All Questions
Tagged with polygons combinatorics
66
questions
1
vote
1
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52
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Maximal irregular polygon inside a regular polygon
Problem:
We have a regular $n$-gon. We want to choose some of it's vertices ($A_1, A_2, \ldots, A_m$), so these vertices form a completely irregular $m$-gon. Meaning that all of it's sides have ...
- 47
0
votes
1
answer
100
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Understanding a proof on IMO shortlist 2016 C3
The problem goes as follow:
Let $n$ be a positive integer relatively prime to 6. We paint the vertices of a regular
$n$-gon with three colours so that there is an odd number of vertices of each colour....
- 779
7
votes
2
answers
238
views
How to enumerate unique lattice polygons for a given area using Pick's Theorem?
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of integer points interior to the polygon, and let $b$ be the number of integer points on ...
- 1,913
2
votes
0
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78
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How many ways to glue a $4n$-gon to a genus $n$ surface?
In this question :
Two Fundamental Polygons for the Double Torus?
Lee Mosher says
There are four octagon gluing patterns (up to rotation and relabelling) which give a double torus.
It is a very ...
- 21
0
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2
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124
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Number of diagonals in polygon connecting different vertices
I run into a combinatorics problem recently. Let's imagine we have a n-sided polygon, the number of diagonals is easily
$$N=\frac{n(n-3)}{2}$$
However, for my work, I need to group the vertices in ...
- 29
1
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0
answers
36
views
Proving Lagrange four square theorem from the "sum of three triangle" theorem
It was proven by Gauss that every integer is the sum of at most three triangle numbers, and by Lagrange that every integer is the sum of at most four squares.
The triangles are the set $\big\{1,3,6,...
- 702
3
votes
1
answer
78
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Maximal cliques on graphs on the vertices of regular $2n$-gons
Consider the vertices of a regular $2n$-gon in the plane, and label the points clockwise $1,\dots,2n$. For any vertex $v$ let $-v$ denote its opposite vertex on the polygon.
Define a graph $G$ which ...
- 2,454
1
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1
answer
69
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What is the largest possible value of the smallest angle between the diagonals of an $n$-gon for even $n$?
In this answer, I showed that the maximum value of the smallest angle between two diagonals of a $21-$gon is $8\frac{4}{7}$ degrees. The method given generalizes to all polygons with an odd number of ...
- 4,057
1
vote
2
answers
297
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Consider the diagonals of a 21-gon. Prove that at least one angle of less than 1 degree is formed.
I think it should be solved using the pigeonhole principle. The answer is:
A $21-$gon has $189$ diagonals. If through a point in the plane, we draw parallels to these diagonals, $2 × 189 = 378$ ...
- 11
0
votes
0
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31
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Collinear points in the Happy Ending Problem
Recalling the statement of the Happy Ending Problem, we see that
For any $k \in \mathbb{N}$ we may find a $n=n(k) \in \mathbb{N}$ such that every $n$ points in the plane, where no 3 are collinear, ...
- 456
5
votes
1
answer
287
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Efficient way to find all polygons of the same shape within a set, regardless of position, scale, or rotation
I've got a big set of 2D polygons described as a set of points. I would like to take this set of polygons and find any that are the same shape, regardless of rotation, translation, or scale.
Each ...
- 295
4
votes
0
answers
84
views
For which natural numbers 𝑛 ≥ 3 is it possible to cut a regular 𝑛-gon into smaller pieces with regular polygonal shape?
I have been working on this question and I found that any regular polygon with n sides works.My claim is that we can cut any regular polygon of n sides into smaller regular polygons with n sides.And ...
- 81
0
votes
1
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387
views
A polygon has 60 diagonals. How many sides does it have?
I've used this formula to calculate the solution.
The number of diagonals of a polygon is $(n(n-3))/2$, where $n$ is the number of sides of a polygon.
But I'm getting answer as decimal point. Is the ...
user956073
4
votes
1
answer
884
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Recursivley count triangulations of a convex polygon
I am trying to find a recursive number of different triangulations of a convex polygon with $n$ vertices.
After some searching I found that the number can be expressed using catalan numbers, this ...
1
vote
0
answers
78
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Convex hull of combinatorial Zonotopes
Given a set of vectors/generators $V \subset \mathbb{R}^2$, one can obtain a Zonotope via Minkowski Sum $Z = \bigoplus_{i \in V} i$.
On the other hand, Given a set of sets of generators $C = \{ V_1, ...
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