3
votes
1answer
180 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
147 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
112 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 ...
2
votes
1answer
62 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 ...
1
vote
1answer
148 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
36 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 ...
7
votes
0answers
327 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
137 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
298 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
287 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 ...
27
votes
4answers
734 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 ...
6
votes
2answers
169 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. ...
11
votes
0answers
380 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 ...
23
votes
2answers
706 views

The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square? By "nonoverlapping" I mean: not sharing an interior point. By "touch" I mean: sharing a boundary point. ...
8
votes
1answer
359 views

Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$. Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if there is a continuous rotation of $P$ such that its orthogonal-projection shadows ...
0
votes
1answer
213 views

Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
12
votes
1answer
287 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 ...
6
votes
3answers
332 views

Herringbone partitions of regions and surfaces

Let $R \subset \mathbb{R}^2$ be a region of the plane bounded by a Jordan curve. The boundary $\partial R$ could be a polygon, or a smooth curve—there are variations depending upon boundary ...
11
votes
1answer
320 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 ...
9
votes
5answers
485 views

Minimal blocking objects with shadows like a cube

This is a more geometric version of the previous question, "Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$. View an $n \times n \times n$ cube $C_3(n)$ as ...
5
votes
2answers
240 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 ...
2
votes
2answers
486 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 ...
4
votes
2answers
350 views

Conformal structure does not see conical singularities

the conformal structure does not see the conical singularities of a polyhedral surface. This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli). The sentiment is ...
10
votes
1answer
472 views

Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that, Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$. In a simple arrangement, every pair of lines ...
11
votes
1answer
491 views

Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question, "Löwner-John Ellipsoid: incribed and circumscribed," I've been wondering what might be the maximum ratio of this quantity? Let $P$ be a convex polygon of $n$ ...
17
votes
2answers
779 views

Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...
1
vote
0answers
136 views

When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...
1
vote
2answers
293 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 $2\pi$. ...
1
vote
1answer
297 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
221 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
756 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 ...
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
881 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 ...
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 ...
4
votes
1answer
309 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 ...
1
vote
2answers
596 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
651 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 ...
15
votes
5answers
882 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 ...
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
1answer
632 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 ...
6
votes
2answers
571 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
800 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 ...
4
votes
2answers
463 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 $\mathbb{R}^3$ ...
4
votes
2answers
258 views

Centralizing four red vectors in six green sectors

Four red vectors are given, one per quadrant, $[0,90^\circ)$, $[90^\circ,180^\circ)$, etc. A rigid star of six green vectors separated by $60^\circ$ can be positioned at $(\theta, \theta+60^\circ, ...
5
votes
4answers
980 views

Coloring Points in the Plane

Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed? I heard this problem when I was a kid. Back then the most ...
38
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 approximatively equal to the area of the disc. Does the complement of S necessarily contain ...