preserve a regular polygon ...
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On Bishop's 'Tensor Analysis on Manifolds' Problem 5.10.6 (definition of Ricci tensor)
Here the complete problem statement (slightly modified):
The Ricci tensor $R_{ij} \mathrm d x^i \otimes \mathrm d x^j$ of a connexion $D$ on a manifold is the tensor of type (0, 2) obtained by ...
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Symmetry group of an hourglass shape, the molecule Ferrocene.
I am starting to learn (Visual) Group Theory and I saw this hourglass shaped molecule Ferrocene:
I am wondering what group is described by its symmetries. On top of the five rotations I think there ...
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The set of unique 2D shapes generated by 90º rotation and reflection
In my project to write a reinforcement learning agent to solve the puzzle piece game Kanoodle, I found myself needing to determine how many (and which) unique shape variants can be generated by any ...
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Translation of box type ball to center it in a convex set but maintaining the covering properties
I have the following problem. Suppose to have a cube $Q$ and a symmetric box type ball $B$ whose center is outside Q, but such that $Q\cap B \ne\emptyset$. Furthermore $B$ is not oriented like $Q$. I ...
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How to detect if a function has symmetries
let's say we have the function
$$y = \frac{x}{x^2+1}$$
we see that y':
$$y' =\frac{-x^2+1}{(x^2+1)^2}$$
by the second derivative test, we see that the points $x=1$ is a local maximum and $x=-1$ is a ...
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What do orders of group elements have to do with stabilizers?
In our group theory course, we are learning about stabilizers, applied to solid geometric bodies. I do not fully understand what stabilizers are for solid bodies, how you can find them and why the ...
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Degree of $\zeta(3)$ as a period
Consider the following integral representation of $\zeta(3)$:
$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xyz}dxdydz$$
The integrand is a rational function of $d$ variables and the domain of ...
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Prove that a rotationally invariant function depends only on the length and relative angles of its arguments.
I have a function $f(v_{1}, v_{2}): (\mathbb{R}^{3} \times \mathbb{R}^{3}) \rightarrow \mathbb{R}$ which maps two vectors in 3D space to a scalar. I know that this function is invariant under ...
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Symmetrical counting problem
A circle is divided into three sectors: $S_1,S_2,S_3$. The central angle of each sector, in degrees, is a nonnegative integer. What is the number of distinct circles that can be made if:
$S_1$ is ...
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Count the number of unique permutations of this puzzle
I have a puzzle that has 9 square slots. It looks like a rubik's cube face. In each square (except the middle one) you can put a coin in the middle.
The goal is to count the number of unique puzzles ...
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Symmetric sub-planes in multidimensional case?
Consider the following function:
$$f(c_1,c_2) = \min((y-c_1)^2, (y-c_2)^2),$$
where $y$ is a fixed value, while $c_1, c_2$ are scalar arguments. The function is symmetric along $c_1 = c_2$ plane. The ...
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Reflective/Inversion Symmetry of Triangle of Powers
I was fascinated by this 3b1b video about the Triangle of Powers and was tinkering around with it myself. Specifically, I noticed (after the timestamped part) that if I reflect the triangle while ...
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What is the minimum number of spheres required to create a larger approximately sphere-like structure having symmetry? [closed]
Suppose you are asked to create a larger sphere using smaller spheres. You can create only something that can be just contained inside a large sphere. If the smaller spheres are closely packed and ...
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What do the dotted lines represent in the hexagonal lattice system?
The above is one of the four two-dimensional lattice systems. See this article for some info.
The parallelogram is the primitive unit cell, because it can be translated in order to recreate the entire ...
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Closed form for matrix with interesting pattern.
I have the following matrices $M_n$ with a pattern:
(shown $M_{25}$)
\begin{pmatrix}
0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&...
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2D and 3D finite rotation groups are very well behaved, what about 4D?
I've recently read up on 2D and 3D finite rotation groups. But I'm struggling to find resources on the analogous results for 4D and I'd appreciate some help finding answers/resources.
For context: In ...
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Could you provide me with an example of a group which has elements $a, b$ and $c$ such that the order of $abc$ is different from the order of $acb$? [closed]
I have tried messing with different symmetries and permutations but I am definitely stuck. What is the catch? Do I need to consider shapes that are more than 2 dimensional? Do I need to consider ...
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Let $K/F$ be a Galois entension, and $Gal(K/F)=S_n$. Prove K is a split field of n degree irreducible polynomial over F. [duplicate]
Here are some of my thoughts.
Use group theory to show contradiction.
Suppose K is a split field of an irreducible polynomial $f$, $deg f=m>n$. Then $S_n$ isomorphic to a transitive subgroup of $...
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Getting symmetry of a function graph
Consider we have an function $f(x)$ and it has a inverse relation and consider that function starts from a point somewhere else in $+x$ axis. When we want to draw its $f^{-1}(x)$ (inverse function) we ...
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Non-symmetrical relation: a mistake while asserting "not implies" with increasing levels of "strength"?
In this section of video from a book group I'm following we consider the lack of symmetry of the relation $\lt$ on $\mathbb{R}$. The condition for symmetry, which does not hold, would look like this
$$...
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Function with same symmetry as planar hexagonal lattice
I'm currently trying to define a function which has the same symmetry as a planar hexagonal lattice. For a quadratic lattice the solution $f(x,y) = \cos(x) + \cos(y)$ is quite simple but I wasn't able ...
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isotropy of the cotangent lift of a group action
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the phase space (i.e. coordinates and momentum), which is the appropriate cotangent bundle of the ...
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Justification for a step in the derivative transform of infinitesimal transformation
When I was looking through p23-24 of Symmetry Analysis of Differential Equations by Daniel J. Arrigo, I wasn’t sure why we can do this:
$$\frac{\frac{dy}{dx} + [Y_x+Y_yy’]\epsilon+O(\epsilon^2) }{1+[...
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Exploit the symmetry of a probability vector
I have a probability distribution over $n$-bit strings $p_{X^n}$. This is the optimal input that achieves the capacity of $W_{Y|X}^{\otimes n}$, which is $n$ iid copies of a channel $W_{Y|X}$ acting ...
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Is there an explanation for this symmetry of the absolute value metric?
Consider a metric space $(\mathcal{X}, d)$ and four points $x, y, z, u \in \mathcal{X}$.
When $\mathcal{X} = \mathbb{R}$ and $d(x,x') = |x - x'|$, then the metric has the following interesting ...
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action of cyclic group $n$ on $2n$ elements
Let an 8-set $X$ represent the vertices of a cube: {$1,2,3,4$} is the label assigned to the four vertices at the "top"
and {$5,6,7,8$} is the label assigned to the four vertices at the "...
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Why don't these hexagons tile seamlessly-- without space in between? Where in my math have I gone silly?
Intro to the Problem
Hi there!
I've been trying to seamlessly tile hexagons on an XY plane-- without spaces in between.
I've currently accomplished this, however:
which as you can tell, isn't ...
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