0
votes
0answers
16 views

Find the normal of a polygon with vertices that are not linearly independent in 3d

For example, take the vectors: $(1,2,3) (4,5,6) (7,8,9) (10,11,12) (13,14,15)$ What would the normal to the polygon be? I'm guessing it would be $(0,0,0)$? For vertices that are linearly ...
1
vote
2answers
32 views

Relation among the diagonals of a regular heptagon

The question is about this problem (it is from a Math' Olympiad in Germany): Prove that if a regular heptagon $ABCDEFG$ has side 1, then $$\frac1{AC}+\frac1{AD}=1$$ I have found something: ...
2
votes
2answers
60 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 ...
0
votes
1answer
25 views

Finding the counter-clockwise direction of points in 3d

I have a set of 5 points of a polygon in 3d. I want to order these points in a counter-clockwise direction. How do I do this? In 2d, to check if two points are ordered counter-clockwise or ...
0
votes
2answers
21 views

Help TOm to find the number of black, white and green pieces?

TOm has a spongy boom ball that is made of 32 pieces of polygon figures: 12 black pentagons and 20 white hexagons. Each pentagon adjoins 5 hexagons and each hexagon adjoins 3 pentagons and 3 hexagons. ...
1
vote
1answer
15 views

Sides of a quadrilateral

In a triangle, with sides say $a,b,c$ we know that $a+b\geq{c}$ and $|a-b|\leq{c}$. What are the inequalities we can form given the sides of the quadrilateral say $a,b,c,d$ where these are unknown to ...
1
vote
1answer
25 views

How does one solve arbitrary polygons, in the same sense as one solves a triangle?

Let us say you are given a polygon, and also are given some, but not all, of its angle measures and side lengths. How would one compute the following: If there is a finite number (zero inclusive) of ...
0
votes
1answer
62 views

How to find the circumcircle radius from this following regular hexagon?

Given a regular hexagon $ABCDEF$. We draw diagonals $AC$ and $CE$. Then, we choose two random points inside the hexagon, call that $M$ and $N$, such that: $\frac{AM}{AC} =\frac{CN}{CE}$. If $B, M$ ...
2
votes
1answer
37 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 ...
0
votes
0answers
29 views

Is there a way to compute the empty area between a group of touching polygons?

Given a bunch of convex polygons layed out like a house truss, is there a way to compute the empty area, or get a polygon for each of those "holes" between the polygons? I tried starting from any ...
0
votes
1answer
45 views

Equation of a polygon

I need a parametric equation for a filled polygon defined by 3 or more points. The closest I've got is by using 3 points in this equation - $polygon = p1 + u(p2-p1) + v(p3-p1)$. But by using points ...
0
votes
1answer
22 views

I need help doing a basic geometry problem

I think it can be solved using parallels but I have no idea how. Solve for "X"
2
votes
2answers
47 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 ...
0
votes
1answer
129 views

Constructing a regular right angled hyperbolic hexagon

I would like to construct a regular right angled hexagon in a klein model. I'm having a hard time understanding why this method works, here is what my professor did in class. Any additional comments ...
0
votes
1answer
34 views

Determine missing angle in polygon

I'm trying to figure out this question: Determine the measure of angle a I'm guessing $a=96\unicode{0186}$ using the following work: $$a = 180 - 84 = 96 $$ ...
0
votes
1answer
18 views

Minkowski difference of two convex polygons

I just want to make sure that the following algorithm is correct for computing the Minkowski difference of two shapes $A,B$: $\text{Minkowski}(A,B) = \text{ CH } \{x: x = a - b \text{ for } a \in ...
0
votes
0answers
73 views

Finding the area of non-standard polygons missing measurements

How do you find the area of non-standard polygons missing a few measurements? Here's a replica of a polygon i had to find the area of on my 9th grade final exam. I understand the fact you have to ...
0
votes
1answer
65 views

Calculating perimeter of n-sided regular polygons using only height?

Say you were drawing n-sided regular polygons on a square grid (first side you drew being flat always). You wanted any polygon you drew to be 100 units high, aka the uppermost point being y = 100. ...
0
votes
0answers
20 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 ...
1
vote
1answer
26 views

Relation for hyperbolic pentagon.

I am trying to get a relation between the length of the sides and the angles of a hyperbolic pentagon. In literature I can find relations for pentagons which has at least three Right angle. So my ...
2
votes
0answers
39 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 ...
6
votes
2answers
61 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 ...
0
votes
2answers
43 views

What kind of shape?

To construct this shape, draw a circle. Place the compass on a point on the circle and draw an arc of the same radius as the circle. Now place the compass at the intersection of the arc and the circle ...
2
votes
4answers
50 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 ...
1
vote
0answers
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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
71 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 ...
0
votes
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. ...
1
vote
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 ...
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?
0
votes
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?
0
votes
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 ...
3
votes
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
votes
1answer
591 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 ...
1
vote
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 ...
0
votes
1answer
136 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) ...
1
vote
1answer
87 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
votes
0answers
37 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 ...
0
votes
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 ...
1
vote
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 $3$). ...
3
votes
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, ...
1
vote
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 ...
2
votes
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 ...
-1
votes
1answer
134 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 $7\times 128.57° = 899.99 = 900°$ ...
1
vote
1answer
95 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 ...
5
votes
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 $A_n$? ...
0
votes
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
votes
1answer
292 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 ...
1
vote
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 ...
1
vote
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 ...