Message #1394

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Interesting object
Date: Tue, 08 Feb 2011 17:14:18 -0800

Thank you Roice. We aim to please! :-)

I had a strong feeling that Andrey was working on your problem at the
same time that I was. I must admit that I didn’t understand his answer
at all, so I’m just glad that I had something complementary to add to
the discussion.

I had never heard of the term "ultraparallel" before. Maybe it should be
obvious in hindsight but running into it this way was very surprising
and really quite magical for me. One thing that they seem to suggest is
that it should be possible to create surfaces of arbitrarily high genus
this way. They almost certainly won’t have regular polygonal
counterparts but it seems like there’s no topological reason why you
can’t have more than three sets of ultraparallel lines repeating across
a hyperbolic surface of sufficient curvature. I’d love to see what a
genus-20 puzzle looks like in Magic Tile.

I don’t understand what you mean about the symmetry of this {8,3} not
allowing 1/8th face twists. It looks to me like any uniform tiling with
a triangular vertex figure should make for a natural magic tile so I
must be missing something. Is there a simple diagram or something that
shows this?

Even though I don’t understand the problem, I wonder if you might be
able to create one very fun puzzle here by allowing only 180 degree
twists? Instead of coloring each face a solid color like usual, imagine
painting half of each face red and the other half green to match my
screen shot. The puzzle is solved when each sphere is a solid color. If
this works then it might make for some beautiful patterns. Solved magic
tile puzzles sometimes look like like unorganized patches of solid
colors, but this sort of two-colored puzzle would really highlight those
ultraparallel lines and make for some really beautiful patterns.

-Melinda

On 2/8/2011 9:05 AM, Roice Nelson wrote:
>
>
> Wow, I pose a question before falling asleep and wake up to two great
> answers. Thanks! I’m grateful for both. Andrey’s modular arithmetic
> approach seems like it can be very helpful with the code calculations
> that will need to be done, and Melinda’s explanation provides a great
> deal of insight into the connections between faces.
> With Melinda’s description, I think I know why my existing coloring
> rule could not generate this tiling. It is because the pattern arises
> from these hyperbolic translations of 6 ultraparallel
> <http://en.wikipedia.org/wiki/Ultraparallel#Non-intersecting_lines>
> lines (with 4 faces on each line), whereas my tiling rule is based on
> patterns that arise in a rotationally symmetric way. I need to study
> further, but if I’m right, one can not do a 1/8th, face-centered
> rotation of a 24 colored {8,3} - it does not appear to be a symmetry
> of the object.
> A coincident aside related to these hyperbolic translations is that I
> found out yesterday I missed a class of checkerboards on KQ which are
> due to symmetries of this kind. You can have six
> translational 4-cycles of KQ, much like what we are discussing here
> (KQ is a more symmetric object though, having the rotational
> symmetries as well). There is no analogous checkerboard of Megaminx,
> since there are no parallel lines in spherical geometry. I’ll likely
> make the actual checkerboard this weekend, and will email if I do. It
> will indeed be the last possible type, because I found this out
> reading part of this paper
> <http://library.msri.org/books/Book35/files/karcher.pdf> (see p32),
> which enumerates all the possible KQ symmetries.
> And on the mystery puzzle, that’s exactly it Melinda. The image
> turned up unexpectedly when I was playing with some code this weekend,
> and I was so pleased that it looked very similar to the stereo
> graphically projected Megaminx. You can do the same inversion on KQ
> to get a similar looking picture of that as well.
> seeya,
> Roice
> P.S. I hope this email came across ok, as the formatting of my last
> one seems to be all messed up on the yahoo group site. Gmail and
> Yahoo groups appear to be a little incongruous lately.
>
>
>