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

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9
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2answers
209 views

Do maximal polyhedra have algebraic volume?

Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number? Update: What ...
5
votes
2answers
261 views

Covering convex polygons with inscribed disks

The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...
3
votes
0answers
119 views

On understanding Discrete-Valued Stochastic Processes( time series, panel data )

It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
11
votes
1answer
509 views

Tiling the square with rectangles of small diagonals

For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...
2
votes
0answers
74 views

Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...
2
votes
0answers
124 views

A question on discrete numerical simulation on fluids mechanics

I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have ...
3
votes
1answer
185 views

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 ...
5
votes
1answer
420 views

Interesting behaviour of Brion's formula under a degenerate change of variables

This is, probably, a question for those knowledgeable on the subject of Brion's theorem and its applications. Lately, I've been dealing with situations of the following sort. Suppose we are given a ...
8
votes
1answer
216 views

Fold-and-cut problem in three dimensions

The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...
8
votes
1answer
271 views

Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...
6
votes
2answers
189 views

Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
6
votes
1answer
288 views

Pick's Theorem for rational points of bounded height

I wonder if the various lattice-point theorems, such as Pick's Theorem or Minkowski's Lattice Theorem, have been generalized to the collection of points with rational coordinates no more than height ...
5
votes
1answer
152 views

Matching on sphere to create cycle with chords

Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through the center of $S$, in such a way that no pair of chords intersect:       I would like ...
5
votes
2answers
229 views

Solving for special rational triangles

I ran into a need for isosceles triangles that (1) have the two equal integer side lengths $a$ (but the base $x \in \mathbb{R}$), and (2) the apex angle $\gamma$ is a rational multiple of $\pi$. ...
3
votes
1answer
125 views

What are interesting 3-colorings of the plane without rainbow lines?

This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...
3
votes
1answer
116 views

Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov]. Is the same true ...
14
votes
3answers
246 views

Covering a hexagon

For $\epsilon > 0$ sufficiently small, can a regular hexagon with sides of length $1 + \epsilon$ be covered by seven equilateral triangles with sides of length $1$? Motivation: Conway and Soifer ...
7
votes
0answers
75 views

Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area? (I'm editing after realizing that my conjecture that a ...
11
votes
3answers
601 views

What fraction of n-point sets in the unit ball have diameter smaller than 1?

This question is inspired by a recent talk by Matt Kahle on random geometric complexes. Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
2
votes
1answer
67 views

Covering the annulus of d-cube

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~\lambda c+x|c\in C$} for some $x\in R^d$ is called homothetic copy of $C$. The number ...
3
votes
1answer
108 views

Covering a convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
3
votes
2answers
326 views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
45
votes
5answers
2k views

If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds. Monsky's famous and amazingly tricky proof says that if we ...
1
vote
1answer
149 views

Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
0
votes
0answers
37 views

Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$. Now ...
4
votes
1answer
300 views

Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...
5
votes
0answers
396 views

N-balls covering n-balls

This question is a follow-on question from: Covering a unit ball with balls half the radius The questions are these: Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
7
votes
0answers
332 views

Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve. I started by posting this ...
2
votes
1answer
138 views

Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
9
votes
1answer
302 views

Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ...
8
votes
1answer
299 views

Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedron for which this ...
2
votes
1answer
139 views

Is this cube packing possible?

I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length ...
2
votes
1answer
243 views

Regularity of Delaunay triangulation of a hypercube

First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations: (A) (B) We say the lower triangulation is more "regular" than upper ...
3
votes
1answer
152 views

Simplex with edges of length at least s having smallest circumradius

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...
9
votes
1answer
275 views

Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
0
votes
1answer
159 views

Counting integer points in a Minkowski sum

We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of degree $d$. We also ...
3
votes
1answer
121 views

The discrete theory of compressible fluids dynamics

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...
11
votes
0answers
346 views

Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly $$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor ...
27
votes
4answers
748 views

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 ...
2
votes
0answers
87 views

Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime, on neither the $x$- nor the $y$-axis. So $a^2+b^2$ is a prime. Construct a region $D(z)$ surrounding $z$ which is the largest ...
2
votes
1answer
174 views

Maintaining boundary of unit circle arrangement

I have a process which in each step creates a new unit circle and I am interested in maintaining the boundary of the resulting arrangement in linear time. Is there anything known about computing this ...
4
votes
0answers
119 views

Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$. An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that ...
3
votes
2answers
264 views

Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics: (Discrete Exterious Calculus is the newly developed ...
2
votes
2answers
197 views

Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice. A set of integer vectors ...
6
votes
2answers
172 views

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. ...
3
votes
1answer
146 views

Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...
7
votes
0answers
140 views

Fractional Helly for more than one piercing

Fractional Helly Theorem says the following: For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...
1
vote
1answer
235 views

Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized. ...
11
votes
0answers
391 views

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 ...
8
votes
0answers
117 views

Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...