[rcthompson]

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.

[metalliqaz]

Does that property imply a Klein bottle? I was not aware they were synonymous.

[rcthompson]

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.

[MrEldritch]

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.

[metalliqaz]

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.

[rcthompson]

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.