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 $-$...
36 votes
Accepted

Tiling the plane with incongruent isosceles triangles

Q1: Yes. Any acute non-isosceles triangle can be tiled by three pairwise incongruent isosceles triangles, by connecting each vertex to the circumcenter. Start from some isosceles $T_0$ with repeated ...
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
Accepted

Why is it so hard to prove Toeplitz' conjecture?

Let me elaborate on Sam Hopkins' comment. The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-...
27 votes

Which theorems have Pythagoras' Theorem as a special case?

Parseval identities in the theory of Fourier series and integrals.
24 votes

Term for "uncheckable constructions"

Your question amounts to treating construction problems in geometry as decision problems, and so it makes sense to me to adopt the terminology of computability theory. This same kind of distinction ...
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 $\...
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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 + \...
21 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 ...
  • 17.7k
21 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.
19 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 ...
  • 191
18 votes

Which theorems have Pythagoras' Theorem as a special case?

So far no one has mentioned the original generalization! Early in Euclid's Elements, the Pythagorean theorem is stated by comparing square areas: Book I, Proposition 47: In right-angled triangles the ...
17 votes

Tiling the plane with incongruent isosceles triangles

Google soon finds that Q2 is problem C11 in Unsolved Problems in Geometry by Croft, Falconer, and Guy. But perhaps it's been solved during the intervening decades. URL
17 votes

Tiling the plane with incongruent isosceles triangles

Simpler construction: a non-isosceles right triangle $T_0$ can be divided into two isosceles triangles not congruent to each other, or into two right triangles similar to $T_0$. Reversing the latter ...
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 ...
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16 votes
Accepted

Is there a subset of the plane that meets every line in two open intervals?

Let $E$ be a set of the claimed form. Call a direction $\omega \in S^1$ a limit direction of $E$ if there exists a sequence $p_n$ of points in $E$ going to infinity whose argument goes to $\omega$, ...
  • 95k
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

Tiling the plane with incongruent isosceles triangles

Illustration of Noam's construction:          
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 ...
  • 15.7k
15 votes

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

Yes, $\arcsin(\frac14)/\pi$ is irrational. Suppose $\arcsin(\frac14)/\pi = m/n$, where $m$ and $n$ are integers. Then $\sin(n \arcsin(\frac14))=\sin(m \pi)=0$. We analyze this usng the formulas from ...
  • 18.6k
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 $C_k$, ...
  • 13.3k
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
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13 votes

Is there a subset of the plane that meets every line in two open intervals?

The answer to the question is no. Rather than typing a new answer, I've just edited my old one (which contained some partial progress). I think this is OK since the main idea of the old answer (or at ...
  • 15.7k
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 ...

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