All Questions
70 questions
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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 ...
- 5,979
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 ...
- 4,898
0
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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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3
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1
answer
94
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Convex polygon shadows: Shortest equivalent segments
Let $P$ be a convex polygon.
Q1. What is the shortest collection of line segments $S$ inside $P$
with the property that both $P$ and $S$ have the same sequence of orthogonal shadows
as $P$ and $S$ ...
- 151k
16
votes
1
answer
888
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Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k
3
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1
answer
152
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Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
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2
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130
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On convex planar regions that can be cut into only a specified number of mutually congruent and connected pieces
References:
https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts
On congruent partitions of planar regions
https://research....
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3
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1
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238
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Least area and least perimeter triangles that contain a convex planar region - how different can they be?
Is there a planar convex region whose enclosing triangles of least perimeter and least area have different areas and different perimeters? And if so, which region maximizes the difference between the ...
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2
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1
answer
84
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What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
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5
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2
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307
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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 ...
- 3,153
15
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2
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863
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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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1
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124
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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 On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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1
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208
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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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235
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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 ...
- 2,773
2
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1
answer
273
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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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3
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175
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Cutting convex polygons into triangles of same diameter
This question continues 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 ...
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2
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1
answer
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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6
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2
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544
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On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
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22
votes
1
answer
886
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Happy ants never leave compact domain?
I am curious if the following seemingly simple question has an easy answer?
Consider an ant population of $N$ ants that lives in $\mathbb R^2$. Each ant can be labeled by some coordinate $x\in \mathbb ...
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1
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1
answer
89
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Vertices of 2 self-polar triangles lie on conic
I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?
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3
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1
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190
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On some centers of convex regions based on partitions
These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'.
Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
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12
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1
answer
373
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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 ...
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4
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1
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266
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A closed chain of $2n+1$-gon around $2n+1$-points
I posed a generalization of Theorem 3.2 In my paper
Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;...
- 1,776
5
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1
answer
156
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On folding a polygonal sheet
Consider a polygonal sheet $P$ of area $A$ with $N$ vertices (it material is not stretchable or tearable). Let $n$ be a positive integer >=2.
Question: Let $P$ lie on a flat plane. We need to fold ...
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9
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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
5
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1
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177
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Orientations of triples of points in the plane
Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements ...
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4
votes
2
answers
94
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Finding a not too slim triangulation with prescribed vertices on $\mathbb R^2$
Let us fix a constant $r>1$. Let $d(x,y)$ denote the distance between points $x,y\in \mathbb R^2$. Suppose we have a discreet subset $X\subset \mathbb R^2$ such that
1) For any two points $x,x'\...
- 14.3k
5
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139
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On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
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1
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2
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153
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Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
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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 $\...
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7
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1
answer
768
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To minimize the Hausdorff distance between convex polygonal regions
Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given two convex polygonal regions P1 and P2 on the ...
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9
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1
answer
338
views
Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
- 151k
2
votes
0
answers
98
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8-partition of a planar convex body by 4 concurrent lines
It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...
- 151k
3
votes
2
answers
323
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Minimum weight triangulation of lattice points in a circle
Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
- 151k
11
votes
1
answer
499
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Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
- 151k
6
votes
1
answer
429
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Bichromatic pencils
A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...
- 19k
6
votes
0
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164
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Sets of points avoiding small angles
(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...
- 151k
11
votes
2
answers
455
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Dodecahedral rolling distance
Let a dodecahedron sit on the plane,
with one face's vertices on an origin-centered unit circle.
Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal.
For any $p \in \mathbb{R}...
- 151k
9
votes
0
answers
237
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Herding sheep in a polygon
Imagine sheep fill a simple (simply connected) polygon $P$, except
at one vertex $x$ there is no sheep.
One convex vertex $g$ of $P$ is a gate through which the sheep should pass.
A herding dog sits ...
- 151k
10
votes
3
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537
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Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
- 151k
10
votes
2
answers
280
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Monochromatic point sets in two-colored plane
Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
Property M (for Monochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic ...
- 10.7k
2
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1
answer
189
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What is the maximal diameter of a cell in a particular partition of the simplex?
Consider a standard simplex with points $(p_1, \dots, p_n)$, $p_i \ge 0$, and $\sum_i p_i = 1$. Fix a set $\{q_k\}_{k=1}^K$ with $0\leq q_k \leq \infty$ and $i,j\in\{1, \dots, n\}$. Partition it via ...
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10
votes
1
answer
277
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Optimization of points on a plane
Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...
- 1,129
0
votes
0
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89
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What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?
I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
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4
votes
1
answer
159
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Best polygonal approximation to a polynomial $\pm$ c
Let a planar region $R$ be defined
by the vertical range bounded by
a polynomial $f(x) \pm c$ with $c>0$ a constant,
and with $x$ varying between the smallest and largest
roots of $f(x)$.
For ...
- 151k
6
votes
2
answers
410
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Existence of finite set of points in the revolving circles
Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...
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8
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2
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2k
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What's the name of this geometric mathematical modeling problem?
There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
- 549
5
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1
answer
307
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Panning for gold nuggets: a type of isoperimetric problem
Let $C$ be a unit-radius circle in the plane.
Suppose you have a total length $L$ of string available, and
your task is to connect chords of $C$ using no more
than $L$ of string to minimize the ...
- 151k
8
votes
4
answers
530
views
Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
- 151k
8
votes
2
answers
371
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Are angles between points enough to decide the realizability?
Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable ...
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