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

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34
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3answers
2k views

Is the “Napkin conjecture” open? (origami)

The falsity of the following conjecture would be a nice counter-intuitive fact. Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
13
votes
2answers
931 views

Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection). I am wondering if each of the five ...
2
votes
1answer
67 views

Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)? What is known about ...
-3
votes
0answers
76 views

What is a discrete shape [closed]

I saw this term in a paper on tiling using shapes. Can you give the definition for discrete shape? I googled on the web, and could not find any explanation for this concept...
1
vote
2answers
150 views

Generalization of Bracketing (or one of its many equivalences)

I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here. *"Is ...
0
votes
0answers
51 views
3
votes
1answer
74 views

What is Known About the Complexity of Calculating Minimal Surface Polyhedra?

I am currently ruminating about ways of generalizing Minimum Spanning Trees to Minimum Spanning "Hypertrees", where the cost is associated with simplex volumes and, where certain topological ...
6
votes
0answers
171 views

Minimal “basis” in $n$ dimensional unit cube

Let's $$ B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots $$ and let's $$ C\subseteq B^n, $$ $$ S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...
1
vote
0answers
45 views

Can any Delone set be approximated by a model set?

Let $\Lambda \subset \mathbb{R}^d$ be a Delone set (uniformly discrete and relatively dense). I would like to know whether $\Lambda$ can be approximated by a model set in the Hausdorff distance. ...
25
votes
5answers
2k views

Nonconvex manhole covers

One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...
1
vote
0answers
125 views

Find m most distant points from a set of n points [closed]

I would like to find the $m$ (where $n$ $\geq$ $m$ > 1) maximally distant subset of points from a collection of $n$ $d$-dimensional points. Maximally distant means the sum of the pairwise distances ...
17
votes
0answers
243 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
6
votes
0answers
102 views

How big can a family of pairwise intesecting affine spaces be?

I apologize if this question might seem to be a bit too elementary. Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...
2
votes
2answers
2k views

Best fit for multiple shapes inside an area

Is there a forumla to come up with the best fit for multiple shapes inside a rectangular area, so that none of the shapes are overlapping?
-2
votes
0answers
75 views

Minimum distance in unit cube

Given the unit cube $I^n=[0,1]^n$, let $x_1,\cdots,x_n\neq (0,0,\cdots,0)$ be $n$ linearly independent vertices (corners) of $I^n$, and $p(x_1, \cdots, x_n)$ be the projection of $(0,0,\cdots,0)$ onto ...
9
votes
1answer
356 views

Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three non-overlapping parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedra ...
2
votes
1answer
85 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 $\lambda C + x = \{ \lambda c+x \mid c\in C \}$, for some $x\in R^d$, is called a homothetic copy of $C$. The ...
1
vote
1answer
194 views

How to compute the number of regular spheres needed to fill a rectangular space

Computing the volume of a sphere is straightforward 4/3*pi*R^3 As is the volume of a rectangular space length*width*height (e.g. 10*10*6) How might I go about determining how many spheres would fit ...
1
vote
3answers
4k views

Compute the Centroid of a 3D Planar Polygon

Given a list of 3D coordinates that define the surface( Point3D1, Point3D2, Point3D3, and so ...
22
votes
6answers
2k 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 ...
0
votes
1answer
335 views

Tetris in 3D with 5 units [closed]

Background: There are 7 "bricks" used in the game of Tetris. These are the 7 unique combinations of 4 unit squares in which every square shares at least one edge with another square. ("unique" in this ...
4
votes
1answer
128 views

The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
4
votes
1answer
193 views

Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane. Suppose you have a total length $L$ of string available, and your task is to connect chords of $C$ using no more than $L$ of string to minimize the ...
3
votes
3answers
271 views

Vertex-transitive polytopes in any dimension with any number of vertices?

Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...
3
votes
1answer
115 views

Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let ...
8
votes
0answers
118 views

Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden. At some point, the region is "saturated," ...
6
votes
4answers
179 views

Inside-out polygonal dissections

A dissection of a polygon $P$ is a partition of $P$ into a finite number of pieces, which can then be rearranged (via planar translations and rotations) and joined (without overlap) to form a new ...
6
votes
2answers
226 views

Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...
7
votes
2answers
375 views

Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ... The fraction of $\mathbb{Z}^2$ lattice points visible from the origin $1/\zeta(2)=6/\pi^2 \approx 61$%. The fraction of $\mathbb{Z}^3$ lattice points visible from the ...
4
votes
3answers
550 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.
6
votes
2answers
314 views

Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know. Assume, however, that for any triple of the points we know the angle. Question: Can we decide whether the n points are realizable ...
3
votes
2answers
205 views

A problem on chains of squares — can one find an easy combinatorial proof?

Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection $$ A = \left\{ \left[ \frac{i}{n},\frac{i + 1}{n} \right] \times ...
2
votes
1answer
64 views

Visibility kernels of embedded graphs

Let $G$ be a connected graph embedded in the plane with all edges straight segments. For $\alpha \in (0,\pi)$, define an $\alpha$-path as a path in $G$ with all turns at vertices within ...
6
votes
1answer
217 views

Thinnest 2-fold coverings of the plane by congruent convex shapes

It is an unsolved problem to determine the "thinnest" $2$-fold covering of the plane by disks. The $2$-fold coverage problem by disks is to find the minimum number of congruent (unit-radius) disks ...
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 ...
12
votes
3answers
601 views

Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram ...
8
votes
3answers
10k views

Dividing a square into 5 equal squares

Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
3
votes
1answer
85 views

Points with pairwise integer distances in the plane

Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a ...
9
votes
0answers
295 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...
40
votes
12answers
2k views

Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...
3
votes
1answer
344 views

Convex n-polytope general position vectors to general position vectors of tetrahedron

I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself. Define a set of general position vectors ...
19
votes
2answers
901 views

Four Dimensional Origami Axioms

What are the axioms of four dimensional Origami. If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
3
votes
1answer
49 views

Average vertex degree in finite Delaunay triangulations in high dimensions

In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6. Does anyone know of a similar result in higher dimension? I ...
4
votes
2answers
109 views

(non-)existence of the aperiodic monotile

The aperiodic monotile problem asks whether there exists a single tile that every tiling made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? ...
8
votes
1answer
327 views

Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...
3
votes
0answers
61 views

A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.) Think of $T$ as a physical, rigid ...
2
votes
0answers
152 views

Dissection of a polygon into convex polygons

Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons. I would be very grateful for any information on this problem. Remark 1. There ...
8
votes
4answers
418 views

What is the right way to think about / represent general tilings?

For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
6
votes
2answers
116 views

Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes. What is the maximum number of edges of a linklessly embeddable graph? A more general question is the following. Given $\mu$ what is the maximum number of edges of ...
7
votes
1answer
137 views

How many maximal triangulations of a rectangle?

I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow. Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with ...