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 ...
- 10.3k
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 $-$...
- 73.8k
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 ...
- 73.8k
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-...
- 63.5k
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 ...
- 205k
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 $\...
- 54.8k
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
- 73.8k
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
...
- 73.8k
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 ...
- 13.3k
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 ...
- 146k
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 ...
- 52.2k
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 ...
- 36.9k
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
- 92.2k
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 ...
- 7,110
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 ...
- 73.8k
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