122
votes
6answers
8k views

Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer, moving initially $\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left $90^\...
71
votes
4answers
2k views

Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
18
votes
5answers
5k views

Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
15
votes
7answers
2k views

Unexpected Occurences of the Sierpinski Triangle

The probably most wellknown occurance of the Sierpinski Triangle is as the odd entries of the Pascal triangle Some month ago however, there was an article about mathematical models of sandpiles along ...
9
votes
2answers
1k views

sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
5
votes
1answer
220 views

Are lattice points in thin spherical shells uniformly distributed?

Consider the spherical shell (annulus) $$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq | x|\leq R+r \}.$$ Think of the limit $R \to \infty$. Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are ...
15
votes
0answers
797 views

Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
6
votes
1answer
208 views

map from 6-vertex model to domino tiling

I am trying to find a correspondence between 6-vertex model and an Aztec Diamond tiling. Here are the building blocks of the 8-vertex model: There seems to be more than one correspondence. I found ...
3
votes
2answers
456 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many problems/questions ...
14
votes
1answer
201 views

Product of a Laver table and a Hadamard matrix has mostly 0 rows

I recently noticed (while playing around) that the product of a Laver matrix with a Hadamard matrix gives a very sparse matrix. In particular, all but logarithmically few rows are all zero. The ...