Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

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2
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1answer
430 views

Lipschitz constant of Laplace-Beltrami Operator

I already asked this question at stackexchange with no response - so I'll try here. I'm reading a paper on discrete differential geometry: Meyer et.al. They define the Laplace-Beltrami operator at ...
1
vote
2answers
296 views

Can a set of tetrahedra glued together by a common vertex be isometrically embedded in R^4?

A collection of triangles with a common vertex $A_1VA_2$, $A_2VA_3$, ... $A_NVA_1$ with specified side lengths can be isometrically embedded in $R^2$ provided the angles around $V$ add up to $2\pi$. ...
20
votes
1answer
474 views

Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$, the function $p_V\colon \mathbb R_+\to \mathbb R_+$ $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$. Definition 2. Define ...
12
votes
1answer
663 views

Random polycube shapes

I am wondering if it is hopeless to obtain any firm results on the following model of a "random polycube shape." First, a polycube in $\mathbb{R}^3$ is a connected face-to-face gluing of unit cubes. ...
13
votes
1answer
530 views

Which sets of lattice points have rational generating functions?

Let $P$ be a subset of $\mathbb N^d$ (or of some normal pointed affine semigroup), and suppose that $f:=\sum_{p\in P}\ t^p\in\mathbb Z[[t_1,\ldots,t_d]]$ is a rational function. What can be said ...
3
votes
2answers
503 views

Torus in $\mathcal{R}^3$

Hi I'm interested in packing the 3 space as dense as possible using equally sized tori whose major radius is much bigger than their minor radius in. Do you have any idea how to attack this problem? ...
20
votes
3answers
1k views

Rolling-ball game

The analyses in two recent MO questions, "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-Ball Game, as follows. A unit-radius ball sits on a grid ...
10
votes
2answers
576 views

Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$. Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$. It seems that not ...
6
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0answers
211 views

balls in arrangements of hyperplanes

The following theorem is from Aronov, Naiman, Pach and Sharir's An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ...
1
vote
1answer
301 views

Pointers/Papers on subdivision of planar quadrilateral meshes (PQ-Mesh) in 3D?

I'm interested in the subdivision of planar quadrilateral meshes (PQ-Meshes). Meshes consisting only of planar quadrilaterals, like discrete Voss surfaces and alike. I've been searching the web for ...
7
votes
0answers
227 views

Coloring toroidal polyhedra with convex faces?

Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 ...
7
votes
2answers
765 views

The straightest possible path embeddable in a path of polygons

I'm studying a problem involving the sets of discrete curves that can be embedded in a non-trivial polygon, from a source to a target point, as shown below. Initially my interest was limited to ...
33
votes
1answer
1k 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 ...
20
votes
2answers
903 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 ...
55
votes
5answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
3
votes
1answer
538 views

Voronoi polygons of general point patterns forced to honeycomb?

This question comes out of a simple wish for graphical displays of people (say members of the Wikimedia chapter for whom I now work) that will display demographic clustering. Obviously in typical ...
8
votes
3answers
2k views

n-dimensional voronoi diagram

Hi, 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 ...
3
votes
1answer
209 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 ...
7
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0answers
268 views

3-dimensional Cayley graph

I would like to see Cayley graphs drawn in 3-dimensional Euclidean space such that the vertices are represented by points and various shadows display the actions of the generators. For example, ...
4
votes
1answer
314 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 ...
42
votes
1answer
2k views

two tetrahedra in R^4

It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $R^3$, such that their union has diameter 1, then they must share a vertex. I wonder whether we have an ...
1
vote
2answers
607 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 ...
8
votes
2answers
679 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 ...
22
votes
6answers
1k views

Shortest grid-graph paths with random diagonal shortcuts

Suppose you have a network of edges connecting each integer lattice point in the 2D square grid $[0,n]^2$ to each of its (at most) four neighbors, {N,S,E,W}. Within each of the $n^2$ unit cells of ...
4
votes
2answers
228 views

Finding the Boundary Faces of the Zonohedron

A zonotope is a linear combination of m vectors with coefficients in [0,1]: $Z = \{ \sum \lambda_i v_i : 0 \leq \lambda _i \leq 1 \}$. The fancy way is to say it's the Minkowski sum of line segments ...
3
votes
1answer
500 views

Gluing Polygons

Consider all polygons whose vertices are lattice points and edges are parallel to the axes such that no more than two edges meet at a vertex. For two polygons A and B, define A+B be to the set of ...
7
votes
3answers
439 views

“incidental” intersections of a complete graph in the plane

Given a complete graph of n vertices (no three of which are no collinear) in the plane and straight edges, what is the maximal possible number of "incidental intersections" of edges, i.e., number of ...
15
votes
5answers
898 views

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 ...
10
votes
3answers
1k views

A random walk on random lines

I am wondering if this random walk remains finite with positive probability. Start with three lines $A,B,C$ that are extensions of an equilateral triangle. Let $p_0$ be one corner. Generate a line ...
4
votes
1answer
246 views

Maximum number of points in two disks

Given two closed disks of unit radius, such that center of one lies on the circumference of the other, let M denote their union. We want to place the maximum number of points in M such that their ...
18
votes
3answers
2k views

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 ...
5
votes
3answers
353 views

Path connected coloured sets on the squared paper

Colour small squares on the standard squared paper in two colors A, B. Name two small squares with common side as "neighbor". Let every colored set be "path connected": for any two small squares of ...
15
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0answers
1k views

Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
11
votes
2answers
712 views

covers of $Z^\infty$

Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
5
votes
1answer
644 views

Which (semi)regular polyhedra are combinations of two others?

The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and ...
5
votes
2answers
338 views

Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?

Background I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More ...
3
votes
0answers
125 views

Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A. Has ...
10
votes
2answers
1k views

Discrete Hairy Ball Theorem

This question is inspired by Math puzzles for dinner The arrow compatibility conditions in that problem can be considered an attempt to discretize the notion of a continuous vector field. The ...
29
votes
3answers
4k views

Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } ...
4
votes
7answers
1k views

How to generate a net on a 8-dimensional sphere

Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere? Thanks for any helpful answers!
10
votes
1answer
464 views

Largest pair of homometric Golomb rulers?

A Golomb ruler is a set of $n$ integers that determines $\binom{n}{2}$ distinct differences. Two sets are homometric if they determine the same (multiset) of differences. For example, ...
12
votes
2answers
1k views

Random walk is to diffusion as self-avoiding random walk is to …?

One can view a random walk as a discrete process whose continuous analog is diffusion. For example, discretizing the heat diffusion equation (in both time and space) leads to random walks. Is there a ...
9
votes
2answers
782 views

covering a square with unit squares

Can some square of side length greater than $n$ be covered by $n^2+1$ unit squares? (The unit squares may be rotated. The large square and its interior must be covered.)
4
votes
4answers
2k views

Tetrahedron splitting/subdivision

Given a regular Tetrahedron A (i.e. each edge of A has same length), is it possible to split A into several smaller regular tetrahedra of equal size? I.e. smaller tetrahedra should completely fill ...
7
votes
2answers
466 views

A question about tilings of the plane

Let C be a compact subset of the Euclidean plane E whose boundary is a Jordan curve J. If C tiles the plane, can J be such that it has a unique tangent line at each point and none of its sub-arcs is ...
10
votes
1answer
394 views

Random geometric graphs and spanners

I would grateful to learn of work mixing random geometric graphs with random graphs under the Erdős-Renyi model, and in particular concerning spanners. Select $n$ points uniformly at random from the ...
6
votes
2answers
579 views

Lattice Stick Number vs. Stick Number of Knot

Can the lattice stick number of a knot be bounded by the stick number of the knot? The stick number $S(K)$ of a knot $K$ is the fewest number of segments needed to realize it by a simple 3D ...
20
votes
5answers
804 views

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 ...
5
votes
2answers
468 views

Generalization of Hamiltonian cycles to “Hamiltonian spheres”

One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$ ...
5
votes
2answers
408 views

Making 'circles' on a lattice/ Making distinct fractions from partitions of a number

First formulation: discrete geometry Pick your favourite 2D square lattice (I'm sure we all have one...) and try to place n points 'in a circle' (that is: in general position [no 3 should be ...