Questions tagged [symmetry]
Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry
1,534
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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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Does every convex planar set contain a centrally symmetric subset with at least $2/3$ its area?
Let $S$ be a bounded convex subset of the plane of unit area. Can we guarantee the existence of a centrally-symmetric subset $C⊆ S$ of area $2/3$?
If $S$ is any triangle, this bound is tight, attained ...
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Why this definition for "symmetry transformation"?
This question concerns section 8.5.1 in these notes:
I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total derivative? ...
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When does a linear combination of trigonometric functions have an axis of symmetry?
I am trying to find out when a linear combination of $\sin(ax)$ and $\cos(bx)$ has an axis of symmetry.
Clearly, $\sin(x)+\cos(x)$ has an axis of symmetry at $\pi/4$. It seems as if $\sin(3 x)+\cos(...
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Is there a classical analog of Bloch's theorem?
In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect ...
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Are symmetric matrices necessarily positive-definite / positive semi-definite?
I am trying to prove this just to be clear about this but I don't have enough conditions to force this idea to be true, so I doubt it is.
Are symmetric matrices always at least positive semi-definite?...
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An expected value puzzle
While working on a larger problem, I encountered this smaller problem that I’ve enjoyed thinking about, but have yet to solve.
Shuffle the numbers 0 to 24 into a 5 by 5 matrix. Sort each column in ...
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On convex hulls of polyhedra and transitivity
It's easy to prove that the convex hull of any vertex-transitive polyhedron is vertex-transitive. Specifically, any symmetry of the original polyhedron that moves any vertex to another will also move ...
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Is the cube the largest polyhedron whose subsets are all symmetric?
Given a polyhedron $P$ with vertex set $V$ of size $n$, say that it is subset-symmetric if, for every subset $S\subseteq V$, there is a non-identity element of the full reflectional symmetry group of $...
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Which reflection groups contain central inversion?
Question:
Which finite irreducible reflection groups $\Gamma\subseteq\mathrm O(\Bbb R^d)$ contain the central inversion $-\mathrm{Id}$, and how can this be spotted from the Coxeter diagram?
The ...
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Is there a general theory of when certain polynomials are integrable due to symmetry tricks?
Consider the functions $x^2$ and $x^4 + 2x^2y^2$ on the unit sphere $S^2$.
The surface integral of these functions over the sphere can easily be calculated by symmetry via
$$3 \iint_{S^2} x^2 \mathrm{...
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Are all symmetries of the Dirichlet functional isometries?
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $E_{M,N}(f)=\int_M \|df\|^2 \operatorname{Vol}_g$.
(...
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Visualizing tuples $(a,b,x,y)$ of the extended Euclidean algorithm in a four-dimensional tesseract. Are there hidden symmetries?
I am trying to visualize the possible symmetries in the Euclidean four-dimensional space of the $4$-tuples of points $(a,b,x,y)$ generated by the extended Euclidean algorithm, where $ax+by=gcd(a,b)$. ...
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Can a L-shaped figure be divided into 5 congruent shapes?
Given a square, remove one quarter of it. Can the resulting L-shaped figure be divided into 5 congruent shapes? If not, how can we prove that fact?
I tried using circles, triangles and smaller ...
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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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Fastest way to show that $D_6 \to S_5$ is an injective homomorphism
I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this ...
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$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$
I'm having trouble with this question, I'd like someone to point me in the right direction.
let $A$ be a n by n matrix with real values.
show that there is another n by n real matrix $B$ such that $...
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Expanding Arcus Tangens as Taylor Series around 1: Symmetries
Expanding $\arctan$ around 0 as a Taylor series yields
$$
\arctan x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}
$$
for $x$'s that are in the region of convergence. All even terms vanish, which is ...
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Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric
I am stumped on a question and am looking for some guidance on how to get it done.
The problem gives you:
$x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$
$...
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Symmetric-decreasing rearrangement of a function
I'm studying section 3.3 of Analysis by Lieb and Loss, about symmetric-decreasing rearrangement of functions.
Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define
$A^*$ ...
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Why is reflection in a plane an automorphism?
I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism.
Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as
...
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Automorphisms of a structure as a powerful tool for studying the structure
This is just an arbitrary testimony of an often repeated slogan:
"The group of automorphisms of a given structure is often a powerful
tool for studying this structure." D. Lascar, On the Category ...
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Can we symmetrize connections on tangent bundles?
Let $M$ be a smooth manifold. Denote by $\mathcal{A}$ the space of all affine connections on $TM$, and by $\mathcal{S}$ the affine subspace of all the symmetric connections.
Is there a projection $...
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Does $|f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|$ imply $f$ is a logarithm?
Let $f:\mathbb R^+ \to \mathbb R$ be a continuous function, and suppose that
$$ |f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|, \tag{1}$$
holds for every $x,y \in \mathbb R^+$.
Suppose also that $f(1)=...
- 25.1k
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Symmetric Direct Product Distributive?
This comes from the context of chemical group theory, so I apologize if I'm using terminology incorrectly. For that context, see Determining the symmetry of overtones of degenerate modes on the chemSE....
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Dihedral groups: Relationship between symmetries and rigid motions
In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
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Symmetry vs isometry
In context of geometry and points in a plane Wikipedia describes symmetry as a type of invariance - the property that something does not change under a set of transformations.
Isn't isometry the ...
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Symmetry of point about a line in 3d
How would I go about finding the symmetrical image of a 3d point $t = (t_x,t_y,t_z)$, about a 3d line given with the equation $\frac{x+1}{4}=\frac{y+1}{-3}=\frac{z-15}{16}$?
Edit: To clarify: The ...
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Can reflections always be represented as rotations in higher dimensions?
If we think of reflection in $\mathbb{R}^1$ (multiplication by $-1$), this can be represented as $180$ degree rotation in euclidean plane (assuming a "natural embedding" notion of $\mathbb{R}...
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Show that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero
I need some help on showing that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero. $A$ is rectangular and can have dependent columns in general. I can show that it cannot have ...
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Connection Between Fundamental Solution and Symmetries of PDE
The typical derivation of the fundamental solution of Laplace's equation is to look for a radially symmetric solution because the Laplace equation has radial symmetry, and a similar heuristic can be ...
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Highly symmetric colorings of the sphere
If you're dropped onto a planet with an exact map of whatever's on the surface, then by comparing the map to your surroundings, you learn something about where you are (and which direction you're ...
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Proving that given a triangle $ABC$, $P$ is the intersection of $p,q$, $P$ is the midpoint of $EF$
Considering an acute triangle $ABC$, $E,F$ are the feet of the altitudes onto $BC$ and $AC$. $M$,$N$ are the midpoints of $BE$, $AF$. $p$ is perpendicular to $AC$ and passes through $M$. $q$ is ...
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Does Ricci flow preserve symmetries?
What does the following sentence mean: (see p.66 The Ricci Flow in Riemannian Geometry By Ben Andrews, Christopher Hopper)
The Ricci flow preserves any symmetries that are present in the initial ...
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Smoothness at the origin of a radial function obtained by rotating an even function
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth even function. Define $g:\mathbb{R}^n\to\mathbb{R}$ by $g(x)=f(|x|)$. How to show that $g$ is smooth at the origin?
We can calculate
$$\frac{\partial g}{\...
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Is every geodesic-preserving map of the sphere an isometry?
Let $\mathbb{S}^n$ be the $n$-dimensional unit sphere, equipped with the standard round Riemannian metric.
Let $f:\mathbb{S}^n \to \mathbb{S}^n$ be a diffeomorphism and suppose that for every (...
- 25.1k
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How to see the symmetry in this trigonometric equation
Consider the equation $2\cos^2x-\cos x-1=0$. We can factor the LHS to obtain: $$(2\cos x + 1)(\cos x-1)=0,$$ leading to three solutions in the interval $[0,2\pi)$, namely $x=0, \frac{2\pi}{3}, \frac{4\...
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Inversion in a circle as radius goes to infinity.
I am trying to show that the in the limit case as the circle gets very large, inversion in it is equivalent to reflection in a line. I have the transform $z \to c+ \frac{R^2}{ (\overline z -\overline ...
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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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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 ...
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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 ...
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What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]
I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
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Determining the symmetry group of an infinite horizontal line.
I believe I have a satisfactory answer to the following question:
Imagine we have a infinite horizontal line running through the origin, what is the associated symmetry group?
Now thinking ...
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Dissect a square into two contiguous congruent shapes. Must the dissection be rotationally symmetric?
Every dissection I can think of that cuts a square into two contiguous congruent shapes seems to be rotationally symmetric. (Allowing disjoint shapes allows for dissections that aren't.) Is there a ...
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If you throw a dice 5 times, what is the expected value of the square of the median?
My question: If you throw a dice 5 times, what is the expected value of the square of the median of the 5 results?
A slightly modified question would be: If you throw a dice 5 times, what is the ...
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The rotational symmetry groups of the $5$-cell and the icosahedron are isomorphic. Is there a geometric proof of this fact?
The rotational symmetry group of the $n$-simplex (not permitting reflections) is always the alternating group $A_{n+1}$. When $n=4$, this coincides with the rotational symmetry group of the ...
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Minimize $|a-1|^3+|b-1|^3$ with constant product $ab=s$
Let $0<s$, and define
$$ F(s):=\min_{a,b \in \mathbb{R}^+,ab=s} \left(|a-1|^3+|b-1|^3\right). $$
I would like to find proofs for the claim
$$
F(s)=\begin{cases}
1 - 3 s - 2s^{3/2}=F\big(a(s),b(s)\...
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Does COVID-19 fit into the Caspar-Klug (Quasi-Equivalence) Theory for virus architecture?
The following is compiled largely from my "Applications of Group Theory to Virology" module I took at The University of York as an undergraduate back in 2012.
The icosahedral group $I$ with ...
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Constructing a metric with discrete isometries on the sphere
Let $g$ be a Riemannian metric $\mathbb{RP}^n$, with no isometries except the identity.
Let $\pi:\mathbb{S}^n \to \mathbb{RP}^n$ be the natural projection, and consider the pullback metric $\pi^*g$ ...
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How do you prove you've found all symmetries of an object in 3D?
In group theory class we studied the example of rotational symmetries of the regular tetrahedon. The teacher showed us 12 symmetries and then said "if you stare long and hard you can convince yourself ...
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