Ok, that is one of the best math + lasercutter = art links that I have seen in a while. And the concept of being able to keep moving the puzzle pieces around is pretty cool.
There is also a link to upload your own art work and create your own puzzle, which is what I will be doing for this mother's day.
Overall, a pretty cool link.
If I counted correctly, you should be able to make a roughly rectangular shape out of the provided pieces. (Even if I miscalculated, I think they make a good exploration toy for the P2 tiling anyway.)
I've played with one of these puzzles in person. They're incredibly cool but pretty difficult. I worked for 20 minutes and was able to tesselate a single piece into a different position.
I might just be bad at it though. On the other hand, the challenge is part of the appeal to me.
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.
The soundtrack appears to be these NASA space recordings that were released in the 80s. They translated probe data into audible frequencies and released them as a CD box set.
How does one map an existing locally-similar pseudorandom pattern like the galaxy image onto a torus, Klein Bottle, or other closed shape? I know that with a generated pattern (e.g. Perlin noise) you automatically get that by taking the value of the noise function at the surface coordinates, but I have no clue about using existing planar images.
Could someone who understands the topology of this more fully say - if I had a set of two or more of these, could I solve each separately and put the solved puzzles together into a larger pattern?
> 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).
Nice! I haven't seen these before. There's also a brand of puzzles called Schmuzzles that are based on an escher lizard and tile. I would say the difference between these and our torus-based puzzles is that they employ a tessellation cut with repetitive piece shapes such that the image is guiding the construction (as all pieces fit in all places). In our puzzles, each piece only goes one place as they each have a unique shape. Our Klein Bottle and Cross-cap puzzles is are a new idea (to the best of my knowledge).
well if anything from the left side can be flipped 180 and attached to the right side of the klein puzzle you should be able to assemble two copies in the same configuration, then flip one whole puzzle over 180 and attach it to the side of the other.
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.
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.
I can buy a 1000 piece puzzle for $5 at walmart. Sure it's not nearly as cool as this but 236 pieces for $120? That's outrageous. I'd rather just have my money, thanks.
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.
Agreed? I mean, I think my comment makes it clear this way too expensive premium market isn't really the demographic that I fit into, though I know you're trying to prove a point.
I do find it a little annoying that the internet has made it common to price based on the people who will pay the most for things. It is still just cardboard. Just because it's an interesting idea doesn't necessarily make 236 pieces of cardboard worth $120.
To counter your point, this is like paying $120 for chick-fil-a nuggets instead of $5 for McDonald's nuggets. It's still just cardboard. It's slightly better made cardboard with maybe a little extra care, but it's just cardboard.
They appear to be hand-made. Manufacturing niche toys is very much a chicken and egg problem. At scale, these could be made for $1. But that implies volume sales which would likely result in a bunch of people complaining that the puzzles were "broken" and "some idiot left out all the edge pieces -- 0 stars!"
Fair point sorry I totally missed that. I still think it's too pricey but you're right that that is a more expensive material and definitely deserves to be reasonably more expensive.
