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.
Not by itself. It could also imply a torus, depending on how the pieces must be oriented in order to fit each other. There may also be other possibilities.
A Klein bottle can be defined topologically as a Mobius strip that's connected on both axes. So if the left side is connected to the right side with a mirror twist, and the top is connected to the bottom with a mirror twist, it's topologically a Klein bottle.
okay, I'm just not seeing how a puzzle with pieces that can be placed on the other side meets that property. The pieces would have to be elastic, and if the pieces are allowed to change shape, it's not really a puzzle anymore.
That's why I wrote that you have to imagine the surface formed by making every possible connection simultaneously. The point is that the "completed" puzzle is topologically a Klein bottle. It can't be completely constructed in 3 dimensions.
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.