Questions tagged [discrete-geometry]
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
2,000 questions
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Local uniqueness of Penrose tilings
Oskar van Deventer asks, if we remove finitely many tiles from a Penrose tiling, can we always uniquely reconstruct the original tiling?
(Actually he asks whether something slightly stronger is true: ...
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Chromatic number of Davenport-Schinzel graphs
Given a finite collection of planar segments, define a graph whose vertices are the segments and $uv$ is an edge if segments $u$ and $v$ intersect on the lower envelope, i.e., such that there is no ...
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Like circle packing but not exactly
I am looking for bibliography on the following problem.
Given $N\in\mathbb{N}$ find $N$ points $p_1,...,p_N\in\mathbb{R}^2$ which
(1) maximize $\min_{i,j} |p_i-p_j|$
(2) subject to the constraint $\...
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Paths in the space of tessellating shapes
This is somewhat related to this old question of mine, but is hopefully easier.
Let $\mathcal{H}$ be the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric, and let $\...
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High chromatic numbers of triangle-free graphs of convex polytopes
I am looking for some examples of triangle-free graphs/1-skeletons of convex $d$-polytopes with $d\ge 4$ whose chromatic number is at least 4, specially in dimensions 4, 5, 6. I know of one 7-...
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Characterizing polyhedra via "finitely many faces"
I recently tried to see in how far polyhedral geometry can be reduced to the study of convex sets with finitely many faces. In other words, I tried to replace "finitely generated" by "...
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Volume of empty lattice simplices and number of integer points in their dilations
Let $S \subset \mathbb{R}^n$ be an $n$-simplex with integer vertices. Suppose that $S$ does not contain any integer point other than its vertices and that $2S$ contains at least one integer point in ...
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Prove rigorously a constraint on the efficiency of intermediate segments in a polygonal chain over a regular $k$-grid
Let $k, r \in \mathbb{N}$ with $k \ge 2$ and $r \ge 3$, and let $n_1, n_2, \ldots, n_k \in \mathbb{N}$ satisfy $3 \le n_1 \le n_2 \le \cdots \le n_k$.
We define
${G_n}^k := \{0,1,\ldots,n_1-1\} \times ...
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Do $\binom{n}{2}$ pairs of equi-distances among $n$ points imply a regular $n$-gon?
There are $n$ points on the plane, satisfying that for any two points $A, B$, there is a unique point $C$ lying on the perpendicular bisector of $AB$, i.e. $CA=CB$.
Prove or disprove: $n$ is odd, and ...
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The smallest set of polygonal regions that can all together form 2 different convex polyhedrons
We add a little to On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
We call a set of polygonal regions that all together form a convex polyhedron a ‘...
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Properties of a set of infinitely stacked regular tetrahedra
If two regular tetrahedra $S_1$ and $S_2$ in $\mathbb{R}^3$ share a triangular face $f$, we will say that each of them is obtained from the other by stacking over $f$. Now, choose some regular ...
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On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
Ref: https://arxiv.org/pdf/1307.3472
It is well known that given only a set of convex polygonal regions (call this set of polygons a 'face set') and no further information, one cannot uniquely ...
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To choose a set of $n$ triangles which together form the largest number of different triangular layouts
Ref: To choose a set of $n$ rectangles which together form largest number of rectangular layouts
We present a variant of above question:
General Question: given an integer $n$, how do we find $n$ ...
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Triangulating the cube with small Hamming distance
For a triangulation $T$ of an $n$-dimensional cube, whose vertices are the $2^n$ original vertices, let $d(T)$ be the largest Hamming-distance of two vertices that are in the same simplex.
How small ...
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To build Heesch-like configurations of 'coronas' around a central triangle with triangles all of same area and perimeter
Ref 1: https://arxiv.org/pdf/1711.04504
Ref 2: On 'Walls' with 'non-tiles' - a variant of the Heesch problem
It is known that we cannot tile the plane with triangles that are pairwise ...
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On 'Walls' with 'non-tiles' - a variant of the Heesch problem
Ref: "Non-tiles and Walls - a variant on the Heesch problem" (https://arxiv.org/pdf/1605.09203)
Definitions (adapted from above doc): A non-tile is any polygon that does not tile the ...
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Tiling the plane with pair-wise non-congruent and mutually similar quadrilaterals
We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles
Starting with 2 unit squares and with squares with sides 2,3,5,... (all Fibonacci numbers), one can form an ...
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Changes to the Delaunay Triangulation after deleting a point inside the convex hull
Consider the Delaunay Triangulation $\mathcal{DT}(P)$ of a finite set $P$ of points in the euclidean plane.
let $CH\subset P$ be all points of $P$ that are on $P$'s convex hull, and not just the ...
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Relation of the most distant point-pair to the smallest enclosing circle
I am looking for a counter example to, resp. a proof of the correctness of, the following conjecture:
among all pairs points from a finite set in the euclidean plane, that are at maximal distance, ...
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Beardon's version of Poincaré's theorem for fundamental polygons
I am currently trying to understand Beardon's proof for Poincaré's Theorem, which can be found in his book The Geometry of Discrete Groups. The last condition in the theorem is giving me a headache to ...
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Are these two definitions of an NTA (nontangentially accessible) domain equivalent? If so, is the constant unchanged?
Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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On partitioning convex polygons into kites -2
We add a little to On partitioning convex polygons into kites
In his answer to above post, Tom Sirgedas has shown that a convex polygon having two successive angles acute or right is a sufficient ...
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153
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4-Flower free set family of 3-uniform sets (AKA f(3, 4)) largest construction?
On Polymath, the table shows that the largest known 3-uniform 4-flower free set family has size 38, sourced by this paper. I don't have access to the paper though, and wanted to see the set family ...
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Twisted Rupert property
It has recently been proven that the 2017 conjecture that all
convex polyhedra $P$ are Rupert is false:
"A convex polyhedron without Rupert's property,"
Jakob Steininger, Sergey Yurkevich. ...
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On partitioning convex polygons into kites
A kite is a quadrilateral with reflection symmetry across a diagonal. A kite with two opposite angles right is a right kite. If a pair of angles in a kite are equal and acute (obtuse), we may say the ...
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Outside-the-box in 3D: can the $27$ vertices of $\{0,1,2\}^3$ be visited with $13$ line segments connected at their endpoints, without repetition?
This is a variant of the 3D generalization of the well-known Thinking outside the box Nine dots problem I discussed in my previous MO post Optimal covering trails for every $k$-dimensional cubic ...
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Inequalities Between "m-measure" of Parallelotopes
Let $P \subset \mathbb{R}^n$ be an $n$-dimensional parallelotope generated by $n$ linearly independent vectors $v_1, v_2, \dots, v_n$.
For $m \leq n$, I define its $m$-measure $\mu_m(P)$ as the sum of ...
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On partitioning convex polygons into mutually congruent spiral polygons
Ref: A claim on partitioning a convex planar region into congruent pieces
Definitions: A spiral polygon is a simple polygon with exactly one chain of successive vertices that are all concave. A ...
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Partitioning planar convex regions into non convex and congruent pieces - 2
We add a little to Partitioning planar convex regions into non-convex and mutually congruent pieces.
A rectilinear polygon is a polygon all of whose sides meet at right angles. Thus the interior angle ...
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Partitioning planar convex regions into non-convex and mutually congruent pieces
For all even $n$, a disc or any rectangle can be (very obviously) cut into $n$ mutually congruent and non convex pieces. Now consider partition into an odd number of pieces instead:
Is there any ...
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Properties of almost additive sets in half-planes inside $\mathbb{R}^2$
A curious combinatorial question appeared whilst thinking about a problem in geometric group theory.
Suppose we have an (infinite) set $U \subset \{(x,y) \in \mathbb{R}^2 : y \ge0\} = H \subset \...
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Can a laser beam hit all points of $\{0,1,2\}^k \subset \mathbb{R}^k$ using $\frac{3^k-3}{2}$ mirrors only if emitted from outside the open $k$-cube?
Note: I already posted the $3$-dimensional version of this question on Mathematics Stack Exchange, but no answer has been received so far, so I hope that MathOverflow can be a more suitable place in ...
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139
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Color remapping and solution space intersections in combinatorial Puzzles
Characterizing Solution Space Preservation Under Color Remapping in Combinatorial Puzzles
I'm investigating the theoretical properties of color remapping functions on combinatorial puzzles (think ...
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Is the spectrum of the Hodge-Laplacian unique for a simply connected simplicial complex?
Imagine a 3-dimensional combinatorial simplicial complex, aka "triangulation", which is constructed by simply gluing together "N" tetrahedra via their faces. For simplicity assume ...
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To place n points on a planar convex region with average pairwise distance among the points maximized
Given a planar compact convex set $C$ and a number $n$, let us try to put $n$ points in $C$ such that the arithmetic mean of the $n\choose 2$ distances between them is to be maximized. Does this ...
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Does a convex polyhedron lose vertices when intersected with a translated and scaled copy?
Given a convex polyhedron $P=\{x\in\mathbb{R}^n\mid Hx\leq h\}$, a vector $c\in\mathbb{R}^n$ and a non-negative real number $\alpha\geq0$, is the following statement correct:
$$
\#vert(P\cap( c+\alpha ...
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On packing vectors in d dimensions
We try to add a bit to Packing obtuse vectors in $\mathbb{R}^d$
given an angle α, in d dimensional Euclidean space, how many vectors can be drawn such that the angle between any pair is at least α?
...
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Monsky's theorem in 3d
Monsky's theorem, which has a rather fancy proof, states that it is impossible to triangulate a square into an odd number of triangles of the same area.
I am interested to find out whether the 3-...
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Is there a standardized name for this incidence structure?
Let $m \geq 2$ be a natural number, let $\mathcal{A}$ be any set whose element we call points and let $\mathcal{B} \subseteq \mathcal{P}(A)$ be a set of subsets of $A$ (we call the elements of $\...
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On tiling scalene triangles with congruent polygons
For n not a perfect square, are there scalene triangles that can be tiled by n (mutually congruent) polygons? If so, how to characterise them?
Which are the values of n such that no triangle (of any ...
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A question on distribution of affine flats
Let $T_1 = |\{(U_1,U_2) \text{ $|$ } U_1 \text{ and } U_2 \text{ are affine flats of $\mathbb{F}_2^m$ such that } |dim(U_1) - dim(U_2)| \leq 1 \}|$ and T be the total number of affine flats of $\...
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Are there triangles that can tile at least one convex n-gon for any value of n?
Ref: https://en.wikipedia.org/wiki/Pinwheel_tiling
In a pinwheel tiling of the plane, is it true that for each n>2 there exists some convex n-gon (formed by edges in the layout) that is the union ...
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Ham-sandwich cut for pseudo configuration of points
I have the following question regarding a generalization of the classical theorem of 'Ham-Sandwich Cut' for pseudo point-configuration.
First, a few definitions. A collection of continuous curves in ...
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Optimal triangulation of points distributed on two parallel lines
Question:
what is the fastest algorithm for optimally triangulating a point set $P$ that is given as the union of two sorted sequences $A\cup B\ :=\ ([0,0],[x_{a_1},0],\,\dots,\,[x_{a_m},0],[1,0])\,\...
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On reptiles that are convex and non-parallelogram
A reptile is a shape that can be dissected into smaller copies of the same shape. A reptile is labelled rep-n if the dissection uses n copies.
Given any positive n, any parallelogram with sides in ...
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On ‘special chords’ of convex solids
Ref: On special points within convex solids with all planar sections passing through them having equal area.
Definition: Given any convex solid C. For any 2 distinct points on its surface, we have a ...
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Thrackle for trees
Suppose that we are given in the plane a set of $n$ points, $P$, and $m$ topological trees that pairwise intersect exactly once such that each leaf of each tree is from $P$.
Is it true that $m\le n$?
...
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Flag spheres, h-polynomials and log-concavity
The following is a major conjecture due to S. Gal
Let $\Delta$ be a flag simplicial complex that triangulates a sphere of dimension $d$. Then, the $h$-polynomial, $h_{\Delta}(x) = h_0 + h_1x+\cdots + ...
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Prove that at least two edges of a polyhedron does not intersect a given plane
The same question was asked on SE
https://math.stackexchange.com/questions/5075489/prove-that-at-least-two-edges-of-a-polyhedron-does-not-intersect-a-given-plane
Let $P$ be a polyhedron (not ...
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Convex polyhedra that have a specified set of planar sections
Adding a bit to this question: Regular polygon shadows of convex polyhedra
Can one construct a convex polyhedron that has planar sections that are equilateral triangle, square and regular pentagon? ...
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