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 ...
- 10.7k
42
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 $-$...
- 79.9k
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.
28
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 ...
- 109k
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 $\...
- 63.9k
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 ...
- 18.7k
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.
21
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 ...
- 114k
20
votes
Accepted
A claim on partitioning a convex planar region into congruent pieces
This picture looks like a counterexample with $N=2$ and $R$ a convex pentagon:
This should work more generally starting from an $n \times (n+1)$ rectangles
for any integer $n>1$, removing two ...
- 79.9k
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
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 ...
- 13.6k
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 ...
- 3,137
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 ...
- 18.5k
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 ...
- 151k
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 ...
- 54.2k
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$, ...
- 13.6k
14
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.$
- 283
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 ...
- 39.9k
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
- 109k
14
votes
Is this a new result about hexagon?
All these conditions are equivalent to six lines $AB'C''=a$, $CB'A''=b$, $CA'B''=c$, $BA'C''=d$, $BC'A''=e$, $AC'B''=f$ being tangent to the same conic (by Brianchon theorem and its inverse). For ...
- 109k
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 ...
- 8,945
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 ...
- 79.9k
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
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 $n^2$ ...
- 1,687
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