Questions tagged [polygons]

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A regular $n$-gon contains a regular $m$-gon, with $n,m$ coprime, no sides coinciding. What is the maximum number of contact points between them?

A regular $n$-gon contains a regular $m$-gon, where $n$ and $m$ are coprime, with no sides coinciding. What is the maximum number of contact points between the $n$-gon and the $m$-gon? (I'm not ...
Dan's user avatar
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1 vote
0 answers
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All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$. A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each ...
Dan's user avatar
  • 1,693
10 votes
0 answers
144 views

Minimum reflection paths in a mirror polygon

Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles, and is non-self-intersecting; also known as a rectilinear polygon. Treat every edge of $P$ as a ...
Joseph O'Rourke's user avatar
6 votes
0 answers
218 views

Is there a bicyclic irregular pentagon in integers?

Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well? If it does ...
shabo's user avatar
  • 61
0 votes
1 answer
46 views

What is the most dense sample for which the Crust algorithm returns an incorrect polygonal reconstruction?

The Crust algorithm by Amenta, Bern, and Eppstein computes a polygonal reconstruction of a smooth curve $C$ without boundary from a discrete set of sample points $S$. It is known that if $S$ is an a $\...
M Wright's user avatar
  • 403
4 votes
0 answers
102 views

Do cut-length-minimizing equidissections exist?

Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
Noah Schweber's user avatar
13 votes
0 answers
235 views

Regular $n$-gon with diagonals: bounds on area of largest cell?

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet). I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
Dan's user avatar
  • 1,693
13 votes
2 answers
572 views

How to characterize the regularity of a polygon?

In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
Caio Tomás's user avatar
4 votes
0 answers
176 views

Happy ending problem - why not a proof by induction? (cont)

After sharing ideas on this post, I have been thinking for some time on the problem, and I think that a possible way to prove the Erdös-Szekeres conjecture could be structured as follows: Consider ...
Juan Moreno's user avatar
2 votes
0 answers
223 views

Generalization of the Napoleon equilateral triangle to higher dimention

When I researched the Fermat-Dao-Nhi equilateral triangle in preamble before points X(33602) of the Kimberling triangle center. I discovered the general result for polygon as follows: Let $A_1$, $A_2$...
Đào Thanh Oai's user avatar
1 vote
0 answers
138 views

Stronger conjectured inequality for area of a polygon

Four years ago, I proposed an inequality related to area and sides of a polygon. After computer checking, I conjecture that the previous inequality can be strengthened as follows: Let $A_1A_2\cdots ...
Đào Thanh Oai's user avatar
2 votes
1 answer
240 views

Double integral in a polygon domain

I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides. $$ I(f) = \int_{D} f(x, \ y) \ dx \ dy $$ The vertex of this polygon are $$\vec{p}_{i} = (x_i, \ y_i) \ \ ...
Carlos Adir's user avatar
1 vote
1 answer
117 views

Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?

When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...
Svjetlan Feretic's user avatar
5 votes
2 answers
536 views

First Dirichlet eigenvalue on regular polygons

Assume $\lambda(P)$ is the first Dirichlet eigenvalue of a regular polygon $P$. Let $u$ be the corresponding eigenfunction, normalized by $\|u\|_{L^2(P)}=1$, and $\partial_{\nu}u$ be its normal ...
guest61's user avatar
  • 319
0 votes
0 answers
76 views

Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form $$ M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\} $$ where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. Goal I am looking for an (explicit) representation ...
Felix B.'s user avatar
  • 305
15 votes
4 answers
792 views

Unlinked interlocking planar polygons

Let $P$ and $Q$ be the boundary segments of two planar simple polygons. View these boundaries as rigid wires. Fix $Q$ in, say, the $xy$-plane, and imagine $P$ arranged in $\mathbb{R}^3$ so that $P$ ...
Joseph O'Rourke's user avatar
5 votes
1 answer
213 views

Inserting points into a polygon

I want to insert $n$ points into arbitrary polygon $P$ described by ordered list of its vertexes $v_1, v_2, ..., v_m$. Each inserted point must distanced from the others on distance at least $d$. In ...
Kamil Kiełczewski's user avatar
1 vote
0 answers
24 views

Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
user1747134's user avatar
1 vote
0 answers
52 views

To extend the Steiner-Lehmus theorem

The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles. Question: What could one say ...
Nandakumar R's user avatar
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0 votes
1 answer
256 views

The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\ This seems to be intuitively obvious but I can't ...
Mathlover's user avatar
3 votes
1 answer
56 views

Scissor congruence for foliated polygons

Given two polygons of equal area with horizontal foliations, can one describe the obstruction (if there is any but I suspect the answer to be yes) to scissor-equivalence respecting the horizontal ...
Roland Bacher's user avatar
1 vote
1 answer
113 views

Is there a non-trivial, exact analytic (symbolic) conformal map from some polygon to some rectangle?

I'm looking for any example of a conformal map $m: P \to Q$ where $P$ is some polygon of at least $4$ sides, $Q$ is a rectangle, and and $m$ is not linear (so $P$ and $Q$ are not merely scaled, ...
J. M.'s user avatar
  • 49
6 votes
1 answer
506 views

Sum of the lengths of all diagonals in a regular polygon

I believe that the sum of the lengths of diagonals of a regular polygon ($n$-gon) is always greater than or equal to any other irregular polygon ($n$-gon) inscribed in a circle.. For example for a 4-...
maths123456's user avatar
1 vote
1 answer
77 views

Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
Pedja's user avatar
  • 2,643
2 votes
0 answers
804 views

Happy ending problem – Why not a proof by induction?

I have been thinking for a while on the happy ending problem, looking for approaches to attack the Erdős–Szekeres conjecture: the smallest number of points for which any general position arrangement ...
Juan Moreno's user avatar
1 vote
1 answer
227 views

High-dimensional polytopes

I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex: How can ...
GuyS's user avatar
  • 43
3 votes
1 answer
150 views

The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
Pedja's user avatar
  • 2,643
3 votes
1 answer
264 views

Need help with finding all angles of 11 sided 3D object [closed]

Question: I'm an artist trying to build a hendecahedron for a project (Image below to see the shape). This object consists of 5 pentagons at the base, 1 pentagon on the bottom, then 5 quadrilaterals ...
Jake Mitchell's user avatar
1 vote
1 answer
48 views

On triangulations and "coverage" of circumcircles

Let $P$ be a convex quadrilateral defined by four vertices $a$, $b$, $c$, and $d$. Suppose that the circumcircle of $\triangle abd$ contains $c$.* Let $D(\triangle abc)$ to denote the area enclosed by ...
Scattering State's user avatar
2 votes
0 answers
73 views

Principal diagonals of octagon meet in a single point

Can you provide a proof for the following claim: Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...
Pedja's user avatar
  • 2,643
3 votes
1 answer
163 views

Construct by compactness (Pentagonal tiling – Rao paper)

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that: "... We keep a connected component $H_d'$ of $H_{d}$ such ...
GingerBreadMan's user avatar
1 vote
1 answer
74 views

Equal products of triangle areas

Can you prove the following claim: Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
Pedja's user avatar
  • 2,643
3 votes
1 answer
92 views

Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
Pedja's user avatar
  • 2,643
1 vote
1 answer
219 views

Partitioning a convex $n$-polygon

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \...
TheVal's user avatar
  • 151
6 votes
4 answers
482 views

Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition: Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
Pedja's user avatar
  • 2,643
15 votes
1 answer
589 views

Acute triangles in "obtuse" polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute? I conjecture ...
Scattering State's user avatar
2 votes
1 answer
97 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
Pedja's user avatar
  • 2,643
1 vote
0 answers
164 views

Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
Talmsmen's user avatar
  • 577
3 votes
2 answers
196 views

Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles?

Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat ...
Hailong Dao's user avatar
  • 30.2k
4 votes
1 answer
254 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
Pedja's user avatar
  • 2,643
6 votes
1 answer
201 views

Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim? Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
Pedja's user avatar
  • 2,643
1 vote
1 answer
291 views

A generalization of Harcourt's theorem

This question is closely related to my previous question. Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem. Claim. Let $A_1,A_2 \ldots ...
Pedja's user avatar
  • 2,643
1 vote
1 answer
281 views

A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals. Claim. Given bicentric quadrilateral $...
Pedja's user avatar
  • 2,643
6 votes
0 answers
97 views

Zero-area-free embedding of points on the grid

Given $n$, I am looking for the smallest $m$ such that there is an $n$ element subset $S$ of the $m\times m$ grid (i.e., $n$ points with integer coordinates in $[0,m]^2$) such that no matter how one ...
domotorp's user avatar
  • 18.4k
4 votes
0 answers
220 views

Illuminating a just-barely irrational polygon

As has been discussed earlier on MO,1,2 recently an impressive advance was proved concerning internally illuminating a mirrored polygon. Here is the result: Let $P$ be a rational polygon. Then for ...
Joseph O'Rourke's user avatar
0 votes
0 answers
30 views

Maintain the area of a polygon when offsetting one side

I have an irregular polygon with the a specific area (area_red). How can I get the (parallel) offset value (y) of n selected sides in order to maintain the same area (area _red = area_green) when (...
tyler's user avatar
  • 101
1 vote
0 answers
81 views

Is the mean center of the vertices of a convex polygon always inside the polygon? [closed]

As simple as that. I'm doing an R program where I need to order clockwise a bunch of points that describe a regular polygon and to do that I figured I could find a point inside, change to polar ...
Brais Romero's user avatar
8 votes
1 answer
207 views

Counting polygons in arrangements

For an arrangement of lines $\cal{A}$ in the plane, an inducing polygon $P$ is a simple polygon satisfying: (a) every edge $e$ of $P$ lies on some line $\ell$ of $\cal{A}$, and (b) every line $\ell \...
Joseph O'Rourke's user avatar
3 votes
1 answer
756 views

Brother of Japanese theorem for cyclic quadrilaterals

I am looking for a proof of a like result as follows and Higher-dimensional generalizations? Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...
Đào Thanh Oai's user avatar
4 votes
1 answer
330 views

Construct closed chain of $k$-gon around $n$ points-$n, k$ are odd primes number

Question 1: I am looking for a proof of the conjectures 1, 2, 3 as follows? Question 2: In conjecture 3, in general case, I can not give a formula of $X$. But I think, If $n, k$ are odd primes number ...
Đào Thanh Oai's user avatar