This store, as Celene has hopefully inferred, sells games. The organization isn't perfectly orderly, but there are some vague sections of the store. Does Celene want to look at computer games, board games, strange video game controllers, handheld puzzles, basic household sports equipment?

Oh, she thought it was a store that hosted a single Game, kind of like escape rooms back on Earth, but with only one game instead of several.

She doesn't know where she got this prior from.

Sure, she'll look at the.... handheld puzzles.

As Celene heads over to that section, she passes by a table with some sort of tile-based puzzle on it!

If she looked around the table and inspected some of the tiles, she'd find 10 red and 11 blue tiles. The blue tiles are substantially larger than the green ones, and the reds just a tad larger than the blues.

Oh, is this like a square packing puzzle? Yeah, seems neat, although she wonders if there's a way to make the walls get smaller or larger so that people can adjust how optimal their packing is.

This particular rendition of this puzzle doesn't have such a feature. There's a little wiggle room, but it's tight enough that you have to arrange them just right.

So, how well does Celene know her quasiregular solids?

On Earth we just slap 12 pentagons on and fill up the rest of the shape with hexagons and call that good enough!

Well yes, you could use a truncated icosahedron (those are useful when you want something especially round), but that would be a little misleading as to the mechanics of the puzzle. Each hexagon's edges alternate between being incident to a pentagon and to a second hexagon. So you'd have to turn the hexagons in increments of τ/3, and it'd just be the same puzzle as the version with triangles except with the turning mechanics slightly less clear.

She.... thinks that makes sense? Sure, let's say that makes sense.

Also it's mostly for like, soccer balls, and video games with hexmaps on spherical surfaces (not toruses, though, because sigh toruses. have no curvature.*)

*technically toruses have zero average curvature

The disembodied spirit of the Eifweni game store that is talking to Celene inside her head doesn't know the math behind toroidal hexagonal grid video games, though Eifwen does of course have such things. But at a first pass: the reason you need twelve pentagons to make your sphere hexmap work is that you need to satisfy V-E+F=χ=2. So if you have P pentagons and H hexagons, that's (5P+10H)/3 - (5P+10H)/2 + (P+H) = 2 (where we're assuming three faces meet at each vertex.) 2H - 3H + H = 0H cancels out, so you're left with 5P/3 - 5P/2 + P = 2, and P = 20.

If you do the same analysis in the toroidal case, you get... P = 0. Which, is great, fantastic really, but if you just shove hexagons in you get... a plane, not a torus. So actually what you want to do is mix in some pentagons and also some heptagons. The H term still cancels out, so you have 5P/3 - 5P/2 + P = -11S/3 + 11S/2 - S. Multiply through by six, 14P - 23P + 10P = -22S + 33S - 10S, or P = S.

So the upshot is to make a torus you want a lot of hexagons, and then an equal number of pentagons and heptagons. Presumably you put the pentagons in positive curvature parts and heptagons in negative curvature parts.

Or you could just be willing to distort and bend the hexagons! Like, it's self evident that if you have a plane of hexagons you can roll it up into a tube, and then you can bring the tube's two ends togethers into a torus, and since hexagons can tile on a flat plane it's obviously possible to make a plane of hexagons.

Are you sure you can't make the hexagons form a sphere if you're willing to distort and bend them enough.

Not.... not if you keep all the lines straight, she's pretty sure? And like, you don't cheat by putting two parallel lines end to end and then calling that a single line.

...By "bend" and "distort" you meant affine transformations????

Celene does not know what this phrase means! Maybe! She's not a mathematician!

Affine maps are homeomorphisms of affine spaces, which are like vector spaces where you forget how to add. But you don't forget how to subtract. Subtracting still gives you a vector. A map f is affine if it induces a valid linear map (b-a) ↦ f(b)-f(a).

Wow! Even more words she does not understand! She's beginning to suspect this planet's math education was rather more special than her own's!

It's the closure under composition of linear maps and translations?

Is Game somehow under the impression that this description is elucidating for her?

...affine maps are maps you can make by combining linear transformations and translations?

Ok, yes. She knows what a linear transformation is. It's a rotation, a translation, a dilation, or a reflection. Translations are, of course, already included in that description.

...no!

This is what she was taught in school! Maybe it's a... translation error.

...some math teacher deserves to get bitten.