Message #3205

From: Melinda Green <>
Subject: Physical 4D puzzle progress
Date: Mon, 02 Nov 2015 18:40:10 -0800

Some time ago I said that I’d made some progress on this front and would
soon post about it. Well, now that the topic has come up again I figure
now would be a good time to do that. Just as was recently brought up, my
thinking was that a 2^4 analog is the place to start. My first thought
was of a puzzle with 4-colored pieces that could move along the edges of
a hypercube like beads on hoops, with a central mechanism like a 2^3.
Twists would only happen in that center position, and rotations would be
required to bring faces into that position. Think of a standard 2^3 but
with the pieces hollowed out into a framework such that solid pieces
could slide into and through them to be twisted by the mechanism. I made
a few sketches of the basic idea and then had an email exchange with
Oskar von Deventer about his thoughts on whether something like this
could be built. He was helpful and encouraging, and came up with the
following nice rendering of the basic idea:

The twisting part is no problem, but a rotation mechanism didn’t present
itself to him. One idea I had was to construct the pieces out of some
very squishy material. He pointed out that there are 3D printers that
can print in colored foam rubber which seems perfect, but we couldn’t
think of any way to attach them to each other and to the hoops that
guide them during rotations. That idea also doesn’t sound like it would
be very easy or fun to use anyway, so we dropped it.

I then spent a lot more time thinking about 4D rotational mechanisms and
came up with an approach that I really like. I don’t quite see how to
take it all the way to a working puzzle, but perhaps some of you will
have some ideas. I realized that the puzzle can be thought of as two
cubes that can sort of roll against each other. Imagine two 2^3 puzzles
snapped together at a face. Now imagine breaking them apart so that they
hinge on any one of their four common edges. Continue to flex that hinge
until two other faces meet and snap together. This constitutes a proper
4D rotation! Here is a sketch:

The puzzle is shown in its solved state, and no number of rotational
moves will change its state, just like you would expect from rotations.
Six of the faces are easily seen as embedded octahedra. Look at the red
face in the first rotation. Once snapped in place, it makes a solid
octahedron in the very center. A standard 2^3 mechanism should then
allow twists that cut through its central planes. The final two faces
(green and magenta) are a little more difficult to see and can never be
moved into the center, but I think that all the puzzle states of the 2^4
should still be reachable. This is a compromise I would be very happy to

So now the problem reduces to finding a mechanism that will allow two
cubical frameworks to roll against each other and snapping into place
such that they always create a working 2^3 mechanism in the middle.
That’s still a lot to ask, I know, but the exciting part is that it
shouldn’t require any exotic materials. I think this can even be
considered to be a proof-of-principle. Now, we just need to reduce it to
a proof-of-concept.

Unfortunately, Oskar couldn’t think of a way to do this, but he did say
that this sort of rolling mechanism has been his holy grail for a long
time! I find that fascinating because I don’t think that he came at that
from thinking about 4D puzzles, and this has been our holy grail too. I
don’t like the idea of using magnets to implement twisty puzzles because
I find the idea unsatisfying. Still, I’m open to it if it works and
there is no better solution available. In this case it looks like a
magnet-based solution to the rotational part should work without too
much difficulty. Still, I can imagine instead, a cage-like outer
structure made of cylindrical rods, with clips that grab onto them and
hinge like the hands of Lego mini-figures.

In any case, marrying a rotational mechanism with 2^3 twisting
mechanisms that split apart and join in their centers is still a big
task. Do you have any ideas?