Tagged Questions

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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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Feasibility of a list of prescribed distances in R^3

I am puzzled with the following problem: Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every ...
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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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Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
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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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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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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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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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About MF Atiyah and R Bott's 1983 paper

I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
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The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
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Packing density of randomly deposited circles on a plane

Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$: Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...
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Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem  ...
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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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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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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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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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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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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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Random rings linked into one component?

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (tilted) randomly (...
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When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?

To ask this question in a (hopefully) more direct way: Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ (i.e....
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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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Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, etc.,...
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Relating curvature and torsion of a connection to those of a curve

I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion. I know that an affine connection $A$ on an $n$-dimensional manifold $M$...
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Maximal number of connected components of complement to an affine plane real algebraic curve

Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$. How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
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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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Essentially one random metric on $\mathbb{S}^2$?

I heard it claimed that there is, in some sense, only one random metric on $\mathbb{S}^2$. I would appreciate any pointer to literature that explicates this intriguing claim. So far my own searches ...
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For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
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Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
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A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
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Dense sphere packings which are not lattice packings

This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
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Fitting a mesh to a density function

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
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How can we count lines in an n-x-n rectangular array?

Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
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Untangling entwined rigid chains in 3-space

I am interested in exploring the degree of "tangledness" of two rigid chains in space. A polygonal chain is a simple (non-self-intersecting) path of segments in $\mathbb{R}^3$, viewed as a rigid body. ...
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Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected) pieces (where $k=1$ means just the square). Your task is to construct the largest volume convex body ...
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Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...
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Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]

How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
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Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note) Consider the Bundle theorem configuration : Points $A_1, A_2, A_3, A_4$ lie on a circle, ...
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Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...
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Finding the convex combination of vertices which yields an inner point of a polytope

Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
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How many convex shapes can be made with the pieces of the Stomachion?

Tangrams are a well-known dissection of the square into seven convex polygons. One fun mathematical question is: how many convex rearrangements of the seven pieces are there? Answer: there are 12 ...
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Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the "...
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Subtract Rectangle from Polygon

I'm looking for an algorithm that will subtract a rectangle from a simple, concave polygon and return a remainder of polygons. If the rectangle encloses the polygon, the remainder is null. In most ...
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Is the following two-dimensional graph likely to be globally rigid?

Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that: All vertices within a ...
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cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P: MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...
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Threading pinholes in the wall of cylinder to pass through an internal coordinate

Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...
Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...