# Tagged Questions

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

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### Separating unit disks by circles

This is inspired by the recent question about separating unit disks by lines, which I will refer to as the "line case". Replacing "line" by "circle" adds one degree of freedom, and I'm wondering if ...
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### Separating unit disks by lines

Given $n\ge 2$. For a real $d>2$, consider a constellation $C$ of $2n$ disks of radius $1$ in the plane such that $h(C)$, the minimal distance between any two of their centers, is equal to $d$. Let ...
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### On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
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### Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...
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### How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...
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### Expected length of the shortest polygonal chain connecting N random points in the unit square

N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) polygonal chain connecting all the points. It can be proved ...
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### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
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### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
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### Geodesic convex hulls in a graph; and their properties

This question asks for an analog of the convex hull in a graph that parallels (as far as possible) convex sets in Euclidean space. Let $G$ be a simple, undirected graph, and let $S \subseteq V$ be a ...
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### Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property? Definitions: A subset S of the xy-plane is star-convex if there is ...
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### Higher dimensional Rubik's cube group

Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group ...
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### Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
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### Hidden points in polygons

Let $h(n)$ be the largest number of mutually invisible points that can be located in a polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment $xy$ contains a point ...
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### n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
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### Discrete Laplace operator and its eigenvalues

I wonder is there any geometric interpretation of the eigenvalues ​​of the discrete Laplace operator on graphs? Maybe there is a relationship between the eigenvalues ​​and combinatorial properties of ...
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### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
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### Sphere - Symmetry and Triangulation [closed]

The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...
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### Optimal radiating $(d{-}1)$-flats within a sphere

Permit me to revisit an earlier unresolved MO question, "Chord arrangement that avoids confining small or large disks" with a (very!) specific version, inspired by radiation therapy. The main idea is ...
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### Number of 3-tuple partitions of a multiple of three which follow the triangle inequality

Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ . If I chose any one partition randomly from ...
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### Softwares to draw euclidean polyhedral surfaces

Let $S$ be an abstract euclidean polyhedral surface. By this, I mean a orientable compact 2D topological surface obtained by gluing together some (convex) euclidean polygons (arbitrary genus). ...
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### Is there a 3d equivalent of this picture?

This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin: Fitting a mesh to a density function The picture below is the image of a regular ...
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### What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows: Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...
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### Embedding points in 2D based on distance estimates?

Suppose we have a collection of exactly $N$ points (say $N=1000$), with each point belonging to 2-dimensional Euclidean space $\mathbb{R}^2$, but we don't know the coordinates of the points. Suppose ...
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### Curvature flows for PL closed curves in the plane?

I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane. There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth ...
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### Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, ...
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### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows: Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...
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### Points in general position on a small grid

A point set $P$ is said to be embedded in $\mathbf{Z}^2$ in general position, if no three points lie on a common line. Assume that $|P|=n$, I am interested in the smallest $k \times k$ integer grid in ...
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### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
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### Estimates on the number of vertices of reflexive polytopes

Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...
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### Rational points of non-rational curves

An algebraic curve (in this question) is the zero set   $C = f^{-1}(X\ Y)$ of any polynomial   $f\in\mathbb R[X\ Y]$;   we say then that   $f$   represents   $C$.   ...
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### A problem in Galois Geometry

Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...
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### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s). Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: (a) Trivial ...
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### Is there a nice way to “unravel” a great dodecahedron?

The great dodecahedron is a non-convex regular polyhedron bounded by 12 pentagonal faces, crossing each other, arranged in a star-shaped manner around each of its 12 vertices (see the Wikipedia page ...
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### Erdős, Harary, Tutte's “dimension of graph”: Progress in last 48 yrs?

I just ran across this delightful paper by an amazing triumvirate: Paul Erdős, Frank Harary, and William Tutte. "On the dimension of a graph." Mathematika 12.118-122 (1965): 20. (Cambridge link) ...
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### Sets of evenly distributed points in the Euclidean plane

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite? If the answer is yes, can ...
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### Can a tangle of arcs interlock in plane?

This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here: Can a (finite) collection of disjoint circle arcs in ...
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### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
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### Partitioning the Projective Plane

Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$. Side Note: If there are more standard definitions for any of the ideas presented here, ...
Pardon my ignorance of this topic. Q1. In which dimensions $d$ is it the case that, for every natural number $n$, there exists a sphere having exactly $n$ lattice points on it ...