All Questions
4,827 questions
4
votes
1
answer
488
views
Are there non-tiling polyhedra that pack arbitrarily well?
The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...
- 131
49
votes
4
answers
12k
views
Volumes of n-balls: what is so special about n=5?
I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for.
The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
- 22.3k
2
votes
1
answer
802
views
Points on circumsphere of n-simplex
Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the center. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots ...
- 29
8
votes
1
answer
673
views
Estimating the Volume of the Metric Polytope
A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}_+^{n \choose 2}$.
For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x_{i,j}$. The set of all metrics is ...
- 794
2
votes
1
answer
1k
views
Sphere packing in a sphere
Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number ...
- 3,217
2
votes
0
answers
261
views
Existence of partitions of $S^{n-1}$ with hypercubes
For which value of the integer $n$ does there exist a partition of $S^{n-1}$, the unit sphere of $\mathbb{R}^n$ for the euclidean norm, by a family of images of the standard hypercube $C=\{ (e_1, ..., ...
10
votes
2
answers
689
views
Do elongated convex objects all have long simple geodesics?
Let $S$ be a closed convex surface, the boundary of a compact
convex body in $\mathbb{R}^3$.
I am interested in whether there are conditions on its shape
that ensure that it supports a long, simple (...
- 151k
5
votes
1
answer
636
views
analogues of Cayley plane as homogenous spaces
The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach ...
- 8,597
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
- 151k
6
votes
1
answer
2k
views
Continuous bijective way of representing a line on a plane
Is there a function $f(a,b)$ which maps ordered pairs to lines in a plane in a continuous, bijective manner?
Here is the definition I am using for the limit with lines: a sequence of lines $L(1), L(...
- 435
40
votes
7
answers
15k
views
How might M.C. Escher have designed his patterns?
I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...
- 461
11
votes
2
answers
1k
views
Floating polyhedra with fair equilibria
Is there a homogeneous convex polyhedron
which floats so that some subset (perhaps all) of its faces
is distinguished as "up" (above the water line)
in stable equilibrium, each face with equal ...
- 151k
5
votes
3
answers
548
views
Quadrics containing many points in special position
Suppose $n$ quadric hypersurfaces cut
out $2^n$ distinct points
$p_1,\ldots,p_{2^n}$ in
$\mathbb{P}^n$. What is the maximal
number of points $p_i$ a quadric can
contain without containing ...
- 11.6k
0
votes
2
answers
251
views
3d width and cross section
Greetings,
We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We ...
- 3
21
votes
2
answers
1k
views
Forbidden mirror sequences
Let $\cal{M}$ be a finite collection of two-sided mirrors,
each an open unit-length segment in $\mathbb{R^2}$,
and such that the segments when closed are disjoint.
A ray of light that reflects off the ...
- 151k
13
votes
5
answers
2k
views
Is there a complete classification of constant mean curvature surfaces?
I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and ...
- 325
0
votes
2
answers
1k
views
Degenerate case of linear programming duality?
Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize $0\vec{...
- 2,019
3
votes
1
answer
697
views
Which motion is exclusive in 3D or higher dimensions?
Hi guys,
I have a simple question
Linear movement can be found in 1D, 2D and 3D world objects
Rotation can be found in 2D and 3D world objects.
Now, are there any kind of motion can only be found ...
- 101
88
votes
2
answers
7k
views
8
votes
2
answers
383
views
Do singular values of a point set determine its shape?
Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind ...
- 2,242
6
votes
4
answers
752
views
Algorithm for k-medians in a convex polygon
Are there any known approximation algorithms or exact solution schemes for the k-medians problem in a convex polygon? That is, placing a collection of points $p_1,\dots,p_k \subset \mathbb{R}^2$ in a ...
16
votes
1
answer
883
views
Hearing the 17 planar symmetry groups
Though I'm sure it's not really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups. ...
- 17.6k
1
vote
0
answers
146
views
Are spherical codes algebraic?
Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
- 11
15
votes
3
answers
9k
views
$n$-dimensional Voronoi diagram
I need to compute the Voronoi diagram of a set of points in $R^n$.
I'm quite unschooled on the topic, could someone point me to the right references so that I can
a) understand the theory behind it;
b)...
- 253
7
votes
1
answer
686
views
Regular simplex in projective space
Is there a reference or a very short argument proving the following statement?
Let $C$ be a set consisting of $r$ points in the real projective space $\mathbb RP ^k$
with its usual round metric. ...
14
votes
7
answers
3k
views
Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
- 17.6k
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
1
vote
3
answers
688
views
How to show the cardinality of nonisometric compact metric spaces is the continuum
It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that
there can be no more than continuum of mutually nonisometric compact spaces
How is this proven?
Its clear that there ...
- 7,197
17
votes
4
answers
823
views
Sweep-segment bot: Will this random walk sweep the plane?
This model is inspired by the random behavior of the
Roomba sweeping robot.
Let a unit segment $ab$ in the plane be placed
initially with $a=(0,0)$ and $b=(1,0)$.
The segment is first rotated a ...
- 151k
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 8,512
5
votes
1
answer
750
views
Geodesic rays and horofunctions
Let $(X,d)$ be a metric space.
Let $(x_n)_{n\in\mathbb N}$ be a geodesic ray: $d(x_n,x_m)=\vert n-m\vert$.
Is it true that, for all $y\in X$, the sequence $(d(x_{n+1},y)-d(x_n,y))$ converges to $...
- 815
5
votes
2
answers
625
views
Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model?
I posted this question at math.stackexchange.com but didn't get an answer.
Motivation
Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with a &...
- 9,720
7
votes
2
answers
611
views
Escher, Conway, Kali, etc.
One can express the symmetry types of, say, Escher's "Circle Limit" prints using
Conway's orbifold notation, best known in the context of symmetries of Euclidean
plane patterns.
For example, Circle ...
- 17.6k
11
votes
1
answer
506
views
"minimal" embedding of bipartite graphs on a sphere
Here is an easy to pose problem I've encountered (but haven't been able to solve or disprove):
Let (V,E) be a bipartite graph with the following property –
the girth of the graph (i.e. the length of ...
- 1,163
3
votes
1
answer
236
views
Non-inherited symmetries of shadows of point sets
Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...
- 1,277
15
votes
3
answers
7k
views
A metric for Grassmannians
I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for ...
user11178
5
votes
1
answer
2k
views
Is Riemannian distance function equivalent to Euclidean one?
Consider the Riemannian manifold $\mathbb{R}^n$ and a smooth Riemannian metric $G:\mathbb{R}^n\rightarrow\mathbb{R}^{n\times{n}}$. We know that if $w_1I_n\leq{G}(x)\leq{w_2}I_n$ for some $w_1,w_2\in\...
- 101
10
votes
1
answer
1k
views
Characterizations of Euclidean space
I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...
- 9,720
4
votes
1
answer
377
views
Discrete gradient ascent cycles
I am wondering what can be inferred when a discrete
gradient ascent algorithm gets stuck in a cycle.
Here is the situation.
A function $f(x,y)$ is defined over a range $[0,n]^2$,
and the algorithm ...
- 151k
45
votes
1
answer
3k
views
two tetrahedra in $\mathbb R^4$
It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $\mathbb R^3$,
such that their union has diameter $1,$ then they must share a vertex.
I wonder whether ...
- 1,359
15
votes
3
answers
2k
views
orientations for zero-dimensional manifolds
I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...
- 6,224
1
vote
2
answers
1k
views
Is there always a parallelogram cross-section of parallelepiped contained in the smallest box
Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...
9
votes
6
answers
2k
views
Classification of surfaces composed of circles
Define a circle as a geometric circle of positive, finite radius:
a set of points in $\mathbb{R}^3$ congruent to the
set $x^2 + y^2 = r^2$ in the $xy$-plane. [Edited as per BMann's comment.]
I am ...
- 151k
1
vote
1
answer
273
views
Alexandrov's theorem analogue for Galilean kinematics
Let $\mathbb R^4_A$, $\mathbb R^4_B$ be spacetime as seen by two inertial observers $A$, $B$ respectively, and call $f:\mathbb R^4_A \to \mathbb R^4_B$ the change of coordinates.
We assume that $f$ ...
- 123
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
- 33
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
- 4,922
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
- 744
1
vote
1
answer
541
views
What is the definition of product of ideal sheaves?
Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I
is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right?
Thanks.
- 873
4
votes
1
answer
736
views
In search for isotropic graphs: Straight lines and parallels
I wonder why I can find only so little attempts of concisely defining "directions" and "isotropy" of graphs.
In Euclidean spaces "directions" can be identified with ...
- 9,720
52
votes
3
answers
5k
views
Is the "Napkin conjecture" open? (origami)
The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
- 1,306