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.$$ Is it possible that in the minimal tilings, patterns like in a 'Fibonacci rectangle tiling' occur frequently? … Moreover, does there even exist a minimal tiling of a $km\times kn$ rectangle such that not all square sides are multiples of $k$? …

This is a follow-up of my recent thread about tiling a $m\times n$ rectangle with squares: I'm wondering to what extent a minimal tiling is essentially unique, that is, up to reflections of the whole rectangle … But now: We'll call a rectangle (or a minimal tiling of it) coprime-reducible if the rectangle can be split into two (tiled) rectangles that have coprime sides each. …

Given three naturals $a<b<c$. We consider polyominos, connected or not, which are composed of three squares of sides $a,b,c$. How can one characterize all triples $a,b,c$ for which such a polyom …

always a square (or at least a bigger rectangle) that can be tiled with $s\times t$ rectangles in an irreducible way (i.e. any grid line splitting it cuts at least one of the $s\times t$ rectangles of the tiling … E.g. for $(s,t)=(1,2)$, it is well known that a $\underline{6\times 6}$ square has no irreducible tiling, but a $\underline{8\times 8}$ one does, and so do in fact all other rectangles with even area and …

So I would like to push it further by asking: Is there a tiling such that not only the number of positions is infinite, but that also all positions (or to start with, say just ONE position) occurs …

In the case that $a=\sqrt{\frac{5-\sqrt{5}}2}$, we are lucky and can split one of those into two smaller ones, obtaining a tiling into 5 different isosceles triangles (with all occurring angles being multiples … I am quite sure the answer to the initial question is no, and it may even be interesting to restrict it to the following: For which other rectangles is such a tiling known to exist? …

My question: How many different positions can occur in a tiling $\mathcal T$ of the plane? … The picture shows a tiling of a certain pentagon with $p(\mathcal T)=12$, where the $12$ marked tiles form a fundamental domain. It comes from the bottom right tiling of this picture quoted here. . …

What if we exclude the possibility of self-similarity by requiring the tiling to be periodic? The two tilings I used as initial examples are both periodic with $p(\mathcal T)=12$. … So I will squeeze this out by sharpening the conditions as follows: If we require moreover that the tiling has a fundamental domain which contains only one tile of each position, how big can such …

Penrose-style) tiling of the plane, but also to grids / tilings in higher dimensions, which looks like hitting a whole beehive. … Is it possible that the average component size for given $k$ and any tiling depends only on the average number of tiles meeting at each "corner"? …

Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. … (Note that up to boundary effects, in the hat tiling on the right, the union of all the dark blue, light blue and white tiles also appears to be a simply connected region with the same "fractal quality …