[forkerenok]

I still maintain this naive belief that all of math is elegant, and from that perspective 13 sides sounds and looks oddly specific and feels contrived to me, at least without knowing specificities of the field.

Where do 13 sides come from? Is it related to a number of transformations?

[OscarCunningham]

This is just the first tile we've found with this property. There might be a tile with the same property and a smaller number of sides. Perhaps the such tile with the smallest number of sides will be especially elegant.

[Jarmsy]

Plenty of mathematics is ugly. Just look at optimal packing. People like to focus on the elegant parts.

[gus_massa]

A lot of the ugly part of math is hidden inside lemas and theorems, so the main part of the proof can be hopefully straightforward.

Anyway, I agree that people like to focus on the elegant parts. Math popularization materials have too much kawaii math.

[GuB-42]

A notorious "ugly" proof is that of the 4-color theorem.

It has been proven using a computer. The problem was first reduced to a few hundred cases, then a brute force algorithm was used to solve each case.

[NeoTar]

Well, from one perspective, since the requirement is that a tile must be able to tile the plane only irregularly; - All three-sided shapes can tile the plane regularly, - All four-sided shapes can tile the plane regularly,

So we know the minimum number of sides is at least five.

Assuming 13 sides is established as the minimum, I suspect there is no 'nice' reason for it; its just that this may be the minimum number to give you sufficient degrees of freedom.

[Laremere]

See also:

The monster: https://youtu.be/mH0oCDa74tE

Ideal packing of squares (some are elegant, 17 is not) https://kingbird.myphotos.cc/packing/squares_in_squares.html

The 4 color theorem was solved by reducing it down to 633 cases and just using a computer to find a coloring for each case: https://en.m.wikipedia.org/wiki/Four_color_theorem

[paulddraper]

I recall hearing a mathematician say "0 and 1 and infinity are the only sensible numbers. Everything else is quirky."

[uoaei]

It's based on a tiling of kites. You combine multiple kites together to create the aperiodic monotile. It just happens that the perimeter of the tile has 13 sides.