16
votes
5answers
806 views
+50

What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below). Question A. How does one arrange $n$ unit cubes ...
1
vote
1answer
349 views

Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...
5
votes
2answers
197 views

Genus of Tutte-Coxeter Graph

What is the genus of the Tutte-Coxeter graph -- the incidence graph of the GQ of order 2? Seems like it should be well known, since nearly every other parameter for that graph is known, but I can ...
12
votes
2answers
431 views

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 ...
2
votes
2answers
166 views

Three questions concerning lattice points on sphere surfaces

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 ...
1
vote
1answer
185 views

Request for some references exploring the connections of Riemann surfaces with medical imaging

I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical ...
8
votes
1answer
209 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
269 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
1answer
286 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
150 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 ...
3
votes
1answer
115 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 ...
5
votes
0answers
386 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 ...
8
votes
1answer
292 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
230 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 ...
11
votes
0answers
337 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 ...
3
votes
1answer
394 views

Combinatorial geodesics

[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.] I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics ...
13
votes
1answer
339 views

The optimal constant in Vitali covering lemma

Let me restate Vitali covering lemma. Let $\{B_i\}_{i\in F}$ be a finite collection of balls in the $\mathbb{R}^n$. Then there is $S\subset F$ such that the balls $\{B_i\}_{i\in S}$ are disjoint and ...
1
vote
0answers
151 views

Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references: Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...
12
votes
1answer
293 views

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 ...
12
votes
1answer
560 views

The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?            The above is a photo of foil I flattened to reuse. It might be ...
0
votes
2answers
130 views

Disks Packing Variant

Usually disk packing problems require that no two disks of the packing intersect. Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...
0
votes
3answers
273 views

The symmetry group of $\mathbb Z^d$

Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm. I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
3
votes
1answer
206 views

Empty convex polytopes for random point sets

I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane (the Happy-Ending Problem), and I know that there are higher-dimensional extensions. A great source ...
11
votes
1answer
322 views

Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere $S$ in $\mathbb{R}^3$, sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$, with $\alpha < 1$ the fraction of ...
5
votes
2answers
243 views

Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$. Define a blocking set for a lattice cube to be a set of points in ...
10
votes
4answers
738 views

Optimal pebble-packing shape

Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes. Q. ...
35
votes
6answers
996 views

Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$, one by one, and create a random jumble of disks above the $x$-axis. When a falling disk hits another, it stops and sticks there. Otherwise, ...
2
votes
2answers
490 views

Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood. I am considering a variant, which I call midpoint lattice polygons. Start with a sequence of ...
21
votes
1answer
960 views

Building a genus-$n$ torus from cubes

I wonder if this has been studied: What is the fewest number of unit cubes from which one can build an $n$-toroid? The cubes must be glued face-to-face, and the boundary of the resulting ...
20
votes
6answers
2k views

Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of radius ...
3
votes
2answers
310 views

Touching-tetrahedra graphs

Have the graphs representable by touching tetrahedra been explored? Let $\cal T$ be a collection of tetrahedra in $\mathbb{R}^3$ with pairwise disjoint interiors. Define a graph $G_{\cal T}$ to have ...
2
votes
1answer
188 views

Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a ...
3
votes
1answer
310 views

Transience of self avoiding random walks on $\mathbb{Z}^d$

I'm finishing up a masters thesis in computer science and want to say a bit in the introduction about self-avoiding walks. My thesis looks at a random process which arose in computer science and my ...
6
votes
0answers
736 views

Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$, and I would like to connect them with a curve $C$ whose curvature is as low as possible. Let $\kappa_\max(C)$ be the sharpest (largest ...
7
votes
3answers
250 views

Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$

Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one of a number of models: (1) the convex hull of $n$ points randomly and uniformly ...
4
votes
4answers
910 views

Delaunay triangulations and convex hulls

This is a reference request. I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
8
votes
2answers
353 views

Limit shape for fixed-perimeter lattice polygons

Let $P$ be a simple polygon defined by $n$ unit-length segments connecting lattice points of $\mathbb{Z}^2$. I have two operations that preserve the perimeter of $P$. The first is the "pop" of a ...
18
votes
3answers
794 views

Growing random trees on a lattice $\rightarrow$ Voronoi diagrams

Imagine growing trees from $k$ seeds on a square $n \times n$ region of $\mathbb{Z}^2$. At each step, a unit-length edge $e$ between two points of $\mathbb{Z}^2$ is added. The edge $e$ is chosen ...
12
votes
1answer
650 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. ...
10
votes
2answers
569 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 ...
7
votes
0answers
223 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 ...
30
votes
1answer
1k views

Pach's “Animals”: What if genus $> 0$ ?

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
889 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 ...
15
votes
5answers
886 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 ...
5
votes
1answer
636 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 ...
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
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, ...