All Questions
Tagged with polygons combinatorics
17
questions with no upvoted or accepted answers
7
votes
0
answers
372
views
When is it possible to find a regular $k$-gon in a centered $n$-gon?
For $n \geq 3$, say that a centered $n$-gon with $L$ layers is given by the origin, $(0,0)$ together with the points $$
\left\{\alpha\zeta_n^j + \beta\zeta_n^{j+1}\ \middle\vert\ 0 \leq j < n, 1 \...
- 5,072
6
votes
0
answers
297
views
Counting Regular polygons in Complete Graphs
The figure shows the correct $24-$gon, which held all the diagonals.
a) Find out how we got right triangles and squares (question for arbitrary $n$)?
b) How this problem can be generalized (if it is ...
- 17.9k
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
3
votes
2
answers
64
views
Need a hint regarding this question...
In how many ways can we select three vertices from a regular polygon having $2n+1$ sides ($n>0$) such that the resulting triangle contains the centre of the polygon?
- 31
2
votes
0
answers
80
views
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
1
vote
1
answer
54
views
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
1
vote
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
1
vote
0
answers
80
views
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, ...
- 1,271
1
vote
0
answers
243
views
Minimal diagonal intersections in a convex polygon
OEIS A006561 gives the number of intersection points in the diagonals of a regular polygon. There's a paper by Poonen. For 4 vertices to 12, the number of intersection points is:
$$1, 5, 13, 35, 49, ...
- 21.4k
1
vote
0
answers
80
views
All polygons satisfy the "normal" property.
A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
- 31
1
vote
0
answers
67
views
What is the probability to pass through $1\le m\le n$ vertices of an $n$-sided polygon after $t$ seconds?
Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and ...
- 7,299
1
vote
0
answers
190
views
Triangulations of the concave polygon
It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
- 11
1
vote
0
answers
91
views
Rectilinear polygons winding around a torus
A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($ 90^{\circ}$) and that of interior concave angles ($ 270^{\circ}$) is always $4$.
Consider a ...
- 1,394
1
vote
1
answer
195
views
Corresponding Triangulations of an (n+2)-gon to n Segments Connecting n+1 Collinear Points
So I'm asked to count the number of ways of connecting n+1 collinear points with n line segments subjected to the following constraints:
If the line is L
1) No segment passes below L.
2) Starting at ...
- 4,093
0
votes
0
answers
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, ...
- 466