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

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Max number of points in a grid with distance to the origin between $d$ and $d+\sqrt{2}/2$, for some distance $d$? [closed]

Consider the square grid centered at the origin $\{-n,\ldots,n\}\times\{-n,...,n\}$. What is the value or an upper bound, as a function of $n$, of $\max f(d)$, where $d$ is the distance from some ...
3
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2answers
157 views

Existence of lattices whose circles have bounded number of points

For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as $$\mathcal K(\Lambda) = \...
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1answer
662 views

On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$. Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...
8
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1answer
270 views

Set of balls which the number of the ball intersects lines on the plane is bounded

Does there exist the set of balls(may be not disjoint) $X=\{B_i\subset\mathbb{R^2};i\in I\}$, satisfing following properties?(Note that the ball has a positive real radius) Let the set of all lines ...
9
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1answer
238 views

How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of Garrett Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...
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1answer
311 views

Status of Zeeman's collapsability Conjecture

Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision. What is the status of this ...
2
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1answer
85 views

Transformation inverting distances between two sets of diameter 1

Let $S_1, S_2 \subseteq \mathbb{R}^2$ be two finite disjoint sets of points in the plane with $\texttt{diam}(S_1) \leq 1$ and $\texttt{diam}(S_1) \leq 1$. Does there always exist a transformation $f: ...
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1answer
111 views

Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence: \...
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2answers
2k views

Subdivision of triangles into congruent triangles

Recently some old notes of mine have gotten me to thinking about the problem of subdividing a triangle into $N$ smaller triangles, all congruent to one another. A little thought shows the following ...
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0answers
61 views

Number of polyhedra with N faces?

A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...
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2answers
585 views

Lattice n-gons with ordered side lengths 1,2,3,…,n

Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8). Are there other (infinitely many) polygons, such as this, lying entirely in ...
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473 views

Dissecting Ramanujan´s Cuboid: 1729 = 19 x 13 x 7

Consider the cuboid of dimensions 19 x 13 x 7 whose volume is 1729, the Hardy-Ramanujan number. What is the least number of smaller cuboids into which it can be dissected so that the resulting pieces ...
22
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4answers
1k views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
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120 views

Computer searches for the $g$-conjecture

McMullen's $g$-conjecture aims the classify possible $f$-vectors of simplicial $d$-spheres. The $g$-conjecture has been proven for polytopal spheres and for simplicial spheres of dimension $d < 5$. ...
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0answers
44 views

Connectedness of semi algebraic set by c.a.d

I do not know whether there is a standard or some traditional ways to decide whether a semi algebraic set is connected or not. One way I know is c.a.d algorithm. I have read some papers of c.a.d ...
7
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2answers
207 views

Graphs with prescribed numbers of k-cliques

Let $(a_1,a_2,\dots, a_n)$ be a sequence of non-negative integers. Q. When does there exists a simple graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_1$ vertices, $a_2$ ...
2
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1answer
90 views

How many distinct sets of n collinear points are there in an evenly-spaced two-dimensional grid of m x m points?

I'm seeking the definition of some function $f(n,m)$ which evaluates to the number of distinct sets of $n$ collinear points which are selected from an evenly-spaced two-dimensional grid of $m \times m$...
2
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1answer
133 views

How many lines of exactly n points can be placed in a discrete, square grid of size m x m?

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of ...
2
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0answers
46 views

lower bound for sum of (squared) distances under a minimum distance restriction

I am trying to solve a packing problem in discrete geometry and it would be useful to know the answer to the following problem. Let $A_1$, $A_2$,..., $A_n$ be $n$ points in the Euclidean plane $\...
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3answers
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Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems. Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ ...
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0answers
33 views

Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the $[0,1)\...
3
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2answers
107 views

Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...
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2answers
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Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?

Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of the ...
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1answer
351 views

Banach-Mazur distance between the cube and the octahedron

The Banach-Mazur distance $d(X, Y)$ between two normed spaces $X, Y$ of the same dimension is defined as $d(X, Y) = \log\inf \|T\| \cdot \|T^{-1}\|$, where the $T:X \to Y$ is a linear and invertible ...
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0answers
55 views

Efficient CW structures on squarefree semi-algebraic set

General Setup Given a collection of $k$ polynomials (with real coefficients) in $n$ real variables, say $f_i(x_1,\ldots,x_n)$, let $V \subset \mathbb{R}^n$ correspond to those $x$-values for which ...
5
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1answer
127 views

Decidability of convex rearrangements of polygons

Triggered by the MO question, "How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question: Q. Given $n$ polygons in a set $S$, say each with integer ...
4
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2answers
132 views

Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
7
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1answer
77 views

minimum number of bases of a matroid, that comes from a convex polytope

Given a d-dimensional polytope P with n points, then what is the minimum number of simplices that are spanned by vertices of P? This question led my research to matroids and so my question is: what is ...
3
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0answers
108 views

Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...
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2answers
227 views

(non-)existence of the aperiodic monotile

The aperiodic monotile problem asks whether there exists a single tile that every tiling of the plane made with it results non-periodic. What is known about this problem? If this tile exists, how can ...
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0answers
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What do you call the collection of all sets shattered by $F$?

The proof of Pajor's lemma uses the collection of all sets $S\subseteq X$ shattered by some $F\subseteq 2^X$. Is there a standard term for the former object? I've been privately referring to it as the ...
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1answer
151 views

Penrose tiling substitution is bijective

Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles. It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...
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Minimal subset of an $n$-dimensional grid intersecting every ($n$-dim) arithmetic progression of length $k$

Let $$A=\{1,2,...,l\}^n \subset \mathbb{R}^n$$ for some positive integers $l$ and $n$, and $B \subset A$ be a set such that $|B| \ll l^n$. I am interested in determining how small must $B$ be in ...
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0answers
154 views

Precise estimate for probability an $n$-point set has diameter smaller than $1$

This question was inspired by an earlier question that I answered but would like a more precise bound for. Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
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0answers
90 views

Balanced partitions of vector sets

We are interested in the following Lemma. Let $V\subset [0,1]^n\subset\mathbb R^n$ be a set of $n$-dimensional vectors. Then for each $r\le |V|$ there exists a partition $$V=V_1\cup V_2\cup\dots \cup ...
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1answer
72 views

Testing whether two vertices are neighbours

I face the following problem: I am given a high-dimensional, convex, bounded polyhedron in both vertex description: $X = \mathrm{conv} \, \{ v_1, \ldots, v_K \}$ and halfspace description: $X = \{ x \...
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2answers
145 views

Minimum length of a convex lattice polygon containing k lattice points?

Let $f(k)$ denote the minimum length of a convex lattice polygon containing exactly $k$ lattice points (including lattice points on the boundary). It is not too hard to show that $k = \frac{1}{4\pi} ...
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1answer
163 views

Constant hole density on the area of a circle

I need to create about 100 (small) holes in a distributor plate (hole diam = 0.5 mm; plate diameter = 100 mm). The sm. holes should be distributed in such a way that the density (hole/area) is nearly ...
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455 views

Maximal set on hypersphere that does not contain pairs of orthogonal vectors

Let R be a region on a hypersphere. Each point A of the hypersphere is associated with a vector pointing to A and with origin at the centre of the hypersphere. So let me identify each point with a ...
3
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0answers
273 views

Generalization of the “double cap conjecture” to a vector space with complex field

The conjecture that I proposed in Maximal set on hypersphere that does not contain pairs of orthogonal vectors is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun. See for ...
3
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1answer
301 views

Radial tilings with variable area ratios

I was looking at this neat page on logarithmic spiral tilings when a question popped up: http://www.uwgb.edu/dutchs/symmetry/log-spir.htm It seems that in all of the tilings shown, the area of each ...
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164 views

An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes. An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities: $$ \forall i \; 0\leq x_i \leq 1 $$ and $ ...
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107 views

Rational inscribed realization of the regular dodecahedron

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...
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154 views

Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...
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1answer
360 views

Tiling by regular simplices

The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional ...
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1answer
60 views

Constructing a polygon of $n$ facets from a set of positive values representing the length of the facets [closed]

The input of my problem is a set of positive values $a=\{a_1,...,a_n\}$ where $n\geq 3$. I want to construct an $n$-gon where the lengths of the $n$ facets are the values $a_i$ for $i=1,...,n$. My ...
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235 views

Reduction to some physical interpretation of this formula

Problem Given N 3-dimensional points which are {$p_1,p_2,..,p_n$} where $p_i = (x_i,y_i,z_i) $ . I have to find the value of the formula $$ \sum \limits_{i=0}^n \sum \limits_{j=i+1}^n \frac{ \mid \...
8
votes
1answer
239 views

Integer sets with forbidden differences

Given a finite set $S$ of positive integers, and a positive integer $n$, let $F(n,S)$ be the largest possible cardinality of a subset of {$1,2,\dots,n$} no two of whose elements differ by a number in $...
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25 views

Homology of the subcomplexes of the “diamond shaped” sphere under 1-norm in $R^n$ as a simplicial complex

The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$. The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...
12
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258 views

Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks on $\mathbb{Z}^2$ that turn left or right at each step (i.e., they cannot continue the direction of the preceding step) have been ...