Questions tagged [polygons]
For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit
1,326
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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 ...
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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 ...
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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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2
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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 ...
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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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Construction of a pentagon in which an angle bisector at a certain vertex is also a perpendicular bisector of a side opposite to that vertex
Construct a pentagon $ABCDE$ if lengths of its sides are $a,b,c,d$ and $e$ ($|AB|=a,|BC|=b,...$), and the bisector of the angle at vertex $D$ is also the perpendicular bisector of $AB$.
I know that $...
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Proof Concerning homeomorphisms of $\mathbb{P}^2$
Is the following proof valid?
CLAIM:
The space obtained by attaching a disc to a Mobius Strip along the boundary is homeomorphic to the projective plane.
PROOF:
We begin by showing that the boundary ...
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3
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length of side of a regular $n$-gon is less than length of any diagonal
In any regular polygon with $n\ge 4$ sides, why is any side length strictly length to any diagonal length? (A diagonal is defined as the line segment joining non-adjacent vertices)
This is intutitvely ...
user10575
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Finding safe points in polygons
Let $P$ be an axes-parallel polygon. A point $(x,y)\in P$ is called safe if for any pair $d_x\in[0,1],d_y\in[0,1]$, either $(x+d_x,y+d_y)$ or $(x-d_x, y-d_y)$ or both are in $P$.
Figuratively, suppose ...
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Area of region of $n$-sided regular polygon made from points that are closer to center than perimeter
If we have an $n$-sided regular polygon with side lengths of $a$, what is the area consisting of all the points whose distances to the center of the polygon are shorter than their distances to the ...
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Decomposing NURBS curve into piecewise Bezier segments
I have a question concerning a paper. In it the authors try to approximate a NURBS curve by biarcs and to do so, they first need to find a polygonal approximation of the NURBS curve. They chose a ...
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For an arbitrary vertex in an arbitrary polygon how to determine which of the vertex's angles lies inside the polygon knowing only the coordinates?
Could someone explain to me how to do what the title states?
For an arbitrary vertex in an arbitrary polygon how to determine which
of the vertex's 2 angles lies inside the polygon knowing only the
...
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On average, how many times must a circular pizza be randomly cut, to get a piece with no curved edge?
On a circular pizza, we make a random straight cut by choosing two uniformly random points on the perimeter and cutting through them.
On average, how many times must the pizza be randomly cut, to get ...
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Girard's Formula for Arbitrary Spherical Polygons (concave, self-intersecting, etc.)
In every reference I've seen, the area of an $n$-sided spherical polygon is given by the sum of all interior angles, minus some amount of spherical excess, or
$$\text{Area}(\text{polygon}) = \sum_{i} ...
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Is there a formula to calculate the number of In-between Points for each adjacent $2D$ lattice polygon vertices?
Lemma
For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions.
For example, here are 3 circles for $k=1,2,3$. The blue dots represent the vertices (integer solutions), then ...
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How to calculate dispersion measure for a set of rectangles?
I want to calculate the dispersion measure of a set of rectangle(30) in such a way that the rotation is considered in the calculation. All the rectangles have same dimension. A rectangle $r_i$ is ...
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Is there a question that contains no numbers except $1$, whose answer is $\pi/7$?
Is there a question that contains no numbers, except possibly $1$, whose answer is $\pi/7$ ?
There are plenty of questions with no numbers except $1$, whose answer is $\pi/n$ for small integer values ...
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A regular $n$-gon contains a regular $(n+1)$-gon, with no sides coinciding. What is the maximum number of points of contact between them?
A regular $n$-gon contains a regular $(n+1)$-gon. That is, they are in the same plane, and no part of the regular $(n+1)$-gon is outside of the regular $n$-gon. None of their sides coincide. There are ...
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Trajectory of light rays in a mirror polygon
Given a general polygon and we are given a ray of light bouncing between the sides of the polygon where each side is a mirror. they hit at points $P_1,P_2...$, we define $\alpha_i$ to be the smaller ...
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Golden number, diagonals of polygons and continued fractions
It is well know that, for a pentagon with unit side, the diagonal $\delta$ is such that
$$
\delta : 1=1:(\delta-1)
$$
so that its length is the positive solution of the equation $x^2-x-1=0$.
i.e. the ...
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Conjecture about expected distance between two points in a regular polygon
Point $P$ is a uniformly random point on the perimeter of a regular polygon.
Point $C$ is a fixed point somewhere on the circle inscribed in the polygon.
Is the following conjecture true:
The ...
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Are circles related to the sum of exterior angles of convex polygons equalling 360 degrees?
The sum of all the exterior angles of a convex polygon equal 360 degrees, and a full circle is equal to 360 degrees. The more sides you add to a convex polygon, the closer you get to a circle.
So do ...
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How to find the area of a rhombus with two given side lengths and given height?
Link to the Rhombus: Rhombus with given side lengths of 11 and a height of 8
Edit: I just realized that this rhombus is a parallelogram so because the 8 inches is the height and that the 11 inches is ...
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Proof only three types of hexagonal, convex monohedral tessellations
Where can I get a proof that there exist exactly three types of convex, hexagonal, monohedral tessellations of the plane?
I'm already aware of Reinhardt thesis and Bollobás paper (1963). I was ...
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Reference request: polygons with an odd amount of sides
Is there a name for polygons with an odd number of sides or are they just simply $(2n-1)$-gons?
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Number of edges vs number of vertices in $\mathbb{R}^2$?
I was thinking about the name "Triangle", when I realized that although we usually think of polygons in terms of the number of their sides. However, when I searched the origin of the word &...
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Given the circumscribed circle diameter and the number of points of a regular polygon, what is its side length?
Is it possible to determine the side length of a regular polygon if you know only the number of points it has, and the "width" (or possibly more geometrically accurately, the circumscribed ...
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Every triangulation of a simple closed polygon in the plane has a shelling
This is exercise 2 from Ch. 1 of "Computational Topology: An Introduction" by Edelsbrunner & Harer:
Consider a triangulation of a simple closed polygon in the plane, but one that may ...
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Are there any names yet for this kind of polytope? [closed]
Are there any names yet for this kind of polytope (either the 3D or 4D instance): A half of the hypercube $[-1,1]^n$ cut by the hyperplane $\sum_{i=1}^n x_i = 0$?
The 3D instance is a polyhedron with $...
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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 ...
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2
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Find the area of pentagon $PQRST$.
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 $...
- 2,274
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2
answers
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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 $...
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How can I prove that the sum of exterior angles of an n-gon is 360 degrees?
I have a question regarding the sum of the exterior angles of an n-sided polygon (n-gon). I understand that for a cyclic n-gon, the sum of its exterior angles is always 360 degrees. However, I'm not ...
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2
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Transformation between space partitions
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 ...
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