A puzzle that tiles infinitely across both sides, based on the Klein Bottle (opens in new tab)

I recently stumbled across this old site: http://www.mathpuzzle.com/ctrules.html

I reverse-engineered the tiles in OpenSCAD. Let me know and I'll upload them.

I'm at work right now, but give me a day and I will design that.

This one is a Klein Bottle but we just released a cross cap one yesterday: https://n-e-r-v-o-u-s.com/blog/?p=8068

No cross-caps here. The first puzzle is a torus, the second a Klein bottle. Informally: the Klein bottle has on pair of edges is glued with a twist and one without; with the cross-cap, both pairs of edges are twisted then glued.

from the article:

> Multiple infinity puzzles can be combined to create a larger continuous puzzle. The image above shows some of the creative combinations possible with two infinity puzzles of different colors ($75, for two).

here is the image from the quote: https://i2.wp.com/n-e-r-v-o-u-s.com/blog/wp-content/uploads/...

That would be INCREDIBLY awesome, considering that it would require pieces that have a curve in to the 4th dimension (or perhaps just pieces that could pass through each other). A Klein bottle is a 2-dimensional surface cannot be embedded in a 3-dimensional space without crossing.

I think the flat version is a more authentic representation of the 4-dimensional object. If you were a 2D creature embedded in the surface of a Klein bottle, space would seem flat in every direction... at least until you went out for a walk and found the mirror-image of your house.

I have a 3D jigsaw puzzle that makes a globe (i.e. the Earth) that consists of curved plastic pieces. It shouldn't be too hard to make such a Klein puzzle, though solving it is harder. The globe puzzle includes a small bowl (of the same radius as the finished globe) to assist putting the pieces together; this wouldn't be possible with a klein bottle due to there not being a consistent radius.

Well you’re looking at it. If you think of the puzzle only tiling in one direction, the flip needed to move the piece from one side to the other is analogous to the twist in the Möbius strip. Because it works in two dimensions, you have a Klein bottle, much the same way a looping plane is isomorphic to a torus.

Just take the Klein bottle one and change the shapes along the top and bottom (or left and right) edges so they no longer match up. Voila.

It's a Klein bottle as an abstract surface, where an abstract surface roughly speaking is anything that is locally 2D. In this case, it is a "flat" Klein bottle, due to the kinds of straight lines (geodesics) the surface has. The usual immersed Klein bottle you're likely familiar with is not flat.

Every closed surface comes from a symmetry of the sphere, the Euclidean plane, or the hyperbolic plane. For instance, you can get a (flat) torus by taking the Euclidean plane and taking all translations that shift the plane in the x and y directions by integer amounts, where we consider two points to be "the same" if they are translates of each other. So, if you take a path horizontally, you periodically return to "the same" point every unit distance. This is the Asteroids geometry.

The Klein bottle can be obtained by the symmetry generated by two transformations: (1) a vertical translation by 1 unit and (2) a horizontal translation by 1 unit followed by a vertical flip.

The wooden puzzles are from tiling the plane in a way that respects one of these symmetries, and then taking just enough puzzle pieces to cover the fundamental domain. For the Klein bottle, all this together means that in one direction you can take off a piece and put it down on the other side in the same orientation, and in the other direction you have to flip the piece over.

If you imagine the surface that is formed when every possible connection between pieces is made simultaneously, that surface is a Klein bottle. Obviously, making all the connections simultaneously is not possible in 3 dimensions, without allowing the pieces to deform and intersect each other.

I can buy a 1000 piece chicken McNuggets for $5 at McDonalds. Sure it's not nearly as cool as a fresh local meal from a nice restaurant, but dinner for 2 for $120? That's outrageous. I'd rather just have my money, thanks.

They're available on the same site as the article: https://n-e-r-v-o-u-s.com/shop/line.php?code=12