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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29 views

Dodecahedron: How do we get the distance between 2 opposite faces?

I am deciphering a CSS code that Ana Tudor Maria has done. http://codepen.io/thebabydino/pen/qIfbL In her example, she has a formula that calculates the distance between 2 opposite faces. I have no ...
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1answer
33 views

How do i find the exterior angles of an L-shaped polygon?

I'm trying to review exterior angles after many years. It's my understanding that the sum of a polygon's exterior angles must equal 360°. How would you find the exterior angles in this polygon?
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2answers
273 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 ...
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1answer
39 views

Lines formed from vertices of n-gons equate to triangular numbers.

Noticed something neat tonight! The number of unique lines you can form by connecting the vertices of an n-gon is equal to the (n-1)th triangular number. (e.g. in a square all 4 veritices make 4 ...
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1answer
22 views

Rotating and Scaling about centroid.

rotating $x' = x\cos(\text{angle}) - y\sin(\text{angle})$ $y' = x\sin(\text{angle}) + y\cos(\text{angle})$ Scaling $x' = x\cdot sx$ $y' = y\cdot sy$ but all formulas will ...
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1answer
73 views

Furthest point on regular polygon given arbitrary direction

In a circle of radius $r$ centered at $c$, if I want to know the point on the circle that is furthest in a direction specified by a vector $d$ I use the formula $c+(r/||d||)d$. Is there a similar ...
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0answers
20 views

Calculate self-avoiding-filling-polygons

Definition of self-avoiding-filling-polygon In Euclidean graph where each vertex is a point on the $2D$ plane, so the weight of each edge is the Euclidean distance between the vertices. ...
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1answer
53 views

Find self-avoiding-filling-polygon represented by system of linear equations

In Euclidean graph where each vertex is a point on the $2D$ plane, so the weight of each edge is the Euclidean distance between the vertices. I want to find self-avoiding-filling-polygon from my graph ...
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26 views

Where are $A,B,C$ in the regular $n$-gon such that $\min (|AB|+|BC|,|BC|+|CA|,|CA|+|AB|)$ gives the max?

Let $F_n$ be the regular $n$-gon of edge-length $1$. Let us consider taking three points $A, B, C$ in $F_n$. Suppose that you can take a point on the edge of $F_n$. Supposing that $|AB|$ represents ...
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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?
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1answer
36 views

What are the applications? [duplicate]

How can I show that a sequence of regular polygons with n sides becomes more and more like a circle as n→∞? In which fields this concept is applied?
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1answer
72 views

Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation : Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...
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1answer
108 views

Prove using integration "circle is a polygon when number of sides-> infinity

Is there a proof of "if number of sidesof a regular polygon ->infinitythe regular polygon -> circle." using integration?
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1answer
47 views

Creating non self intersecting Quadrilaterals from 4 points

Given 4 points (lP1, lP2, lP3, lP4) ordered from highest y value to lowest, and when y values equal each other, it is sorted from lowest x to highest x, based in a regular mathematical Quadrant I (not ...
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2answers
109 views

How to determine whether a line is at the clockwise side of another line?

This is my first post in math-overflow , I am trying to implement an algorithm where , i am given 2 lines , two lines have one point in common , i need to determine if one line is at the clockwise ...
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2answers
117 views

Calculating integration over polygon (area of polygon)

Given a polygon, I am trying to find the area of the polygon intersected with $\mathbf{x}<x$ and $\mathbf{y}<y$. I reckon this can be viewed as an integration of a function where its value is 1 ...
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0answers
24 views

About the relation between two regular icosahedrons and a regular dodecahedron

Let $C$ be the regular icosahedron, each of whose vertex exists at the centroid of the each surface of the regular dodecahedron $B$, each of whose vertex exists at the centroid of the each surface of ...
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2answers
84 views

About the diagonals whose length is an integer multiple of the edge length of a regular polygon

Are the followings true? 1. In every diagonal of every regular polygon, some diagonals of regular hexagon, whose lengths are twice as long as its edge length, are the only diagonals such that the ...
3
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0answers
51 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 ...
2
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1answer
609 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 ...
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1answer
63 views

When is a convex polygon inscribable?

Defining the diameter of a convex polygon as the maximum possible distance between all pairs of vertices, can we conclude that the convex polygon is inscribable (i.e has all its sides as chords of a ...
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246 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 ...
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1answer
55 views

About intersection points of some diagonals of a regular $n$-gon

Are my expectations true? My expectation 1 : There exist some intersection points, which are not on the center of the regular $n$-gon, of three or more diagonals when you draw all diagonals of a ...
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0answers
61 views

Area of a 2D convex polytope made of halfspaces

For a computer program I am attempting to solve the area of a convex polytope defined by a finite number of halfspaces. I understand that this forms a polygon and given the vertices of a polygon I am ...
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1answer
139 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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1answer
90 views

Shortest path calculation

I have a given set of start points, a given set of end points. Each start point corresponds to one endpoint. I have to visit all start points, and then the corresponding end points, in the most ...
2
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0answers
38 views

How: Determine area painted by a path with width within a polygon

I have a path that represents the movement of some equipment. The equipment has a width so I'd like to determine the approximate area created by this path within a polygon. If I use the distance ...
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0answers
71 views

First event in a straight skeleton

Is there a simple geometric criterion to check whether the first event in (the wave propagation of) a straight skeleton is an edge event or a split event? The literature I could find is computational ...
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2answers
138 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 ...
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0answers
56 views

Finding every $(m,n)$ such that a regular $n$-gon is inscribed inside a regular $m$-gon.

I found the following five types. 1. $(m,n)=(kn,n)$ for $n\ge3, k\ge1$. 2. $(m,n)=(hk,2k)$ where $h\ge1$ is an odd number and $k\ge2$. 3. $(m,n)=(m,3)$ where $m\ge4$ and $m\not\equiv0$ (mod $3$). ...
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1answer
166 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, ...
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1answer
44 views

Sum of interior angles of a rectilinear

For a Rectilinear (concave polygon having all sides parallel to either X or Y axis), how do we find the "sum of all the interior angles" given total number of convex corners (corner whose internal ...
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1answer
133 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 ...
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4answers
282 views

Area of irregular polygon using side edges

I have only lengths for the sides of an irregular polygon, can anyone tell me how I can measure the area of the polygon? Remember only lengths of all the sides , no angles or coordinates. Few forums ...
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1answer
67 views

Generalized Straight Skeleton

The straight skeleton of a polygon can be computed by having the edges of the polygon move inwards at a uniform constant speed. Is it useful to generalize this computation process by varying the ...
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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, ...
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1answer
378 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 ...
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1answer
135 views

the internal angle, and the sum of the internal angles in any N-sided polygon?

We have: triangles have $3\times 60°=180°$ squares have $4\times 90°=360°$ pentagon have $5\times 108°= 540°$ hexagons have $6\times 120°=720°$ heptagons have $7\times 128.57° = 899.99 = 900°$ ...
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1answer
96 views

Smallest square that can be fitted outside the regular hexagon

They have the derivation here http://www.drking.org.uk/hexagons/misc/deriv4.html The figure is In this derivation the second line is $ a^2 = b^2 + b^2 $ .How have they assumed $ AB = AC $ ? . Is ...
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0answers
84 views

Solving the “Library of Babel” puzzle, but for polygons.

The Library of Babel is a story about a universe whose contents are every possible 410-page book that could possibly exist. After a conversation with someone about doing this with images, and coming ...
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2answers
146 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 ...
5
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0answers
134 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 $A_n$? ...
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1answer
80 views

What is the height of a regular polygon?

I have three small circles forming a pyramid. I would like to centre that group in a square but have spent a couple of hours trying to calculate the height of the pyramid. I just can't seem to get ...
2
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1answer
298 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 ...
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1answer
92 views

Computing bounding box of polytope (system of linear inequalities)

Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints. I think ...
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1answer
79 views

sum of exterior angles of a closed broken line in space

I am looking for a simple proof of the following fact: The sum of exterior angles of any closed broken line in space is at least $2 \pi$. I believe it equals $2 \pi$ if and only if the closed broken ...
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1answer
50 views

Mathematical word for geometrical object?

Is there a mathematical word to designate the concept of a geometrical object like: square cube tesseract N-dimensional cube circle sphere hypersphere regular and non-regular polygons regular and ...
0
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2answers
96 views

Calculating interior angles of quadrilateral

stupid question... but: I've a polygon which has the points $(a_x,a_y),(b_x,b_y),(c_x,c_y), (d_x,d_y)$ How can I calculate each interior angle of this quadrilateral? I know that in sum, it has to be ...
2
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1answer
190 views

Draw a regular hexagon in simcity

I know how to make a pentagon (there are youtube videos), and I kinda thought I knew how to do a hexagon using 5 circles: one in the center, shift-drag 2 guide blocks. 2 below it, each shift-drag 2 ...
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1answer
111 views

Rotation angle of regular polygon that has largest taxicab maginitude between all vertices

Firstly just to apologise, I posted this on mathoverflow before realising it was focused on research level mathematics. If I have a regular polygon that is centred at the origin. Then take the ...