Thanks for clarifying, Tom — I read Allan's post too quickly. That's good to know. (Did we ever decide how many different-size squares could tile the plane periodically?) But I'm pretty sure I've never seen a squared square torus with all different size squares aligned orthogonally (meaning, parallel to the sides of the "square" of the torus) and no 4-way intersections (which excludes squared squares). (As I mentioned, I first wondered about this regarding whether the 24 squares of sides 1 through 24 could tile a 70x70 square torus.) —Dan ----- Allan mentioned Pythagorean tilings, which satisfy the conditions outlined: Any two different sized squares tile the torus (with only one instance of each square). There are no 4-way intersections. The sides are not parallel to the sides of the square torus. Here's an image from the wiki page: https://en.wikipedia.org/wiki/Pythagorean_tiling#/media/File:Pythagorean_dis... Clearly this is a minimal solution. If you want to add the constraint that the sides must be parallel to the sides of the square torus, then this solution wouldn't qualify. Tom Dan Asimov writes:

I mean the same thing that a "squared square" usually means: a tiling of a square torus by squares of distinct sizes.

Ideally this would be without any 4-way intersections:

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(and I don't want to exclude the possibility of such tilings having sides that are not parallel to the "sides" of the square torus, in case any such may exist).

But even forgetting oblique tilings, the possibility exists that if there are such things as squared square tori, they may beat the records for squared squares, since they involve fewer constraints.

—Dan

----- Dan, do you mean to exclude the near-trivial family related to the Pythagorean tiling? Any two squares tile a square torus. I don't know of any *other* examples, though.

On Thu, Jul 23, 2020 at 8:19 PM Dan Asimov <dasimov@earthlink.net> wrote:

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Has anyone ever found a squared square torus that wasn't a squared square?

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