Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
- 13.6k
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5
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21k
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Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
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3
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What are the matrices preserving the $\ell^1$-norm?
So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
- 391
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2
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952
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What is the largest possible thirteenth kissing sphere?
It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...
- 626
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Are there space filling curves for the Hilbert cube?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...
- 11.1k
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4
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825
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How much redundancy resides in an $n \times n$ orthogonal matrix?
Suppose one has an $n \times n$ orthogonal matrix $M$:
$$
\left(
\begin{array}{ccc}
0.239326 & 0.846726 &
0.475161 \\
0.768893 & 0.13356 &
-0.625272 \\
0.592897 & -0....
- 151k
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5
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1k
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Lightray trapped between two mirror disks: Computation formulation?
I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...
- 151k
19
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3
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931
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Is the circle in the square best at avoiding random lines?
This question is inspired by a recent one (and takes a great deal from the answers there). Given a convex subset $\Delta$ of the unit square, let $p(\Delta)$ be the probability that a random line does ...
- 30.1k
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3
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2k
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Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
- 151k
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Can every simple polytope be inscribed in a sphere?
It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...
- 13.6k
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2
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569
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Repeated random two-steps in $\mathbb{R}^3$: unbounded?
I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...
- 151k
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3
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725
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Curvature of a finite metric space
I am sorry to ask a very vague question, but:
What are good ways to define the curvature of a finite metric space?
The best way I can think of is: the curvature of a finite metric space $M$
is the ...
- 26k
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3
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Towards a metric characterization of Euclidean spaces
I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry.
Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each ...
- 1,987
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1k
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Four Dimensional Origami Axioms
What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
- 193
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1k
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Intercept the missile
A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal .
...
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806
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Simple connectedness under a metric undistortion condition: on a tricky point in an argument of Gromov
The context
I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following.
Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its ...
- 3,669
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564
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Measure-preserving maps from the square to the cube
There is a measure preserving map from the unit interval onto the unit cube that is Lipschitz of order 1/2, that is $|f(x)-f(y)| \leq A |x-y|^{1/2}$. By considering the image of small intervals, one ...
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2k
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Does this Banach manifold admit a Riemannian metric?
First, the question; after, the motivation.
Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
- 7,831
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1
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448
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Precise estimate for probability an $n$-point set has diameter smaller than $1$
This question was inspired by an earlier question that I answered but would like a more precise bound for.
Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
- 148k
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841
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I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
- 5,215
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577
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"Japanese Theorem" on cyclic polygons: Higher-dimensional generalizations?
A beautiful theorem known as the Japanese Theorem (Wikipedia, MathWorld)
says that, no matter how one triangulates a cyclic (inscribed in a circle) polygon,
the sum of the radii of the incircles is ...
- 151k
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5
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2k
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Definition of area
I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
- 45k
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12
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10k
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Theorems in Euclidean geometry with attractive proofs using more advanced methods
The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...
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2
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1k
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Example of a compact homogeneous metric space which is not a manifold
A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
- 11.4k
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2
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2k
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Integrating over a hypercube, not a hypersphere
Denote $\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube.
It is all too familiar that $\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$.
...
- 43.2k
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2k
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Assistance with understanding parent/child relationships in Pythagorean Triples
I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained elsewhere - ...
- 283
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Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
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3
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405
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Tilting the $d$-cube to vertically separate its vertices
Let $C_d$ be a unit edge-length cube in $d$ dimensions.
I would like to orient it ("tilt" it) so that the vertical (last) coordinates
of its $2^d$ vertices are maximally separated, in the sense
that ...
- 151k
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3
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3k
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Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
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4k
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Turning pants inside-out (or backwards) while tied together
An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this ...
- 32.1k
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641
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Can all convex polytopes be realized with vertices on surface of convex body?
The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited ...
- 907
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627
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Construction of an optimal electron cage
I will describe the question first in 2D, but my interest is in $\mathbb{R}^3$.
An electron $x$ will shoot from the origin along an initial vector $v$. You know the speed $|v|$ but not the direction.
...
- 151k
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2
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667
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Total length of a set with the same projections as a square
Take some convex polygon $P$. I'm mostly asking about the unit square, but would also appreciate thoughts on general polygons. We want to take a family of line segments inside $P$ that have the same ...
- 1,160
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2
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700
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Can all unit-distance graphs have their vertices at algebraic integers?
A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...
- 12.4k
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2
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986
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"Derived" polyhedra and polytopes
The notion of derived polygon is natural and leads to remarkable convergence.
Start with a polygon, and replace it by locating a point on every edge
a fraction $\alpha$ between the two endpoints. For ...
- 151k
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2
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602
views
Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?
What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...
- 458
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1k
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An ellipse through 12 points related to Golden ratio
I am looking for a proof of the problem as follows:
Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$,...
- 1,776
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Does equality of Laplacians imply Kähler?
This question follows on from this one.
Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...
- 19.3k
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644
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Egalitarian measures
A question I got asked I while ago:
If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
- 47.3k
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The group of isometries of a manifold is a Lie group, isn't it?
Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
- 14.3k
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810
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How many unit simplices are needed to cover a unit $n$-cube?
The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.
...
- 151k
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1
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reference for "X compact <=> C_b(X) separable" (X metric space)
I know (and am able to prove via Stone-Čech compactification) that the following is correct:
Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
- 888
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Are the Platonic solids shadows of 4-polytopes?
Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...
- 151k
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841
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Known configurations maximizing the volume of the convex hull of n points on the unit sphere
For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the ...
- 32.5k
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875
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What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?
As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
- 15.3k
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A question about the proof of Mostow rigidity
I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma \...
- 29.2k
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1k
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Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?
There is a folklore in the empirical computer-science literature that, given a tree $(X,d)$, one can find a bi-Lipschitz embedding into a hyperbolic space $\mathbb{H}^n$ and that $n$ is "much ...
- 195
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573
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Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
- 13.6k
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901
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How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?
Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by ...
- 697