All Questions
Tagged with polygons combinatorics
66
questions
23
votes
4
answers
93k
views
How many triangles can be formed by the vertices of a regular polygon of $n$ sides?
How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle ?
I have no idea where I should start to ...
- 335
22
votes
8
answers
3k
views
Number of ways to connect sets of $k$ dots in a perfect $n$-gon
Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
- 821
14
votes
3
answers
38k
views
What is the number of intersections of diagonals in a convex equilateral polygon?
Question: [See here for definitions]. Consider an arbitrary convex regular polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which $\alpha_i$ ...
user180918
10
votes
1
answer
398
views
Intersections of circles drawn on vertices of regular polygons
Using only a compass, draw all possible circles on the vertices of a regular $n$-sided polygon.
(That is, in every ordered pair of vertices one is the center, and their distance is the radius.)
...
- 12.5k
9
votes
2
answers
432
views
Why does the term ${\frac{1}{n-1}} {2n-4\choose n-2}$ counts the number of possible triangulations in a polygon?
In the given picture bellow, it counts the number of different triangloations in a polygon, how do the get to this expression, why is it:
$$
{2n-4\choose n-2}
$$
and why do we multiply it by $${\...
- 1,661
9
votes
1
answer
460
views
Number of chords in a $n$-gon if each chord is crossed at most $k$ times
Consider an $n$-gon where we denote the points by $v_1, \dots, v_n$.
If we allow each chord (internal edge of the $n$-gon) to have at most $k$ crossings, how many chords can we put into the $n$-gon (...
- 61
8
votes
2
answers
541
views
Number of points chosen form a polygon to have no isosceles and equilateral triangles.
Let $\Omega$ be a regular polygon with $n$ sides. Let's choose $\Gamma$ a set of vertices, for which any triangle with the vertices in $\Gamma$ is neither isosceles, nor equilateral. Find $\max |\...
7
votes
2
answers
241
views
How to enumerate unique lattice polygons for a given area using Pick's Theorem?
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of integer points interior to the polygon, and let $b$ be the number of integer points on ...
- 1,913
7
votes
0
answers
372
views
When is it possible to find a regular $k$-gon in a centered $n$-gon?
For $n \geq 3$, say that a centered $n$-gon with $L$ layers is given by the origin, $(0,0)$ together with the points $$
\left\{\alpha\zeta_n^j + \beta\zeta_n^{j+1}\ \middle\vert\ 0 \leq j < n, 1 \...
- 5,072
6
votes
1
answer
599
views
Smallest circumscribed polygon around regular polygons
Given a regular $n$-gon $Q$, there are many polygons $P$ that entirely contain $Q$, and such that all $n$ vertices of $Q$ lie on edges of $P$. These circumscribing polygons $P$ have different numbers ...
- 63
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
5
votes
3
answers
151
views
Lattice embeddings of a polygon
Consider the four lattice polygons below. Each shape is over the coordinates.
If reflected or flipped on the major axes and diagonals, these four polygons remain distinct. However, this is the ...
- 21.4k
5
votes
1
answer
293
views
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
2
answers
9k
views
Number of triangles in a regular polygon
A regular polygon with $n$ sides. Where $(n > 5)$. The number of triangles whose vertices are joining non-adjacent vertices of the polygon is?
- 345
4
votes
1
answer
896
views
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 ...
