All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
2
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1
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202
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To cut a triangle into $n$ $p$-sided polygonal regions
Given any triangular region and two integers $n$ and $p$ which can be large and $p > 4$. It is needed to cut the triangle into $n$ $p$-gons (e.g., cut a triangle into 10 heptagons). Among the $p$-...
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10
votes
1
answer
159
views
For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
- 6,652
10
votes
1
answer
673
views
A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
- 105k
11
votes
1
answer
652
views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is often stated briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
- 13.6k
9
votes
0
answers
144
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
10
votes
3
answers
460
views
Do triple-linked graphs exist?
Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
- 13.6k
2
votes
1
answer
108
views
Discrete isoperimetric inequality involving the diameter of an n-gon
I am interested in discrete isoperimetric-type inequalities that allow one to bound the perimeter of an $n$-gon from above (as opposed to below, as in the classical case when one bounds the perimeter ...
- 1,625
6
votes
1
answer
413
views
How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 506
10
votes
2
answers
255
views
Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 13.6k
13
votes
0
answers
378
views
Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
0
votes
0
answers
176
views
How to find a configuration of lines
In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
- 61
7
votes
0
answers
316
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 5,409
0
votes
0
answers
106
views
Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$?
Any helpful answer would be appreciated. Thank you!
- 63.9k
3
votes
1
answer
484
views
On some infinite planar arrangements with triangles
Background: Given a convex region C. One can define a graph corresponding to a planar arrangement of non overlapping congruent copies of C - each unit C is a node and an edge connects it to another ...
CommunityBot
- 1
5
votes
1
answer
247
views
Question on the exact largest minimum angle
Could anyone help find the EXACT largest minimum angle between any pair of lines among 5 lines passing through the origin in $\mathbb{R}^3$? Additionally, what is the exact largest minimum angle ...
- 61
3
votes
1
answer
239
views
The realization space of non-convex polyhedra - What is known?
The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
CommunityBot
- 1
4
votes
0
answers
66
views
Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 63.9k
10
votes
6
answers
700
views
Tiling with similar tiles
Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...
- 677
4
votes
1
answer
330
views
Billiard circuits in pentagons
A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...
- 2,370
0
votes
0
answers
67
views
Comparing partitions of a given planar convex region into pieces with equal diameter and pieces of equal width
We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
- 5,979
2
votes
1
answer
147
views
Are there polytopes with precisely two realizations?
A convex polytope is projectively unique if it has a unique realization up to projective transformations. Such polytopes are somewhat mysterious but still well-studied. Examples are simplices, the ...
- 732
4
votes
2
answers
341
views
Cutting convex regions into equal diameter and equal least width pieces - 2
This post is a spinoff from Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points in ...
- 5,979
25
votes
1
answer
513
views
Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
- 156k
19
votes
5
answers
21k
views
Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
- 2,370
1
vote
0
answers
68
views
Name of the perspector of the orthic triangle and excentral triangle
The orthic triangle and tangential triangles of a given triangle are in perspective. What's the official kimberling center associated with this perspector?
- 5,429
8
votes
0
answers
149
views
Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
2
votes
2
answers
163
views
References for geometric properties of optimal Euclidean traveling salesman tour
Consider a finite set of points $V \subseteq \mathbb{R}^2 $ as a TSP-instance under the standard $\| \cdot \|_2$ norm. (TSP stands for traveling salesman tour.) We know that every optimal TSP tour $T$ ...
CommunityBot
- 1
3
votes
1
answer
285
views
Name this kimberling center
The lines which connect the vertices of a triangle with the tangent points between the Spieker circle and the medial triangle are concurrent. Which kimberling center does this point correspond to?
- 4,713
14
votes
1
answer
642
views
Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from an ...
- 45k
2
votes
0
answers
63
views
Convex planar regions such that every boundary point has a 'fair bisector' passing thru it
We add a little to On 'fair bisectors' of planar convex regions and A claim on the concurrency of area bisectors of planar convex regions .
A fair bisector of a planar convex region is a line ...
- 3,068
1
vote
0
answers
67
views
Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
0
votes
0
answers
82
views
On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
- 5,979
3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
7
votes
1
answer
498
views
Is there a bicyclic irregular pentagon in integers?
Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well?
If it does ...
- 2,370
12
votes
1
answer
373
views
A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
- 4,713
4
votes
2
answers
1k
views
Polyline averaging
I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...
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1
vote
1
answer
134
views
An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
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3
votes
0
answers
136
views
If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
- 5,979
3
votes
1
answer
253
views
Nagel line of a tetrahedron?
It's well known that there is an analogy for the Euler line in a tetrahedron, but is there also an analogy for the nagel line of a tetrahedron? I can't seem to find any decent literature talking about ...
- 109k
1
vote
0
answers
42
views
On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
1
vote
0
answers
59
views
What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
2
votes
1
answer
302
views
Are two convex solids with all corresponding shadows equal in area congruent?
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 ...
- 5,979
4
votes
1
answer
303
views
On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
- 5,979
1
vote
0
answers
53
views
The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
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15
votes
1
answer
530
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
- 17k
9
votes
1
answer
542
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Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"
I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...
1
vote
0
answers
44
views
On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
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2
votes
4
answers
997
views
Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
- 2,370
1
vote
0
answers
52
views
'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
- 5,979
4
votes
3
answers
347
views
Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
- 13.6k