All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
2
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132
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Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
- 115
-1
votes
1
answer
502
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How to formulate such problem mathematically? (line continuation search) [closed]
I have an array of "lines" each defined by 2 points. I am working with only the line segments lying between those points. I need to search lines that could continue one another (relative to ...
- 1,446
1
vote
1
answer
78
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To optimally wrap convex laminae with paper
Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra
Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ...
- 151k
4
votes
0
answers
222
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What does it mean "parallel"?
I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following
Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear ...
CommunityBot
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3
votes
0
answers
76
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A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
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2
votes
1
answer
151
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On congruent partitions of planar regions
Given any integer $n$, any rectangular region or any sector of a disc (including the full disk as a boundary case) can be cut into $n$ mutually congruent pieces - by equally spaced parallel lines and ...
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1
vote
0
answers
72
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Intercept theorem in $\mathbb R^n$
The celebrated intercept theorem(also known as Thales's theorem) provides the ratios between the line segments created when two parallel lines are intercepted by two intersecting lines.
I'm looking ...
- 193
1
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0
answers
52
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On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
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2
votes
1
answer
504
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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
- 5,979
7
votes
1
answer
248
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Decidability of completing Penrose tilings
Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
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17
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0
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731
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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 ...
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1
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2
answers
232
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What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 35.5k
5
votes
4
answers
540
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Consider a graph $G$ with nonnegative edge weights.
Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight?
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4
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0
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114
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Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
1
vote
1
answer
3k
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Covering an arbitrary polygon with minimum number of squares
I have a problem whereby, given an arbitrary polygon with any number of points, I need to cover the whole area by a number of fixed size squares. I can easily find a set of squares which covers the ...
- 1,446
5
votes
1
answer
397
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How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 13.4k
4
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0
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232
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Illuminating a just-barely irrational polygon
As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Let $P$ be a rational polygon.
Then for ...
CommunityBot
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10
votes
0
answers
177
views
Minimum reflection paths in a mirror polygon
Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles,
and is non-self-intersecting;
also known as a rectilinear polygon.
Treat every edge of $P$ as a ...
- 151k
6
votes
1
answer
435
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On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
- 561
6
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1
answer
631
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On covering convex 2D regions with rectangles
Given a convex 2D region $C$ and a positive integer $N$. We need to cover $C$ with $N$ rectangles such that the sum of the areas of the $N$ rectangles is the least – no further constraints on the ...
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1
vote
0
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57
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Inside-out dissections of solids
We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...
- 63.9k
1
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0
answers
41
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About the number of faces of the conification of a polytope
Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
- 131
1
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1
answer
127
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Smallest trapeziums containing a given convex n-gon
Question: Given a planar convex $n$-gon $C$, to find the smallest area / smallest perimeter trapezium (trapezoid) - a convex quadrilateral with at least one pair of mutually parallel edges - that ...
- 151k
6
votes
3
answers
705
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Minimum space dimension to place n-points knowing pairwise distances
Let $P$ be a set of $n$ points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those $n$ points with respect to ...
- 63.9k
3
votes
0
answers
65
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Cutting triangles into triangles with equal longest side
This post elaborates on a specific instance of Cutting convex polygons into triangles of same diameter .
Question: For any integer n, can any triangle be cut into n non-degenerate triangles all of ...
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4
votes
2
answers
254
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Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?
Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let $\phi: G_{P_1}\to G_{P_2}...
- 156k
0
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0
answers
74
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The closest ellipse and circle to a given triangle - 2
We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...
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5
votes
2
answers
134
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Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?
Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
- 13.6k
1
vote
0
answers
69
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Least number of squares of size N that a set of R rectangles can occupy
Given a set $R$ of rectangles of different positive integer sizes, and any number of squares of the same size $N\in\mathbb{N}$, what's the least number of squares $C$ that all the rectangles together ...
- 119
14
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4
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453
views
Smallest containing simplex
Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...
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18
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2
answers
573
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Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
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4
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0
answers
132
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Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
- 13.6k
4
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2
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260
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Discrete isoperimetric problems
It is well-known that among all planar curves, the circle — invariant under $O(2)$ — has the best isoperimetric ratio. Similarly, among all $n$-gons, the regular $n$-gon — invariant under the dihedral ...
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0
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46
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Kissing behavior of planar regions
This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...
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18
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3
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2k
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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 ...
- 4,713
14
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2
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540
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Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
$M(M(a,b),M(a,c))=...
- 27.7k
3
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0
answers
135
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Intersecting the unit n-cube and (n-1)-planes
(Is this a known problem?)
Question Let $\ 1<n\in\mathbb N.\ $ What is the greatest $(n-1)$-area
$\ S(n)\ $ of $\ L\cap I^n\ $ where $\ I^n\subseteq\mathbb R^n\ $ is the unit cube, and $\ L\ $ ...
- 6,337
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1
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55
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On 'axiality' of planar convex regions
Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
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270
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Regular $n$-gon with diagonals: bounds on area of largest cell?
Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
- 63.9k
3
votes
1
answer
111
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Constrained morphing of polygons
This post continues 'Constrained morphing' of planar convex regions
If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one ...
- 151k
2
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1
answer
84
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'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
- 5,979
36
votes
2
answers
2k
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Bodies of constant width?
In two-dimensional case one can generalize figures of constant width as figures which can rotate in a convex polygon.
Here is one example which can be used to drill triangular holes:
I would like to ...
- 5,971
9
votes
2
answers
310
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Generalized figures of constant width
Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$?
This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
- 45k
2
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0
answers
233
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Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
9
votes
3
answers
1k
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What rectangles can a set of rectangles tile?
(I asked this question first on math.stackexchange, but did not get any responses so I thought I would try here.)
If we have a set of $p_i \times q_i$ rectangles ($p_i, q_i \in \mathbf{N}$), which $m \...
- 63.9k
1
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0
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Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
- 63.9k
3
votes
3
answers
314
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4-polytope with vertices at the binary octahedral group
Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identifying $H$ with $R^4$).
The binary tetrahedral group lies at the vertices of the so-called ...
- 4,713
5
votes
1
answer
230
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Which pyramids fill space?
Let us define a pyramid as a convex polyhedron with one quadrilateral face and four triangular faces.
Question: How many pyramids (or families of pyramids) are known that can fill 3D space without ...
CommunityBot
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14
votes
1
answer
280
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How many distances are required to calculate all distances among $n$ points in the Euclidean plane?
I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
- 45k
15
votes
2
answers
779
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How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
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