# Tagged Questions

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### Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I ...
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### Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth ...
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### Tying knots with reflecting lightrays

Let a lightray bounce around inside a cube whose faces are (internal) mirrors. If its slopes are rational, it will eventually form a cycle. For example, starting with a point $p_0$ in the interior of ...
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### Fair but irregular polyhedral dice

I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$...
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### What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...
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### The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
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### Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...
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### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that every disk can be ...
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### Sheaves and bundles in differential geometry

Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
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### “Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...
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### $\exists$ a shot in ideal pocket billiards?

Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with the game idealized in that no spin is placed on the cue ball in the initial shot, all collisions between billiard ...
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### Why is the Laplacian ubiquitous?

The title says it all. I'm wondering why the Laplacian appears everywhere, e.g. number theory, Riemannian geometry, quantum mechanics, and representation theory. And people seems to care about their ...
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### Why is Fourier analysis so handy for proving the isoperimetric inequality?

I have just completed an introductory course on analysis, and have been looking over my notes for the year. For me, although it was certainly not the most powerful or important theorem which we ...
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### The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square? By "nonoverlapping" I mean: not sharing an interior point. By "touch" I mean: sharing a boundary point.   &...
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### Geometry of complex elliptic curves

Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat?
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### Functions whose gradient-descent paths are geodesics

Let $f(x,y)$ define a surface $S$ in $\mathbb{R}^3$ with a unique local minimum at $b \in S$. Suppose gradient descent from any start point $a \in S$ follows a geodesic on $S$ from $a$ to $b$. (Q1.) ...
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### Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be. "Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that F(a)=F(b) and F'(t) exists ...
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### Why is the half-torus rigid?

The half-torus surface that results from slicing a torus like a bagel, depicted below (left), is isometrically rigid.       I know this from a remark of Alexandrov in Mathematics: ...
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### A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter? Background and motivation The Borsuk conjecture (disproved in 1993) ...
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### What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?

Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay ...
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### Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that \mathop{\rm area} \Sigma\ge ...
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### Iterated Circumcircle

Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no ...
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### 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 ...
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### Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball

It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
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### Is there a combinatorial analogue of Ricci flow?

The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing. However I have ...
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### Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$ embedded in $\mathbb{R}^3$. You do not know anything about $M$. You shoot off a geodesic $\gamma$ in some direction $u$, and learn ...
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### Largest hyperbolic disk embeddable in Euclidean 3-space?

Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the ...
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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 ...
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### Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...
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### Is the ball reducible in some high dimension?

Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$ of the fundamental ...
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### Arrangements of points in the plane

Let $p_1,\ldots,p_n$ be a collection of distinct points in $\mathbb{R}^2$, no three of which lie on a line. For each $p_i$, let $\omega_i(p_1,\ldots,p_n)$ be the following ordered list (well-defined ...
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### What can be said about the Shadow hull and the Sight hull?

This is a question implicitly raised by Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always ...
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### When does the set of isometries form a group?

Motivation Its a classic set up. Take a metric space $M$, with distance function $d:M\times M\to \mathbb{R}$. The set of isometries of $M$ is the set of functions $f: M \to M$ which preserve distance....
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### Is a rhombus rigid on a sphere or torus? And generalizations.

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a ...
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### Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
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### Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
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### 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 ...
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### Integer lattice points on a hypersphere

Is the following statement true? For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice ...
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### Blocking visibility with cylinders

Suppose you have a supply of infinite-length, opaque, unit-radius cylinders, and you would like to block all visibility from a point $p \in \mathbb{R}^3$ to infinity with as few cylinders as possible. ...
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### Hanging a ball with string

What is the shortest length of string that suffices to hang a unit-radius ball $B$? This question is related to an earlier MO question, but I think different. Assume that the ball is ...
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### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
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### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...
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### The Symmetry of a Soccer Ball

Let $P$ be a polyhedron which satisfies the following three conditions: $P$ is built out of regular hexagons and regular pentagons. Three faces meet at each vertex. $P$ is topologically a sphere. ...