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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13
votes
3answers
238 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 ...
11
votes
2answers
138 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 ...
10
votes
0answers
257 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 ...
8
votes
3answers
811 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
158 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
152 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 ...
7
votes
2answers
151 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 ...
7
votes
1answer
392 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
152 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
172 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
2answers
326 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? ...
5
votes
2answers
68 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
358 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
1answer
140 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 ...
3
votes
1answer
257 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
68 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
1k 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
383 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 ...
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
113 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
1answer
180 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
182 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
2answers
117 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
718 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
1answer
63 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 ...
3
votes
1answer
110 views

Prove $\forall a,b,k \in \Bbb Z^+$ such that $a \equiv -1 \bmod 3$ and $b \equiv 1 \bmod 3$, $2^{2k-1}a,2^{2k}b$ are non-trivial polygonal numbers

Below is my original question, which has since been modified to a more general form. Prove that $\forall p,q \in \Bbb P$ and $k \in \Bbb Z^+$ such that $q \equiv -1 \bmod 3$ and $p \equiv 1 \bmod 3, ...
3
votes
2answers
77 views

What's wrong with an irregular digon?

I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a ...
3
votes
2answers
27 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
3
votes
0answers
78 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
0answers
56 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
3
votes
0answers
93 views

Polyonimo Tiling

I came up with the following conjecture the other day, and was wondering if the result was well-known or even true: Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total ...
2
votes
2answers
72 views

Find the area of quadrilateral formed by $4$ (not consecutive) vertices of a $12$-gon inscribed in a circle.

A regular $12$ sided polygon is inscribed in a circle of radius $10$. $A,B,C,D,E$ are its consecutive vertices taken in that order. Find the area of quad. $ABDE$. The angle of $12$-gon is ...
2
votes
2answers
112 views

What is inside and outside of complex polygon?

I am reading this paper http://arxiv.org/pdf/1207.3502.pdf Given a complex polygon. Its edges may intersect. The algorithm finds out if given point is inside of polygon or not. It draws a line from ...
2
votes
3answers
155 views

Very confusing polygon question. Can anyone help?

I was practising questions on principles on mathematics. I stumbled onto this question and I don't know where to start. Can anyone please help?? If $P_1P_2....P_n$ is a regular polygon in the ...
2
votes
2answers
61 views

What's the fewest number of sides required to make a polytope in n dimensions?

In 2 dimensions it takes at least 3 sides to make a polygon, the triangle, and in 3 dimensions it takes at least 4 faces (so far as I'm aware) to make a polyhedron. Can this rule be generalized to ...
2
votes
1answer
264 views

Non Self Intersecting Polygons?

Given a set of n points is it always possible to construct a non self intersecting polygon?
2
votes
2answers
77 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 ...
2
votes
4answers
54 views

Geometry terminology: concrete vs. continuous polygons?

I am trying to find the proper terminologies for 2 kinds of shapes: The first type of shape I'm calling "concrete polygons". They have a finite number of straight sides (connecting at vertices) and ...
2
votes
2answers
32 views

Forming a simple polygon from the extrusion of a polygonal chain

Let's say I have a set of vertices connected by edges to form a polygonal chain. Each vertex may be shared by a number of edges to form various sub-chains. An example is shown below. Each edge has ...
2
votes
2answers
40 views

How to check does polygon with given sides' length exist?

I have polygon with $n$ angles. Then I have got $n$ values, which mean this polygon's sides' length. I have to check does this polygon exist (means - could be drawn with given sides' length). Is ...
2
votes
2answers
394 views

Proving the regular n-gon maximizes area for fixed perimeter.

It is often assumed that, given $n$, the regular $n$-gon will make the most efficient use of perimeter for area. I have never seen this proven. Anyone have something slick? (That is, how can we ...
2
votes
1answer
48 views

Question regarding polygons

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?
2
votes
1answer
369 views

coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times ...
2
votes
1answer
400 views

Formulas for finding out if a number is Heptagonal or Octagonal

I am trying to find the formula for finding out if a number is heptagonal. I am also looking for the formula for finding out if a number is octagonal. I already have the formula for finding out the ...
2
votes
1answer
780 views

How many rectangles can fit in a polygon with n-sides?

I am trying to write an algorithm to solve a problem I have. I have a few ideas of what the algorithm might be like but I am posting to see if anyone else has a better more efficient solution or any ...
2
votes
1answer
20 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 ...
2
votes
2answers
32 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
2
votes
1answer
33 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| ...
2
votes
1answer
48 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 ...
2
votes
1answer
54 views

Symmetry center of hexagon

How to show that the figure has a symmetry center for instance if we have a convex hexagon where opposite sides are of equal lenght and parallel?