A puzzle that tiles infinitely across both sides, based on the Klein Bottle • Hacker News Board

Re: A puzzle that tiles infinitely across both sides, based on the Klein Bottle

#51

Post by metalliqaz » Wed, May 02, 2018, 7:55 PM UTC

MrEldritch wrote: post #46

metalliqaz wrote: post #22

Earlier quoted context omitted.

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

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.

Re: A puzzle that tiles infinitely across both sides, based on the Klein Bottle

#52

Post by rcthompson » Wed, May 02, 2018, 8:03 PM UTC

metalliqaz wrote: post #51

MrEldritch wrote: post #46

Earlier quoted context omitted.

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.

Re: A puzzle that tiles infinitely across both sides, based on the Klein Bottle

#53

Post by chris_st » Thu, May 03, 2018, 12:47 AM UTC

pgreenwood wrote: post #39

chris_st wrote: post #37

So it's many (more than I want to admit :-) years since my Euclidean and Non-Euclidian Geometry class, but isn't this a cross-cap, not a Klein bottle?

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.

Thanks for the reminder.