# Questions tagged [polygons]

For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit

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### What is a gyrational square in this context?

This is a diagram of quadrilaterals and their duals. Within this diagram, there is a square (shown with 8 lines of symmetry) and right below that there is a "gyrational square" shown with no ...
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### How to prove that a figure tiles the plane (for example certain pentagons)

Is there a general way(s) to approach proving that a certain shape tiles the plane? For example the type 4 pentagonal tiling found here: https://en.wikipedia.org/wiki/Pentagonal_tiling
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### Generating fractal outlines

I am looking for an algorithm that will generate natural-looking (as in created by nature) polygonal shapes. The goal is to create 2D colorful art. This might be via parameterized fractals (I found ...
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### 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 ...
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### Given a convex polygon with integer coordinates vertices, how can you transform it so you can always find a point inside it with integer coordinates?

Given a convex polygon, with all its vertices having integer coordinates, you are allowed to perform one transformation T to all of its vertices (which are points), such that: T transforms a point in ...
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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 ...
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### Existence of a simple convex polygon of specified angle measures

Given $n \geq 3$ positive reals $\alpha_1, \dots, \alpha_n$ such that $\alpha_i < 180$ and $\alpha_1 + \dots + \alpha_n = 180(n - 2)$, how do we show the existence of an $n$-sided simple (convex) ...
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### How do we define exterior angle of a concave polygon whose interior is reflex?

How do we define exterior angle of a concave polygon whose interior is reflex? I have seen in few books and websites saying that, sum of all exterior angles of a concave polygon is $360$ degrees. ...
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### Given regular decagon $ABCDEFGHIJ$, and midpoint $M$ of $AB$, prove that $AD$, $CJ$, $EM$ concur

Let there be a regular decagon $ABCDEFGHIJ$, and $M$ the midpoint of $AB$. Prove $AD$, $CJ$, and $EM$ are concurrent. I was going to do this problem by making AD and CJ pass through a point K, and ...
133 views

### Prove four points on a parabola are concyclic

In this question a method is presented for solving a cubic equation using strainghtedge, compasses and a single additional conic section,the parabola $y=x^2$. Briefly, the method starts by introducing ...
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### What is the mathematical technique, or a branch in mathematics that represents fitting a line on to a set of points such that it becomes a polygon?

And what if the line that being fitted has an additional constraint of avoiding certain points while making the polygon? Is there some specific fields I need to be exploring? For instance, set theory. ...
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### Extremal problems for (non-convex) polygons on hexagonal lattices

I am interested in non-convex polygons and extremal properties thereof, specifically on hexagonal (honeycomb) lattices (or three valent and three colorable lattices in general). In particular, I am ...
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### Why is the sequence of sopfr($n!$) so similar to the sequence of the number of diagonals in $n$-polygons?

Could someone please enlighten me on why the sequence of sopfr(n!) (the sum of prime factors of n!) is seemingly related to the ...
1 vote
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### What is the relation between the Reuleaux triangle and the imaginary golden ratio?

The imaginary golden ratio is given by $$\varphi_i = \frac{1+\sqrt{3}i}{2}=e^{i\pi/3}$$ This has many properties in common with the golden ratio and has been adequately described here (Imaginary ...
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### Relax triangles in mesh flattening / UV map

I am working on an application that flattens 3D meshes into 2D injective maps such as this one: Essentially, the map is produced by flattening the object, ensuring that it fits within the 2D ...
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### Finding circumradius of regular polygon such a circle is inscribed

Say we want to inscribe a cicle with a regular polygon: The circumradius is is the distance to the vertices of the regular n-ngon. We can see on the image, this circumradius of a regular n-gon is ...
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### Archimedes: Inscribed regular polygon smaller perimeter than a circle?

When Archimedes found the upper and lower boundary for the value of pi, he used an $\color{red}{\textrm{inscribed regular polygon}}$ and a $\color{blue}{\textrm{encapsulating outer regular polygon}}$. ...
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### Largest circle inscribed in a cube

Given a cube of side length $a$, what is the the radius of the largest circle that can be inscribed in the cube? My Attempt: I just assumed that the largest circle is the incircle of the hexagon ...
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### How to determine the reflex angles in a concave polygon in 3D?

For a concave polygon in 2D, it's easy to use the cross product to determine the reflex angles, which are greater than $180^{\circ}$, but I wonder if there is a simple way to do it in 3D.
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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 ...
1 vote
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### Do there exist (non-Euclidean) equilateral $n$-gons whose angles are all right angles, for $n\geq6$? (via Numberphile)

In the YouTube video "5-Sided Square" from Numberphile, Cliff Stoll states that, if a "square" is a shape with sides of equal length whose angles are all $\pi/2$, then we can find: ...
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### what is the exact rules to figure out the total angle value? [closed]

Today I found a nice geometric pattern. In both of the images, the total value of outside angles (marked in red or the value) are respectively $180°$ and $540°$. But I can't figure out the exact ...
1 vote
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Let $ABCDE$ is a regular pentagon. If the area of yellow star (see the picture above) is equal to 24, then what the area of regular pentagon $PQRST$? My Attempt: To find area of regular pentagon $... 0 votes 2 answers 47 views ### Calculate the angle in the regular pentagon I want calculate the angle of$DAC$in the regular pentagon. But I'm not sure with my attempt. Is it right my attempt? Since$\angle EAB$divide into$\angle EAD$,$\angle DAC$,$\angle CAB$and$...
We have a space S, being partitioned into a set of polygons P containing $n$ polygons $P_1, P_2,..., P_n$. Given $n$ constants $k_1,k_2,...,k_n$. Apply a transformation $T$ from partition $P$ to ...