39
votes

### What is the name of the 65537-gon?

"$65537$-gon" is the name. Likewise "$257$-gon":
writing (let alone saying) something like "diacosipentacontaheptagon"
serves less to communicate $-$ if indeed it succeeds in communicating at all $-$...

- 74.4k

34
votes

### Can one "hear" the shape of a polygon via external reflections?

For question 3, the answer is yes: take a solid disc and excavate half of the Penrose unilluminable room from it. Then, there are boundary arcs which can never be touched, and you can perturb them ...

- 11.9k

24
votes

Accepted

### Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.

- 41.2k

14
votes

Accepted

### Unlinked interlocking planar polygons

As Sam Hopkins commented, 8 vertices are enough. Let $Q$ be the pentagon from the picture and let $\pi$ be the plane containing it. Now we can define the triangle $P$ as a triangle of less diameter ...

- 6,697

13
votes

Accepted

### How to characterize the regularity of a polygon?

Internal angles are not enough to determine the regularity of a polygon. E.g., angles of $2\pi/3$ between sides of length $1,1,4,1,1,4$ make an irregular hexagon.
For a metric of regularity, I suggest ...

- 18.7k

11
votes

Accepted

### Limiting shape for Brillouin zones

Yes, it is. Take a point P, and let us check if it belongs to Nth Brillouin zone. The Bragg planes (rather, lines, as we are in $R^2$) that we have to cross while going from the origin $O$ to $P$, ...

- 1,144

10
votes

### Maximum area of the intersection of a parallelogram and a triangle

I don't know if this is the optimal, but an isosceles triangle
with base and height $\sqrt{2}$ overlaps
$2 \left(\sqrt{2}-1\right) \approx 0.828427$
when placed as below,
and so improves over $\frac{...

- 147k

10
votes

### An inequality related to area and sidelengths of a polygon $Area(A_1A_2....A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

Cauchy–Schwarz tells you that
$$\sum_{i=1}^n \lvert A_iA_{i+1}\rvert^2\geq \frac{P^2}{n}$$
where $P$ is the perimeter of the polygon. Then we need the inequality $$P^2\geq 4n\tan(\pi/n)A$$ which is ...

- 83.2k

10
votes

Accepted

### Strange formula for area of a convex polygon

Denote $A_i=(0,b_i)$. It is a point on $\ell_i=\{(x,y):y=k_ix+b_i\}$, and let $$P_i=\ell_i\cap \ell_{i+1}=\left(\frac{b_{i+1}-b_i}{k_i-k_{i+1}},\frac{k_ib_{i+1}-k_{i+1}b_i}{k_i-k_{i+1}}\right)$$ be ...

- 93.6k

9
votes

### Need help with finding all angles of 11 sided 3D object

There is a problem in the design,
if I understand the phrases "the height I want the spire to be"
and "how tall I choose to make the spire":
these suggest there is freedom to ...

- 147k

8
votes

Accepted

### Necessary and sufficient condition for tangential polygon to be cyclic

$H_i$ lies on the ray $IA_i$ and $IH_i\cdot IA_i=2r^2$ (where $r=IB_i$), since the midpoint of $IH_i$ is the midpoint of $B_iB_{i-1}$. Hence $H_i$ is the image of $A_i$ under the inversion with ...

- 93.6k

7
votes

### Which polygons have *simple* periodic billiard paths?

Consider the simple polygonal billiard path itself. It is a polygon and it is convex,
because all interior angles are less than $\pi$. Now start with an arbitrary convex polygon.
It is a billiard path ...

7
votes

### First Dirichlet eigenvalue on regular polygons

Ignoring the normalization, your inequality writes
$$\lambda(P)\|u\|_{L^2(P)}^2\ge\|\partial_\nu u\|_{L^\infty(\partial P)}^2|P|\qquad?$$
Good news: this is scaling invariant.
Bad news: this is false ...

- 49.2k

7
votes

### How to characterize the regularity of a polygon?

All indices are in $\mathbb Z\bmod6$.
Let $z_k=\frac{\ell_{k,k+1}}{\ell_{k-1,k}}e^{i(\pi-\theta_k)}=\frac{v_{k+1}-v_k}{v_k-v_{k-1}}$, where $\ell$ is edge length, $\pi-\theta$ is vertex exterior angle,...

- 171

6
votes

Accepted

### Angle subtended by the shortest segment that bisects the area of a convex polygon

$\let\eps\varepsilon$This is not a complete answer. I will just show that $\theta$ can be smaller than $\pi/3$, but $\theta>\pi/4$.
1. Take an isosceles triangle $XYZ$ with $\angle Y=\angle Z=\pi/...

- 19.9k

6
votes

### Necessary and sufficient condition for quadrilateral to be cyclic

Regarding the "only if" part, a stronger result actually holds: if the quadrilateral $ABCD$ is cyclic, then $PQRS$ is similar to $ABCD$, and so it is cyclic, too.
This is stated, without ...

- 63.7k

6
votes

### Unlinked interlocking planar polygons

It is not possible with 7 (i.e., with a triangle $T$ and a quadrilateral $Q$). I write a rough proof.
First, any quadrilateral $Q$ lying in a plane $\pi$ can be partitioned in two triangles $Q_1$ and $...

- 370

6
votes

### First Dirichlet eigenvalue on regular polygons

There is a more general reason why any such statement will fail: If you consider a $P$ consisting of $N$ separate copies of the same basic region $P_0$, then $|P|=N|P_0|$, while everything else in ...

- 17.8k

5
votes

Accepted

### What is the name of the 65537-gon?

Following the portuguese nomenclature (I am from Brazil) and translating to english its results:
hexacontakaipentachiliakaipentahectakaitriacontakaiheptagon.
Best regards!!

5
votes

Accepted

### An inequality related to area and sidelengths of a polygon $Area(A_1A_2....A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

Presumably the indices $i$ in $A_i$ are taken mod $n$,
so "$A_{n+1}$" is to be identified with $A_1$.
This must be a known isoperimetric inequality,
but it's easier to prove than to find in the ...

- 74.4k

5
votes

Accepted

### Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?

The set of points that can't see out to infinity around a three-dimensional polyhedral obstacle is not itself a polyhedron.
In general, its boundary surfaces will be patches of hyperboloids, curved ...

- 18.3k

5
votes

### Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

I posed this problem at the 2007 CANADAM meeting in Banff after having discussed it
privately with several mathematicians starting in about 2000. The problem arose for
me because a positive answer ...

5
votes

Accepted

### Point generation in polygon

To generate points with the same distribution as the Halton sequence in a polygon you can just take a rectangle enclosing the polygon - for example take the convex hull of the polygon (assuming it may ...

- 4,512

5
votes

### Strange formula for area of a convex polygon

Fedor's derivation is very slick and elegant - surely the best way to guess the formula if you didn't know it already.
It does however require additional justification to show that the oriented ...

- 4,512

5
votes

Accepted

### A closed chain of $2n+1$-gon around $2n+1$-points

You may easily calculate everything in complex numbers. Denote $m=2n+1$, $w=e^{2\pi i/n}$, $Q_i=A_{i,3}=A_{i+1,2}$ for $i=i,2,\ldots$. We may suppose that $P_{m+i}=P_i$ for $i=1,\ldots,m$ and we have ...

- 93.6k

5
votes

### Necessary and sufficient condition for quadrilateral to be cyclic

This is not an answer but a comment. However, it will be too long and it would be awkward to break it up. It is of course natural to ask what is special about the nine point centre here. One can put ...

- 609

5
votes

Accepted

### The product of the lengths of two line segments that belong to Newton line

Let the incircle be the unit circle. Let $A'$, $B'$, $C'$, $D'$ be the points where this is tangent to $AB$, $BC$, $CD$, $DA$, and let them have coordinates $(\cos \alpha, \sin \alpha)$, $(\cos \beta,\...

- 18.7k

5
votes

### Collinearity in tangential pentagon

This follows from Brianchon's theorem. Note that in order to use that you need to consider the degenerate hexagon, $ABCFDE$. The theorem implies your conclusion.

- 5,535

4
votes

### Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way ...

- 2,926

4
votes

Accepted

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