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What is the maximum percentage of an infinite plane that could be tiled with primitive Pythagorean triangles, if every triangle may be used at most once? My intuitive guess is that it can’t reach 100%, but might approach that value. But I can’t think of a way to approach this quantitatively.

ADDED 7/11/24: To address concerns expressed by a bot and others, I think this is a new, nontrivial and important question with regard to number theory in a couple regards. While the generation, validation, and patterns in Pythagorean triples have been extensively studied, Pythagorean tiling has only been addressed briefly and in the context of repetitions of a few basic structures.

Furthermore, it introduces a deviation from classic tiling problems in that instead of potentially infinite numbers of similar tiles or sets of tiles, it has to be done never using the same tile twice, or even two tiles with the same proportions. I could not find much related to this besides spiraling Fibonacci squares and a lone question on Reddit.

I can think of a few strategies for attempting a complete tiling as proposed. One would be to work outward in a spiraling manner, either consistently clockwise or, if needed, doubling back once or more. A second would be to find a way to predictably build shapes known or that could reliably shown to tile when assembled. A third would be what I think of as the wall or outward method, where you might construct tiling that, for instance, tiles a portion of the infinite plane working from a center such as covering the top or left half, leaving a linear edge, and then repeating in a mirrored fashion. Variations of the latter might include constructing some number of pie slices that fit together.

All of that supports the answer “yes it’s possible” by seeking a logical and reliable method to accomplish the tiling. Strategies for proving “no” could involve showing there are necessary one or more holes that must form which can’t be avoided. And using any triangles of infinite size is against the rules of course. They can be as large as you like but must have some countable integer side sizes.

That’s all I have to offer at this time, and if it’s too troublesome or novel to demand any further investigation here I can seek solutions in the realm of theoretical molecular modeling. Huran and others have touched on related questions in ACS with articles like “Atomically Thin Pythagorean Tilings in Two Dimensions.”

As such, I thank you in advance for all your time, efforts and good intentions.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Jul 10 at 4:17
  • $\begingroup$ I presume a "primitive Pythagorean triangle" is a right triangle whose sides form a primitive Pythagorean triple. $\endgroup$ Commented Jul 10 at 4:35
  • $\begingroup$ Correct, that is the accepted definition. $\endgroup$
    – Nalacram
    Commented Jul 10 at 5:46

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If each such triangle can only be used at most once, any configuration is obviously aperiodic, so it's not at all obvious that it has a "percentage": presumably you're asking for a limit of the fraction of (some shape) that is covered by the triangles as the size of (that shape) goes to $\infty$, but the details of exactly how this is done may be important. Well, I think here's one way.

Given any $\epsilon > 0$, if $R$ is sufficiently large, we should be able to take six different primitive Pythagorean triples $(a_1, b_1, c_1), (a_2, b_2, c_2), \ldots, (a_6, b_6, c_6)$ such that $(1-\epsilon) R < a_i \le R$ and $(1-\epsilon) R < b_i \le R$. Then the corresponding six triangles can be arranged to take up fraction at least $(1-\epsilon)^2$ of three $R \times R$ squares. Arrange those three squares in an L shape, forming three quarters of a $2 R \times 2 R$ square. Next do the same construction with $R$ replaced by $2R$, getting six more primitive Pythagorean triangles taking up at least $(1-\epsilon')^2$ of three $2R \times 2R$ squares, where $\epsilon'$ can be somewhat smaller than $\epsilon$. Proceed inductively, with the $\epsilon$'s going to $0$. The picture after $3$ generations looks something like this:

enter image description here

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Here is an example of six distinct primitive triples that form an isosceles triangle with sides $\,(2600,2600,1456),\,$ and my avatar is a photograph of this figure cut out of cardboard triples. A simple computer program quickly found several of these in the limited range I permitted and with the requirement that 6 triangles form 1.

enter image description here

There are 4 primitive triples with side-$B=2600\,$ and another 4 with side-$B=1456.\,$ so we know that the tiling can extend at least one step beyond this figure. There are an infinite number of primitive triples so it seems logical that an infinite number of infinite combinations could tile an infinite plane.

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