Questions tagged [discrete-geometry]

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

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Packing densities of non-centrally symmetric planar convex regions

Reference: https://en.wikipedia.org/wiki/Smoothed_octagon Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...
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Polyhedrons and their centers of mass

Given a convex polyhedron, one considers 3 possibilities: wireframe - only the edges of the polyhedron have mass which is uniformly distributed. surface - only the surface is massive with uniform ...
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Rigid monohedral tilers

Say that a tile $T$ that alone can tile the plane—a monohedral tile—is rigid if it is not the case that $T$ can be slightly deformed to $T'$ so that: $T'$ can also tile the plane $T'$ is arbitrarily ...
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Variants of the Euler Brick

Reference: https://en.wikipedia.org/wiki/Euler_brick An Euler brick is a rectangular cuboid whose edges and face diagonals all have integer lengths. A perfect Euler brick is one where the longest ...
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1answer
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Has this curious "duality" of weighted $K_4$ already been noticed?

A complete symmetric graph with $n=4$ vertices, i.e. a $K_4$ is the disjoint union of three perfect matchings $M_{\text{min}},M_{\text{mid}},M_{\text{max}}$ of which $M_{\text{min}}$ denotes the ...
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Polytope where each vertex belongs to all but two facets

Let $P$ be a (convex, bounded) polytope with the following property: for every vertex $v$, there are exactly two facets which do not contain $v$. Does it follow that $P$ is (combinatorially) a ...
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Shadows and planar sections of polyhedra – 2

This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal ...
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Shadows and planar sections of polyhedra

By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
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A Variant of the Malfatti Problem

See https://en.wikipedia.org/wiki/Malfatti_circles for an introduction to Malfatti's problem. The above page also states that for n >3, the question of whether a greedy method (at each step, the ...
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1answer
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Embedding linklessly embeddable graphs without Borromean rings

A linklessly embeddable graph is a graph which can be embedded into $\Bbb R^3$ so that no two of its cycles are linked. For example, the Petersen graph is not such a graph. Now, I can think of another ...
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On intersections of several convex regions

Question: Given n convex planar regions. Required to place them (in suitable position and orientation) so that that part of the plane lying under all the regions (their common intersection) is of ...
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Partitioning a set of lattice points in the plane into rectangles

The "long comment" by Pietro Majer on Reference for puzzle on dividing piles and scoring products suggests the following problem. Let $S$ be a finite subset of $\mathbb{Z}\times \mathbb{Z}$. ...
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Set of unit vectors such that among any three there is an orthogonal pair

I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as: Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
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Tiling a Jordan polygon

I saw this problem some years ago, don't remember the source: Let $P$ be a Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) that can be tiled with ...
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combinatoric proof $\sum_{i=0}^{n}(-1)^i\binom{n}{i}\binom{n-i+k-1}{k}=\binom{k-1}{k-n}$ [closed]

I would like help with combinatorial proof , not algebraic proof . Thank you for your time $\sum_{i=0}^{n}(-1)^i\binom{n}{i}\binom{n-i+k-1}{k}=\binom{k-1}{k-n}$
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Uniformization of triangulation on a sphere up to Moebius transformations

This is not the most precise question but rather a hope that someone has seen something like this. I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...
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Planar subsets with many pairs of points on distance $1$

Let $X$ be a subset of $\mathbb R^2$ consisting of $n$ distinct points. Let $d_1(X)$ be the number of pairs of points of $X$ on distance $1$ from each other. Define $$d_1(n)=\sup_{X\subset \mathbb R^2|...
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On planar sections of 3D convex bodies

Consider the space of planar sections of any given convex 3D body. Basic Question: What is the lower bound for the ratio $$\frac{\text{area of section of greatest perimeter}} {\text{area of section of ...
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1answer
109 views

Which convex pentagon gives least packing density?

Among all convex pentagons, does the regular pentagon give least packing density? Further question: For each n > 6, is the regular n-gon the minimum of packing density? An analogous question can be ...
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Facility location and traveling salesman

This question is based on Distributing points evenly on a sphere and Facility location on manifolds The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
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The polytope algebras generated by polytopes with rational vs arbitrary vertices

The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows. Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated ...
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Projective planes over algebraically closed fields

Suppose I am given a projective plane $P \cong \mathbb{P}^2(k)$ over a (commutative) field $k$. With "projective plane," I mean the point-line geometry (and not, for instance, the scheme): $...
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3answers
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On packing axisymmetric bodies in 3D

Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body. Is the following claim valid? Claim: ...
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Cutting of a regular polygon into congruent pieces

Question. For which $N$ it is possible to cut a regular $N$-gon into congruent pieces such that the center of the regular polygon lies strictly inside one of the pieces? For $N=3,4$ there are trivial ...
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How many regular d-dimensional simplices of side length 1/2 are required to cover a regular d-dimensional simplex of side length 1?

For positive integers $n$ and $d$ satisfying $d = n-1$, let the $d$-dimensional regular simplex of side-length $\sqrt{2}$ be $X = \{(x_1, x_2, \cdots, x_n) \in \mathbb{R}^n: x_1+x_2+\cdots + x_n = 1, ...
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What do optimal tours tell about finite point sets?

Let $T_{\mathrm{MIN}}$ and $T_{\mathrm{MAX}}$ denote the shortest, resp. longest Hamilton cycle through a set of $n=2k+1$ points. Let further $S_{\mathrm{MIN}}$ and $S_{\mathrm{MAX}}$ be the "...
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Are there any convex pentagonal rep-tiles?

A rep-tile is a shape that can tile larger copies of the same shape. Question 1: Are there any convex pentagons that are also rep-tiles? Remarks: 15 convex pentagonal tiles of the plane are known and ...
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532 views

Three squares in a rectangle

One of my colleagues gave me the following problem about 15 years ago: Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (...
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A center of convex planar regions based on chords

This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues mathoverflow.net/a/383528/143513. A point $P$ in the interior of a planar convex region $C$ divides ...
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Multi-layered wrapping of polyhedra

This post continues from How big a box can you wrap with a given polygon? and Convex polyhedra that can be folded from convex polygons. One can also mention 'k-fold coverings of the plane' as examined ...
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Convex polyhedra that can be folded from convex polygons

This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf. Therein is stated the theorem: Every convex polygon folds to an infinite number (a continuum) of noncongruent ...
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597 views

Happy ending problem – Why not a proof by induction?

I have been thinking for a while on the happy ending problem, looking for approaches to attack the Erdős–Szekeres conjecture: the smallest number of points for which any general position arrangement ...
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How big a box can you wrap with a given polygon?

Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
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1answer
50 views

Covering radius of a lattice from relevant vectors

Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by $$ \mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|_2\leq \|x-v\|_2~\forall v\in L\setminus\{0\}\big\}. $$ Considering the ...
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Folding polygons into 'vessels'

This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
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Common basis property

Let F be a field and let $U_1, \ldots, U_k$ be subspaces of $F^n$. Say that $U_1, \ldots, U_k$ has a common basis if there is a basis $b_1,\ldots,b_n$ of $F^n$ such that each $U_i$ is a span of some ...
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1answer
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Convex polyhedra with non-congruent faces

Question: Are there convex polyhedra wherein all faces are convex polygons with same area and perimeter and no two faces are mutually congruent? Remarks: If the answer to above is "no", then,...
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Real rootedness of a polynomial with binomial coefficients

It is possible to show using diverse techniques that the following polynomial: $$P_n(x)=1 + \binom{n}{2} x + \binom{n}{4} x^2 + \binom{n}{6} x^3 + \binom{n}{8} x^4 +\ldots + \binom{n}{2\lfloor\tfrac{n}...
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Counting lattice polytopes by volume

For any $n \in \mathbb{N}$ and $B \in \mathbb{R}_{\geq 0}$, let $\mathcal{P}(n,B)$ be the set of $n$-dimensional convex polytopes $\Delta \subseteq \mathbb{R}^n$, taken up to integral, unimodular ...
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On a possible variant of Monsky's theorem

See Wikipedia for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area. Questions: Are there quadrilaterals that allow partition into ...
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Is the unit-stick number of a knot equal to its stick number?

Define the unit-stick number $\sigma_1(K)$ of a knot $K$ to be the fewest unit-length sticks that can realize $K$. Clearly $\sigma_1(K)$ is at least the stick number $\sigma(K)$. It is known that the ...
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Cutting convex regions into equal diameter and equal least width pieces - 3

We add a bit to Cutting convex regions into equal diameter and equal least width pieces - 2. There, we asked, for example: If we divide a 2D convex region C into n convex pieces such that the maximum ...
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Does there exist numerically balanced dice with odd numbers of faces?

This question is motivated by "Numerically Balanced Dice" by Bosch, Fathauer, and Segerman, in which they produced the most numerically balanced d20 and d120. After reading this paper, I ...
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Arrangement of points, lines, and planes

Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties? every line is incident with four points and ...
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What can we say about the intersection of an algebraic and product set?

This question is a bit vague by design. Let $F$ be a field. I'm mostly interested in finite fields, but would also be interested in $R$ or $C$. Let $S \subset F^d$ be an algebraic set and let $A = ...
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An isoperimetric inequality for a Hamming sphere

Let $S$ be a subset of $\{0,1\}^n$ such that every element of $S$ has weight (the number of $1$-coordinates) $k$ (may be not all elements with such weight belong to $S$). Denote by $S_r$ the $r$-...
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1answer
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Triangulations of point sets — obtuse and acute triangles

Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of ...
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Definition of "regular" in Stringham's "Regular figures in n-dimensional space"

I've been reading Irving Stringham's 1880 thesis, "Regular Figures in n-dimensional Space" (only 14 pages!), after it was mentioned by Coxeter in Regular Polytopes (§7.x). I'm confused about ...
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On Convex Regions Contained in Convex Polygons

In this earlier post - On convex polygons contained in convex polygons - the following was asked: Given a convex n-gon C, find the smallest convex region R such that C is the smallest n-gon that ...
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Name for point sets with trivial optimal Hamilton cycle

Question: is there an established name for sets of $n$ points in the euclidean plane whose shortest Hamilton cycles consists of the $n$ pairs of points having the $n$ smallest distances? Names for ...

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