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 ...
Moritz Firsching's user avatar
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 $-$...
Noam D. Elkies's user avatar
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' ...
29 votes

Which theorems have Pythagoras' Theorem as a special case?

Parseval identities in the theory of Fourier series and integrals.
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 ...
Fedor Petrov's user avatar
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 $\...
YCor's user avatar
  • 60.1k
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 ...
domotorp's user avatar
  • 18.4k
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 + \...
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.
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 ...
anon's user avatar
  • 201
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 ...
Terry Tao's user avatar
  • 109k
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 ...
j.c.'s user avatar
  • 13.5k
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 ...
Oscar Cunningham's user avatar
16 votes
Accepted

Packing rectangles: Does rotation ever help?

                    @YosemiteStan's example.                     Detail: Tilt angle $=\sin ^...
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.
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 ...
Will Brian's user avatar
  • 17.4k
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 ...
Joseph O'Rourke's user avatar
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 ...
Robert Israel's user avatar
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 $C_k$, ...
j.c.'s user avatar
  • 13.5k
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 ...
Gerry Myerson's user avatar
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
Fedor Petrov's user avatar
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 ...
Chris Wuthrich's user avatar
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 ...
Noam D. Elkies's user avatar
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 ...
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 $ABC.$
Donatien Bénéat's user avatar
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 $n^2$ ...
user36212's user avatar
  • 1,687
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}...
Ivan Izmestiev's user avatar
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 ...
fedja's user avatar
  • 59.8k

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