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38 views

How to fold a rectangular sheet (with objects at each point) so that it folds into a tube? [closed]

For some reason I can't wrap my head around this. Say I have a flat sheet with atoms at each point as shown in the picture and I only know the coordinates (x,y,z) of each atom in the sheet. Now ...
0
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0answers
14 views

symmetry ratio and symmetry index

Can someone explain how to calculate "symmetry ratio" and "symmetry index" when the following points in the Cartesian plane is given ? ...
2
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0answers
24 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of DEs. I'm presently doing research on identifying financial indicators/phenomena that can be ...
1
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0answers
22 views

Characterizing a function regarding symmetry

Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists ...
3
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3answers
53 views

Definition of Equals

DISCLAIMER: This is a first time Math.SE post from a 30-something who is only now learning math. I have read the Rules and this may not satisfy the "Questions with too many answers" criteria or ...
2
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1answer
37 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
1
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2answers
25 views

Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
0
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1answer
16 views

Mollifying while conserving symmetries

Suppose $B = B(z,v)$ is a function in $L^1_\rm{loc}(\mathbb{R}^d \times S^{d-1})$ depending only on the values of $|z|$ and $|z \cdot v|$ (you don't make any assumptions on how $B$ depends on $|z|$ ...
2
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0answers
17 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
0
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1answer
26 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
2
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0answers
47 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
1
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2answers
44 views

Why must isometry of regular polygon fix origin?

Here is the question: Suppose $\varphi\colon\Bbb R^2\to\Bbb R^2$ is an isometry and $\varphi(\pi_n)=\pi_n$, where $\pi_n$ is the regular $n$-gon with center at origin. Why must $\varphi$ fix the ...
0
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0answers
24 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
2
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0answers
12 views

Ordering binary matrices for reflection/rotation

I have a collection of $n\times n$ binary matrices and I would like to reduce it for symmetry ($D_4$ -- reflections and rotations). The naive method of testing each pair is very slow because the ...
1
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2answers
724 views

Show groups of symmetries of a cube and a tetrahedron are not conjugate in isometry group.

I've shown that the symmetry group of a cube and a tetrahedron are both isomorphic to S4, but I am now trying to show that they are not conjugate when considered as subgroups of isometries of 3D ...
0
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3answers
343 views

center of symmetry formula

How to prove that $I(0,-1)$ is the center of symmetry of the function $$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?
4
votes
2answers
54 views

Intersection of two tetrahedra, point reflexion

We are given a regular tetrahedron $ABCD$ ($ABC$ is its` base and $D$ is its vertex) and we reflect it through the middle of its height (point reflexion) - and thus we obtain a congruent regular ...
1
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2answers
26 views

How to prove that $n$ is prime if an $n$-Venn diagram has $n$-fold rotational symmetry

I was reading this article on "The Search for Simple Symmetric Venn Diagrams" by Frank Ruskey, Carla D. Savage, and Stan Wagon and on the first page page they prove that $n$ is prime if an $n$-Venn ...
1
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1answer
22 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
1
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1answer
29 views

Rotation and Symmetries of Equilateral Triangle

Let $\Sigma=\{ 1,2,3 \}$ be the set of the vertices of an equilateral triangle. Let $f=\sigma$ be the rotation of level with center of rotation $O$ over an angle of $\frac{2 \pi}{3}$ radians or ...
1
vote
1answer
22 views

Symmetries of the set of points $S$

Consider that we have a set of points $S$ in the plane $\mathbb R^2$ or in the space $\mathbb R^3$ and we also consider the one to one mappings $f:S \to S$ which have the following property: They ...
0
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1answer
27 views

Conventional unit mesh

I'm trying to find and outline a non-primitive conventional unit mesh, I'm not sure how to go about it. I'd also like to find any mirrors of planes and rotional symmetry axes. Would this look ...
0
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1answer
50 views

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 ...
0
votes
2answers
40 views

Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
0
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0answers
35 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?
1
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2answers
23 views

Symmetry in Space

Is it possible for a non-co-planar set of points to be symmetric about a point but not symmetric about a plane? I am pretty sure this is true but I can't think of an example. Things that I think ...
4
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0answers
28 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
2
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1answer
60 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
2
votes
1answer
104 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
2
votes
2answers
38 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
0
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0answers
14 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
0
votes
1answer
48 views

Symmetry Definition and Equation [closed]

I need some help to understand Inversian Symmetry, Conformal Symmetry, and Scale Symmetry. Could you give me some guideline?
1
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1answer
28 views

Proving that the Moment Tensor is super-symmetric

The Carathéodory theorem in the image bellow is the one about convex hull, isn't it? Would you please explain why can the tensor F be rewritten as that sum? From that representation the author ...
1
vote
2answers
38 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
0
votes
1answer
85 views

What is the name of the symmetry of a bracelet transposition?

Take a bracelet with colored beads on it. Normally two bracelets belong to the same equivalence class under rotations and reflections. For an example, consider the bracelet denoted by the word ...
2
votes
0answers
152 views

What to call this kind of symmetry in a sphere?

Geometrically, if the two hemispheres of a spherical distribution of some kind (let's say a spherical gas cloud) are similar such that the properties of the gas as seen by a person standing on a ...
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3answers
239 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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5answers
83 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
1
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2answers
43 views

Understanding the shape of $\phi''(x)=F(\phi(x))$

Hello I've got a question and no idea to get a solution. Maybe someone can give me an advice. The following problem is given: There is given a function $\phi \in C^2([a,b])$. Furthermore there is a ...
0
votes
2answers
27 views

Create a new function using symmetry w.r.t a point

Let's say I have a function $f$ which is defined on an interval $[0,1]$ . How can I create a function $g$ defined on $[0,2]$ where: $f(x)=\sqrt{x^{a} - x^b}$ ,$0<a<b$ ,$a$ and $b$ are constants ...
0
votes
0answers
10 views

Name for symmetric irreflexive binary relation

I have an irreflexive relation $\prec$ called unpreference: if $x\prec y$ then I say $x$ is unpreferred (or not preferred) to $y$. I wish to give a name to the symmetric part of the relationship, ...
6
votes
1answer
80 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
2
votes
0answers
54 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
-1
votes
1answer
85 views

How to find the order of the group?

Translation: If $G$ is a finite group in which every element $g \in G$ satisfies $g^2 = e$, where $e$ is the unit element of $G$, then what are the possible values for the order $k=|G|$ of $G$? ...
1
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2answers
89 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
0
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1answer
81 views

Is the shortest path using Dijkstra's Algorithm symmetric?

I am writing a code for finding the shortest path using Dijkstra's algorithm for an unweighted graph. I am wondering if this shortest path is symmetric i.e. if the shortest path from say A->E is ...
8
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1answer
263 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
1
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2answers
114 views

The path to understanding Frieze Groups

What is the "Path" for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or "building blocks" approach to learning something. For example if I know how to ...
4
votes
1answer
54 views

What exactly is meant by symmetry?

This is something that has troubled me for long. Artin describes 4 types of symmetry: translational, rotational, reflective and glide. I somehow understand symmetry to be a "condition" in which a ...
2
votes
2answers
27 views

Symmetrical curve equation in range of [0,1]

I would like to define a function $f(x,a)$ that has the following properties: $f(0,a)=0$, excluding $a=\infty$ $f(1,a)=1$, excluding $a=-\infty$ $f(x,0)=x$ $f(0,a\to\infty)\to1$ $f(1,a\to-\infty)\to0$ ...