
A puzzle that tiles infinitely across both sides, based on the Klein Bottle - merraksh
https://n-e-r-v-o-u-s.com/blog/?p=7613
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baldeagle
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.

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gnarbarian
this is awesome and reminds me of the tiling puzzle from the book Anathem:

[http://anathem.wikia.com/wiki/Teglon](http://anathem.wikia.com/wiki/Teglon)

or

[https://en.wikipedia.org/wiki/Penrose_tiling](https://en.wikipedia.org/wiki/Penrose_tiling)

I would love to have a Penrose tiling puzzle set.

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

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gnarbarian
Woah, Thanks! Does the AK in your name stand for Alaska? If so, I'm in
Anchorage.

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AKrumbach
No, that's my first initial and last name (A Krumbach).

Anyway, "puzzle" is now available at
[https://drive.google.com/file/d/1KYDTZ2AVWQfWCSACUvfl5gAIHzA...](https://drive.google.com/file/d/1KYDTZ2AVWQfWCSACUvfl5gAIHzAWXG7m/view?usp=sharing)

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.)

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phinnaeus
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.

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chris_st
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?

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pgreenwood
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.

~~~
chris_st
Thanks for the reminder.

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oceanghost
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.

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andreareina
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.

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rambojazz
Very interesting! With those curved borders however, it's going to be pretty
hard to find where each piece fits :)

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tobr
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?

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gnarbarian
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/...](https://i2.wp.com/n-e-r-v-o-
u-s.com/blog/wp-content/uploads/2016/11/infinity.png?resize=750%2C290&ssl=1)

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tobr
Sorry! I should’ve both read and expressed myself more carefully. I meant the
Klein bottle topology, not the torus.

EDIT: FWIW torus puzzles are definitely not a new thing - I had this one as a
kid:
[http://img.tradera.net/images/096/270662096_95a37033-d8f9-4c...](http://img.tradera.net/images/096/270662096_95a37033-d8f9-4c3a-b874-37ab02db6e34.jpg)

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nervous_jessica
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).

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tempodox
This would be even more awesome if the puzzle pieces were curved, so they'd
form a Klein bottle when put together.

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mcherm
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.

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tempodox
While this is true mathematically, there is at least one project that emulates
Klein bottles in 3D quite credibly:

[http://www.kleinbottle.com](http://www.kleinbottle.com)

Edit: You can even put in some liquid (the “hose” continues through the
crossing). The fun part is getting it out again :)

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mcherm
There are lots of projects that do this (implement a Klein bottle with a
crossing).

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dancek
Even my dad has knit several Klein bottle hats for fun, inspired by one of
those projects. I got one as a Christmas present.

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Semirhage
The coolest part is if you put a few them together you open a non-orientable
wormhole and pass into an Alice Universe.

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austincheney
I want to see a jigsaw puzzle that results in a mobius strip.

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0xfaded
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.

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metalliqaz
I don't see how this is based on a Klein Bottle.

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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.

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

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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.

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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.

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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.

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werdnapk
A puzzle like this will drive someone with serious OCD nuts. :)

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dwighttk
(2016)

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zodPod
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.

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NickBusey
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.

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zodPod
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.

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NickBusey
To counter your point, it is not just cardboard. It is laser cut wood. A far
more expensive material, and manufacturing process.

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ballenf
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!"

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ryanmarsh
Would some enterprising HN’er please manufacture these? My wife would love
this.

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

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Avshalom
Solid MEH, we bought my mom a jigsaw puzzle with no edges and the image
(kittens) flipped on the opposite side in like... 94?

