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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14
votes
3answers
310 views

Area of the Limiting Polygon

Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon - see the picture below. Next, trisect each of the ...
14
votes
2answers
179 views

Coffee and regular polygons

To save some money, I decided to brew my own morning-fix coffee and skip buying it from the coffee shop. BTW, I drive to work and put my coffee cup in between the two front seats. While driving on the ...
12
votes
0answers
312 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
11
votes
1answer
229 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
9
votes
3answers
297 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
9
votes
4answers
204 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 (dots), in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon and no vertice ...
8
votes
2answers
168 views

How to check if polylines can be untangled?

In a program I'm writing I need to be able to check whether a straight line between two points is homotopic to a polyline between them. For example in the below example the first one is equivalent to ...
8
votes
3answers
2k views

How is the area of a country is calculated?

As countries or states borders are not straight lines but they are irregular in nature. I wonder how does anyone will be able to calculate area of a country or a state. When do you think area of a ...
8
votes
2answers
177 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 ...
8
votes
1answer
194 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the ...
8
votes
1answer
146 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
7
votes
2answers
511 views

Importance of construction of polygons

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? ...
7
votes
2answers
119 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
7
votes
1answer
454 views

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number, where a number $q$ is practical if and only if every integer less than or equal to ...
7
votes
0answers
174 views

What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
6
votes
2answers
256 views

Area of an irregular polygon

I was searching for methods on how to calculate the area of a polygon and stubled across this: http://www.mathopenref.com/coordpolygonarea.html. It does work and all, yet I do not fully understand why ...
6
votes
4answers
222 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. ...
5
votes
1answer
2k views

Why does nature prefer hexagons?

The best ratio of surface to volume in three dimensional space is the ball. This can be easily observed with soap-bubbles, rain-drops and so on. They "choose" this shape naturally. Given restricted ...
5
votes
2answers
82 views

Show that diagonals intersect at common point

Given is octagon where opposite sides are equal length and parallel. Show that diagonals: $AE,DH, BF, CG$ intersects at point $S$ So I have tried to create a parallelograms $AHED$ and $BCFG$ and use ...
5
votes
1answer
430 views

cyclic polygons & trigonometry

At one vertex of a pentagon inscribed in a circle of unit diameter (unit diameter, not unit radius) let the angles between adjacent diagonals be $\alpha,\beta,\gamma$, at the next, ...
4
votes
4answers
659 views

What polyhedron has 11 vertices and 17 edges

On my math test it asked me how many polygons it takes to create a polyhedron that has $11$ vertices and $17$ edges. I'd just like to see what the shape would look like and I can figure out the ...
4
votes
1answer
97 views

Why do rings appear in regular polygons with diagonals?

When looking at regular polygons with all the diagonals filled in, I saw that concentric rings seem to form. Why does this occur? It's not so obvious with small $n$, but for larger $n$ it becomes ...
4
votes
2answers
993 views

Algorithm of cutting a polygon into equal parts

I have a convex polygon. I need to divide it into 4 equal parts using the two slit. For example, if I have a square, I have to cut it along the diagonals. Are there some common algorithm for this ...
4
votes
1answer
40 views

What is the name for a rectangular figure of many sides?

What is a polygon where each edge is at a 90 degree or 270 degree angle to the prior edge (giving both concave and convex vertices) called? Here is one example of such a shape: ...
4
votes
1answer
233 views

Symmetrically splitting an octagon into quadrilaterals

I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
4
votes
0answers
26 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
4
votes
0answers
61 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
4
votes
1answer
110 views

The Erdős-Szekeres problem on points in convex position

The Erdős-Szekeres problem on points in convex position and its proof using Ramsey theorem are well know. The problem goes like this: For every natural number $k$ there exists a number $n(k)$ such ...
3
votes
1answer
288 views

How can I construct a 2^63-gon with a straightedge and compass?

I entered 2^63 as a stand alone value at WolframAlpha. Among the responses was a factoid that 'A regular 9223372036854775808-gon is constructible with a straightedge and compass.' What is such a ...
3
votes
1answer
81 views

Labeling the vertices of a polygon with 0's and 1's

Suppose $P_n$ is the regular polygon with n vertices ($n\geq 5$). Let $V=\{v_1,\ldots,v_n\}$ be the vertex set. I would like to define a labeling function $\ell:V\to \{0,1\}$ so that ...
3
votes
2answers
3k 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?
3
votes
2answers
107 views

Polygon and Pigeon Hole Principle Question

Seven vertices are chosen in each of two congruent regular 16-gons. Prove that these polygons can be placed one atop another in such a way that at least four chosen vertices of one polygon coincide ...
3
votes
1answer
1k views

Polygon Inequality

We know that to form a triangle the 3 sides should obey the triangle inequality . So is there any rule to be followed by the sides of $n$-sided convex polygon. For Eg:- $1,2,4$ cannot form a triangle ...
3
votes
2answers
33 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
3
votes
2answers
28 views

is a line of symmetry going through the center of an octagon parallel to the ground it is on?

I came across this question recently, It wanted me to find angle EDG assuming the line crossing between the octagon/polygon is it's line of symmetry. My answer was 25 degrees, but A few friends ...
3
votes
1answer
56 views

In every polygon circumscribed about a circle, there exist three sides that can form a triangle.

How can one show that in every polygon circumscribed about a circle, there exist three sides that can form a triangle? (This was posted by another user and then deleted while I was typing my answer.) ...
3
votes
2answers
153 views

Construct polygon from random segments

Given an arbitrary amount of ordered segments, with arbitrary lengths is there a way to determine if they can be formed into a simple polygon? And if so, is it possible to work out the angles needed ...
3
votes
1answer
1k views

Concave polygons overlapping test

I have set of $N$ concave polygons, given as list of 2D Euclidean coordinates. How to compute: a. if any of them are overlapping? b. if one arbitrarily selected polygon overlaps with any of the ...
3
votes
2answers
119 views

Is every triangle a quadrilateral?

I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral? More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
3
votes
4answers
422 views

Can anyone give me x,y coordinates for an octagon?

I am looking to draw a octagon and I need $(x, y)$ coordinates.
3
votes
2answers
206 views

splitting polygon in 4 equal parts

I have a convex polygon and I want to divide into 4 equal parts using the two perpedicular splits. Like in a picture. I need s1 = s2 = s3 = s4; I need to get coordinates of point where the lines ...
3
votes
1answer
87 views

Mean curvature flow - implementation fails for some meshes

I am working on piece of software to deal with 3D meshes and I need to smooth some meshes. I have implemented MCF by using this formula $\vec{H} = {{t}\over{2}} \sum_{q \in\ link\ p} \vec{Ne} |e| ...
3
votes
1answer
78 views

Area of the intersection of regular pentagons

I was playing around in Geogebra and came up with an interesting problem. Take an arbitrary point $A$ and draw a circle with center $A$. Then draw any line through $A$. Call the points where the ...
3
votes
2answers
880 views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
3
votes
1answer
204 views

Polygon sine waves

So I came across this picture on Google+ and I wanted to understand further. I created an equation for the second wave, the one with the square. Here it is: $$y=\frac{\sin x}{\cos(\min(x \mod \pi/2, ...
3
votes
1answer
288 views

Find the vertices of the polytope

Let $x,n$ be 2 integers with $x<n$. I need to find the vertices of the polytope $P$ of $2 \times n$ nonnegative matrices $A$ such that: The first row in $A$ is summed to $x$. $$\sum_{j=1}^n ...
3
votes
1answer
771 views

Calculate 'Rectangle' Coordinates Given 2 Points and width

I want to create a rectangular polygon using two points as guides. So let's say a journey starts in Egypt and ends in London, my polygon should have 4 points: 10 miles further from London ...
3
votes
1answer
62 views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which ...
3
votes
1answer
54 views

finding parallel sides from a irregular decagon?

Is it possible to find out that which of two sides are parallel in this irregular decagon.If,it is yes;then how can I proceed? I have tried with "Consecutive Interior Angles".but can't come to a ...
3
votes
1answer
90 views

How do I create a Hilbert curve that is bounded by a polygon?

All images of the Hilbert curve that I've seen show the Hilbert curve as bounded by the unit square: However, if I have a list of vertices that define a closed polygon, how can I create a Hilbert ...