From the Mathologer files: Melbourne, 31 July 2015
HOW TO GET A RUBIK’S CUBE INTO A KLEIN BOTTLE
The following are footnotes to our two YouTube videos
A mirror paradox, Klein bottles and Rubik's cubes
https://youtu.be/4XN0V4xHaoQ
and
Can you REALLY put a Rubik's cube in a Klein bottle? https://youtu.be/ZMC61C5tigA
In these videos I talk, among other things, about inserting Rubik’s
cubes into Klein bottles that by themselves don’t fit through the
bottles’ “tunnels”.
As we said in the blurb and the YouTube annotations of the first video,
the 2x2x2 in the little Klein bottle at the end was meant as a puzzle
for everybody watching the video. A few people stood up to the
challenge and shared their ideas for how this feat can be accomplished.
Basically, there are five different ideas that I and others came up with.
In the following I’d like to discuss the feasibility of these ideas a little
bit:
1. BENDY TOOLS: Apply the same method I use to put Rubik’s cubes in
ordinary bottles (see the video https://youtu.be/pl9lLgfsw5I) just
using the sort of weird bendy tools that I show in the second video
above. I am fairly sure that this is not possible, at least not with the
tools and Klein bottles readily available (none of the tools work for the
small Klein bottles and none are long enough to be of any use with my
monster Klein bottle).
2. 3D PRINTING. Just 3d printing a Klein bottle and a working Rubik’s
cube inside. It would not be a big deal to do this using the sort of 3d
printed mesh plastic Klein bottle that I showed in the first video
and a 3d printed Rubik’s cube that hangs loosely together inside this
Klein bottle. Alternatively, one could consider assembling one of the
little nano cubes I showed in the first video inside one of these 3d
printed Klein bottles (the core presents a bit of a challenge but nothing
insurmountable). Printing in material that looks like glass is just
becoming a possibility, but nested printing of different objects
involving this material on the outside for the Klein bottle and another
material for the Rubik’s cube on the inside does not seem to be
possible yet. 3d printing can also help with some of the other solutions
that were suggested and that we’ll discuss below.
3. INSERTING DURING ASSEMBLY: Inserting a Rubik’s cube into a Klein
bottle while it is being assembled from its different components (see
Cliff Stoll’s site for how exactly they make their glass Klein bottles
http://www.kleinbottle.com/making_a_klein_bottle.html). The
glassblowers I talked to told me that this would not be possible
because of the heat involved which would melt any plastic Rubik’s
cube. They said they could insert metal pieces into the bottles during
assembly. This means that a Rubik’s cube made from steel could at
least in theory be inserted in a Klein bottle in this way. At least one
steel Rubik’s cube exists: http://lulabs.net/machining/ss-cube/. It
would not be a problem to 3d print a Rubik’s cube in steel. I am also
fairly convinced that if one just went for a really oblong Klein bottle, an
inserted plastic Rubik’s cube could survive the assembly process.
4. GROWING CUBES: Inserting a really small Rubik’s cube made from a
special material that grows when water is added. This should be
possible, but probably would not result in anything that could be called
a working Rubik’s cube. I actually experimented with this idea a little
bit.
5. MAGNETS: Well, that’s what I went for in the end. Would the method
I demonstrate in the video really work? Yes, it would. The cube I put in
the training bottle actually has the right size for my monster Klein
bottle: The individual dice that my Rubik’s cube consists of are copies
of the one I threw into the big Klein bottle at the beginning of the
second video. I also checked that two such bluetacked dice can be fed
into the big Klein bottle through its tunnel. ONE PROBLEM: The dice
themselves are made from plastic which resulted in me having serious
trouble getting the magnets and metal inserts firmly embedded in the
dice (turns out that this sort of plastic just does not like to be glued). I
would not dare to use my plastic Rubik’s cube in the real Klein bottle,
mainly because I cannot be sure that the metal bits and magnets will
not dislodge when I try to pull on them hard as I did at the very end of
the second video. Of course, overcoming this problem just means
building another Rubik’s cube, this time from wood or 3d printing one.
I’d also replace the little magnets that I used to lower the pieces into
the bottle by one that is slightly more powerful. Apart from that with a
bit more practice and finetuning I don’t think it would be terribly hard
to get this sort of cube into the monster bottle.
Some final remarks:
1. All the magnet that I used in this video are superstrong neodymium
magnets.
2. I am not sure how well the method I demonstrated in the video
“scales”. I personally don’t have the right sort of equipment to scale
things down to work with the more affordable Klein bottles. Maybe you
do and are interested in giving this a try? Again 3d printing
components might be the answer. I’ll probably give this a try at some
point.
3. If you’d like to build a 2x2x2 or a 3x3x3 magnet Rubik’s cube here
are instructions:
http://www.instructables.com/id/2x2-Magnetic-Dice-Cube/
http://www.instructables.com/id/Magnetic-Rubik-s-Dice-Cube/
There is even a place that sells 3x3x3 magnet cubes:
http://magneticcube.com. I ordered some but have not received them
yet and so cannot comment on how well they work. It might also be
possible to adapt these 3x3x3 magnet cubes to “Klein bottling”.
Finally it’s interesting to note that in “March 1970, Larry Nichols invented
a 2x2x2 Puzzle with Pieces Rotatable in Groups and filed a Canadian patent
application for it. Nichols's cube was held together with magnets. Nichols was
granted U.S. Patent 3,655,201 on April 11, 1972, two years before Rubik
invented his Cube.” (Wikipedia). When you click on the link it takes you
to the schematics for this early 2x2x2 cube.