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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9
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217 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
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81 views

Counting Regular polygons in Complete Graphs

The figure shows the correct $24-$gon, which held all the diagonals. a) Find out how we got right triangles and squares (question for arbitrary $n$)? b) How this problem can be generalized (if it is ...
5
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0answers
98 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
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69 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
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0answers
49 views

Centroids of a polygon

Obviously, for any polygon we can define at least 3 different centroids: C1: mass center of the lamina; C2: mass center of vertices with equal masses; C3: mass center of the perimeter. For the ...
4
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0answers
149 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
3
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0answers
47 views

Centroid and circumcenter — how close?

Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
3
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0answers
65 views

Existence of a polygon with specified interior angle measures

We all know that the sum of the interior angles of a polygon is $180^{\circ} (n-2)$. But is the converse true? Given a sequence of $n$ angle measures whose sum is $180^{\circ} (n-2)$, can it be ...
3
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48 views

Elementary proof of Jordan curve theorem for polygons

Courant described the outline of an elementary proof of the Jordan curve theorem for polygons using the order of points: The order of a point $p_0$ is defined by the net number of complete ...
3
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0answers
186 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...
3
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0answers
54 views

Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my ...
3
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0answers
95 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
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0answers
124 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 ...
3
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0answers
261 views

Formal name for polygon with hole

Is there a formal name for an irregular polygon that has 1 or more holes or cutouts in it? I've heard it refered to as a "swiss cheese polygon" or a "Donut polygon". Is this even strictly a polygon?
2
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25 views

Area covered by fixed perimeter around polygon.

Suppose I have a polygonal field with a post at each vertex and a non-extensible rope threaded through each post around the perimeter but with some slack. How can I determine the perimeter of the area ...
2
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0answers
76 views

Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: ...
2
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0answers
54 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
2
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0answers
215 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
2
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0answers
62 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 ...
2
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0answers
48 views

Determine direction of minimum overlap of convex polygons

Given two convex polygons $P$ and $Q$ what is the minimum intersection polygon $A=P\cap Q'$ where $Q'$ is the polygon $Q$ offset by a vector $\overline r$ of fixed length? Put another way, what is ...
2
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0answers
177 views

Points in the cartesian plane

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. In the $xy$ plane, suppose $M_{i,j}=\{(x,y)\mid i\le x\le j\}$. ...
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0answers
14 views

Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
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0answers
89 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
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0answers
20 views

Maximum number of regular polynom projections of a polyhedron

It is trivial that a cube has both a square and a regular hexagonal projection. We can also easily construct a polyhedron with three perpendicular projection, which are different regular polygons. ...
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0answers
18 views

Non-convex subdivisions of newton polygon of a tropical plane curve

This is probably an elementary question, but how come the Newton polygon of a tropical plane curve can't have non-convex subdivisions? Or can it?
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0answers
30 views

How to generate random boundary programmatically?

What should I read to understand how to draw programmatically random oil 'boundary' like on the picture below? Yes, it should go from the top to the bottom and so I don't need these 'long' drops ...
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0answers
52 views

Minimal diagonal intersections in a convex polygon

OEIS A006561 gives the number of intersection points in the diagonals of a regular polygon. There's a paper by Poonen. For 4 vertices to 12, the number of intersection points is: $$1, 5, 13, 35, 49, ...
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0answers
44 views

All polygons satisfy the “normal” property.

A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
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0answers
54 views

What is the probability to pass through $1\le m\le n$ vertices of an $n$-sided polygon after $t$ seconds?

Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and ...
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0answers
21 views

Formula to determine the number of exterior edges in multiple tiled hexagons

I'm looking for a formula which determines the number of external (that is, non-touching) edges in multiple tiled hexagons. By observation, there are 10 external edges when 2 hexagons are adjacent. ...
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0answers
112 views

Solve for Sides of a 5-Sided Irregular Polygon

I have a 5-sided irregular polygon and I know the lengths of 4 of its 5 sides and 2 of its 5 angles. Is there a way to know the length of the 5th side using this information?
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0answers
40 views

Testing if a number N is prime by its regular polygon's angles

Is it possible to tell if a number N is prime by looking at the angles of a regular N-sided polygon? For example, a regular triangle has 60 degree angles, is there a way to tell that the number 3 is ...
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0answers
80 views

How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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0answers
44 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
1
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0answers
65 views

Finding vertices of a hexagon or pentagon

I have a grid of 150000 x 150000 points, and I have a list of points corresponding the x,y coordinates of a shape that make up a slightly imperfect hexagon or pentagon. I'm trying to figure out a more ...
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0answers
51 views

Decomposition of ball in Banach Tarski paradox and covering a soccer ball

Banach Tarski paradox says that it's possible to decompose a ball in $R^3$ into a finite number of disjoint subsets, which can be then reassembled into 2 identical copies of the original ball. ...
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0answers
118 views

Number of grid points in a polygon

Following problem: I want to approximate the number of grid points in a polygon, based on the condition that the distance of the grid points are variable. What i need is an approximation, i am aware ...
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0answers
18 views

What is the term to define a single point in a polychoron?

I'm looking for any correct term used to define a point in 4 dimensional space. IE: What does a polychoron compose?
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0answers
146 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
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0answers
18 views

create polygon section with equal sides

I have to create essentially these sections of a polygon. I have width(W) and height (H), and number of sides (3 on left abc and 4 on right image ABCD) I need each side to be equal. How can I ...
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0answers
62 views

Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
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0answers
149 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
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0answers
27 views

Best convex bounding polygon from a set of given lines

Given a polygon $P$ and a set of predefined lines, I am looking for the subset of lines that creates the best fitting convex polygon with respect to $P$. In other words the area of an ...
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0answers
58 views

Sufficient condition for a simple polygon

Some background: I'm trying to simulate biological cells as polygons in 2D. Real biological cells have an internal cytoskeleton which enables them to conform to a variety of shapes, many non-convex. ...
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0answers
27 views

Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
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0answers
74 views

Find all polygons in points in plane

I have a set of points in the plane and I want to find all convex polygons without including a point inside them. For example I want to find all triangles, all four sized polygons, all four five ...
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0answers
52 views

Rectilinear polygons winding around a torus

A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($ 90^{\circ}$) and that of interior concave angles ($ 270^{\circ}$) is always $4$. Consider a ...
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0answers
90 views

A convex $n$-gon and the $n$-gon made by its $n$ medians

For a convex $n$-gon $P_1P_2\cdots P_n$, let $M_i$ be the mid-point of the line segment $P_iP_{i+1}\ (i=1,2,\cdots,n)$ where $P_{n+1}=P_1$. Also, let $Q_1Q_2\cdots Q_n$ be an inner $n$-gon made by $n$ ...
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0answers
144 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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0answers
107 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 ...