# Tag Info

### 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 $-$...
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### 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 ...
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### Acute triangles in "obtuse" polygons?

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

### 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 ...
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### Billiard circuits in pentagons

We do not need the pentagon to be cyclic. But we do have a requirement for the angles. Begin with two cases where a cyclic path can be defined. In the first case it does not exactly meet the ...
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### Reordering vertices of a polygon

I'm not certain, but it may be that the expansive motions in the Connelly-Demaine-Rote paper cited below can provide a route to your $Q''$. An expansive motion is one in which the distance between ...
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