Message #1396

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Interesting object
Date: Wed, 09 Feb 2011 10:27:54 -0600

Hi Melinda,

I haven’t fully grokked Andrey’s approach yet either, but my hope is that if
I can understand it, it could help with some of the features I hope can be
supported. For instance, if I could quickly calculate the coloring pattern
from a single starting tile, supporting hyperbolic rotations/translations
might become much easier.

I planned to email out some corrections after further thought yesterday, and
your email beat me to it :) My hypothesis looks mistaken, and I think
you’re right that you can rotate a 24-colored {8,3} surface an eighth
turn about a face. I was talking about rotating the whole object (not just
a twist of a face), and the thought was an attempt to understand the reason
why my face coloring algorithm couldn’t generate this puzzle (but that
turned out to be something different, the details of which I won’t bore you
all with). The simple way to show a lack of rotational symmetry would be to
find copies (of the face being rotated about) that did not move into each
other during the rotation.

Also, I should clarify something else I said, that KQ is only "more
symmetric" in the sense that it is maximally symmetric for its genus, while
the {8,3} is not (see wikipedia’s Hurwitz
surface<http://en.wikipedia.org/wiki/Hurwitz_surface>).
KQ has less symmetries than the 24-colored {8,3}.

Onward to higher genus surfaces, I don’t know if there is a limit or not,
but I did run across an interesting surface a while back which is analogous
to KQ in many ways and is genus 70. Here is a paper on that surface, which
I hope to study more in time, as I can only wonder at the moment how many
ultraparallel lines might compose the object. The relevant dual tilings are
{11,5} and {5,11}, and I do know an {11,5} MagicTile puzzle would have 60
faces (though I don’t know how playable it would be!).

http://www.neverendingbooks.org/DATA/biplanesingerman.pdf

As for any obviousness of ultraparallel lines, I have to say that with all
of this, none of it ever feels obvious to me in hindsight. I could stare at
ultraparallel lines for a while, feel like they make sense at times, but
still never feeling like I fully grasp it. And for that matter, I can still
do the same with a picture of a "simple" dodecahedron. I like this
paragraph from the 120-cell
article<http://www.ams.org/notices/200101/fea-stillwell.pdf>
:

Telling the story in contemporary language
> has the danger that certain connections become
> “obvious”, and it is hard to understand how our
> mathematical ancestors could have overlooked
> them. However, there is no turning back; we can-
> not stop seeing the connections we see now, so
> the best thing to do is describe them as clearly as
> possible and recognise that our ancestors lacked
> our advantages.

I like your new puzzle idea. Something a little like it is possible right
now by setting multiple faces to the same color. So via configuration you
can make a 3-colored KQ puzzle by coloring sets of 8 the same. But it is
still different than you’re describing (since faces are restricted to having
a single color, and twisting is unrestricted). Your idea reminds me of some
of the gelatinbrain puzzle extensions.

Anyway, thanks again for working out the coloring (and to Andrey for that as
well, and to David for starting this thread)!

seeya,
Roice

<http://www.neverendingbooks.org/DATA/biplanesingerman.pdf>

On Tue, Feb 8, 2011 at 7:14 PM, Melinda Green <melinda@superliminal.com>wrote:

>
>
> 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
>