56
votes

### Does this geometry theorem have a name?

Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29)
In the ...

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 $-$...

34
votes

### Which theorems have Pythagoras' Theorem as a special case?

The Law of cosines is the first that comes to my mind:
$$c^{2}=a^{2}+b^{2}-2ab\cos \gamma$$
(source: Wikipedia)
If $\gamma$ is a right angle, its cosine is 0 and all that remains is Pythagoras' ...

Community wiki

29
votes

### Which theorems have Pythagoras' Theorem as a special case?

Parseval identities in the theory of Fourier series and integrals.

Community wiki

27
votes

Accepted

### Does greedy circle packing exhaust the measure of every bounded open set in the plane?

Yes, the reason is the same as in the proof of Vitali covering theorem. It suffices that $U$ has finite measure, allowed to be unbounded.
If $D_i$ are the greedy-chosen (closed, but this is not ...

23
votes

### A question about subsets of plane

(Initial post November 24, 2016, edited November 27, 2016) This does not exist.
The proof that $X$ doesn't exist is a bit elaborate and makes use of ends of coset spaces. I will prove:
(a) Let $\...

22
votes

Accepted

### Can any sequence of consecutive integers be realized as winding numbers?

Isn't this easy by induction? Delete one of the largest numbers, say $m$, from your sequence, realize the remaining numbers, then in the realization pick any region with winding number $m-1$, and make ...

22
votes

### Which theorems have Pythagoras' Theorem as a special case?

The parallelogram law says that if $\mathcal{V}$ is an inner product space and $\mathbf{v},\mathbf{w} \in \mathcal{V}$, then
$$
2\|\mathbf{v}\|^2 + 2\|\mathbf{w}\|^2 = \|\mathbf{v} + \mathbf{w}\|^2 + \...

Community wiki

22
votes

### Which theorems have Pythagoras' Theorem as a special case?

The Pythagorean theorem is a limit of the general formula for spherical or hyperbolic space: $\cos(a\sqrt{\kappa})\cos(b\sqrt{\kappa})=\cos(c\sqrt{\kappa})$, where $\kappa$ is the curvature.

Community wiki

20
votes

### Kakeya crossed-needles problem

I do not know if the minimal area needed to rotate the + is that of the smallest disk containing it. However, I do know that it cannot be done with arbitrarily small area.
This is a special case of a ...

20
votes

Accepted

### Aperiodic monotile without reflections?

The same authors have just released a preprint claiming a positive answer to this question.
EDIT: Here is a picture of the reflection-free aperiodic monotile:
More visualizations and other data are ...

17
votes

### The space of triangles that fit inside a given triangle, parametrized by edge lengths

Here's the abstract of K.A. Post, "Triangle in a triangle: On a problem of Steinhaus", Geom Dedicata (1993) 45: 115; this paper was cited in the one given in the comment by Nemo.
A necessary and ...

17
votes

### How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?

I tried to find a colouring where one of the colours comprised exactly the 'flipped' tiles. So I coloured all of the flipped tiles blue and then found three of the remaining tiles which were all ...

16
votes

Accepted

### Packing rectangles: Does rotation ever help?

@YosemiteStan's example.
Detail: Tilt angle $=\sin ^...

Community wiki

16
votes

### Which theorems have Pythagoras' Theorem as a special case?

This is probably the simplest: $(a+b)^2=a^2+b^2+2ab$, if you take $a,b$ elements of some inner product vector space and $(\cdot)^2$ means inner product with itself.

Community wiki

15
votes

Accepted

### Can two-point sets be Borel?

A two-point set cannot be $F_\sigma$, as Mohammad mentions in his question. Also,
A two-point set cannot contain a dense $G_\delta$ subset of an arc.
This was proved by Gareth Davies in his thesis ...

15
votes

Accepted

### Minimal pizza cutting

Here is recent paper coauthored by Cox (whom Yoav Kallus cited),
with a different focus:
Headley, Francis, and Simon Cox. "Least-perimeter partition of the disc into $N$ regions of two different ...

14
votes

### A question about subsets of plane

I don't think there is such a set for the plane, but I'll point out that there is one for the sphere $\mathbb S^2$. Namely, let $S$ and $T$ be two members of $SO_3$ such that the group $G$ they ...

14
votes

### Dodecahedral rolling distance

Here are a few trivial lemmas. I won't use anything about the rolling motion, just that the distance is defined by gluing pentagons edge-to-edge:
The $dd$-circle of radius $k$, which I'll call $C_k$, ...

14
votes

### How can we find n points on a plane so that as many pairs of points as possible have the same distance?

The number is tabulated at OEIS. It seems that it's only known up to $n=14$ (and some scattered larger values). Links are given there to some papers on the topic. Evidently, no one knows how to do it ...

14
votes

Accepted

### Is $\arcsin(1/4) / \pi$ irrational?

This is a partial case of the classical result.
https://en.wikipedia.org/wiki/Niven%27s_theorem

13
votes

### Two queries on triangles, the sides of which have rational lengths

On the second question: There is the inequality $12\sqrt{3}A\leq P^2$ that needs to be satisfied first of all to get a triangle.
Using Heron's formula for a triangle with sides $x$ and $y$, you are ...

13
votes

Accepted

### On circles and ellipses drawn on an infinite planar square lattice

(1-2) Yes.
For each integer $n > 0$ the circle $x^2 + y^2 = 13^{n-1}$ passes through
exactly $4n$ lattice points, namely those with
$$
z := x+iy = \zeta (3+2i)^a (3-2i)^b
$$
with $a,b$ nonnegative ...

13
votes

### Which theorems have Pythagoras' Theorem as a special case?

Pythagoras' theorem is a special case of the three point identity for Bregman distances: Let $h$ be convex and lower semi-continuous on a Banach space - further assume differentiability of $h$ for ...

Community wiki

12
votes

Accepted

### Do two new special points in any triangle exist?

Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$

12
votes

### Packing rectangles: Does rotation ever help?

The classic answer to this is a paper of Erdos and Graham 'On packing squares with equal squares'. Given a square of side $n+\varepsilon$, where $0<\varepsilon<1$, we can obviously fit in $n^2$ ...

12
votes

### Distance between point inside a triangle and its vertices

Let $a, b, c$ be the side lengths of the triangle, and $x, y, z$ the distances from a point inside a triangle to the respective vertices. Then the numbers $x,y,z$ satisfy the equation
$$\begin{vmatrix}...

12
votes

Accepted

### All saddles in the unit ball have area $<2\pi$?

It is actually next to trivial if you choose the right parameterization (and rather puzzling if you don't, so it can make a decent take-home exam problem in multivariate calculus).
I'll use the line ...

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