All Questions
18
questions
1
vote
1
answer
70
views
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
300
views
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
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
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
0
votes
1
answer
788
views
How many obtuse angle triangles are possible in a regular Heptagon by joining its vertices?
I am only able to make one possible case,
Where we take any 3-consecutive vertices, since one of the vertices contains angle of Heptagon, which is approximately 128.57°, we get 7 such triangles.
I am ...
- 19
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 |\...
1
vote
1
answer
332
views
Polygon Diagonal Combinatorics
A diagonal for a polygon is defined as the line segment joining two non-adjacent points. Given an n-sided polygon, how many different diagonals can be drawn for this polygon?
I know that the number of ...
- 150
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
1
vote
2
answers
772
views
Schlafli symbol determining number of faces
Schlafli symbols are used to describe polytopes, but can also be used to describe more general objects through the use of flags. In particular, some information can be readily 'read-off' from a ...
- 1,903
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
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
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
0
votes
1
answer
67
views
Number of Pieces a regular $n$-gon is cut into by its diagonals [closed]
In how many pieces a regular n-gon is cut into by its diagonals?
I need a general formula.
By inspection, I have the solution to some lower values of $n$.
For $n=3,4,5,6$ solutions are $1, 4, 11, ...
0
votes
1
answer
108
views
Intersections in polygons
I'm having troubles solving the following problem which is about combinatorics:
let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices.
Each vertices are supposed to connect ...
- 285
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