Message #1457

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: slicing up MagicTile puzzles without triangle vertex figures
Date: Sat, 26 Feb 2011 12:53:22 -0600

Thank you Nan, for the feedback and positive comments.

On the {5,5}, yep that was a problem. For hyperbolic puzzles, I incorrectly
allowed the slicing circles to get scaled beyond the boundary of the
Poincare disk, which is not something that really makes a whole lot of
sense, since the circles then lived outside the bounds of the model of the
space. I fixed this when adjusting the scaling I mentioned in my email to
Brandon, and now the maximum radius for those circles will put it on the
disk boundary (meaning they would have infinite radius). These changes led
to the tool no longer defaulting to having the circles go through the
vertices, but I think the new behavior is better. (Btw, using Chrome, I had
to manually clear my browser cache to get the new version to load :S, though
in IE a refresh of the page was enough.)

Thanks for telling me about the Skewb Diamond. With the sizing newly
available on the spherical puzzles, I see now you can do something analogous
with the {5,3}. The midpoint slicing that doubles up slicing circles
results in a circle set that is an icosidodecahedron (looks like this is the
"Pentultimate"). The result on the dual {3,5} looks a little stranger,
having some hexagonal facets (It’s pretty to have both these turned on at
the same time). I don’t know what shape the latter is, but it’s not in
wiki’s list of Archimedean solids, so maybe those hexagons aren’t regular or
are skew or something.

One last thought your email sparked. I like that gelatinbrain has labels
for all his puzzles that people can point to. I only wish they were somehow
more algorithmic and/or descriptive of the puzzle. This has me trying to
think of some natural naming scheme which could generally cover
possibilities. My initial thought (which I don’t like, but may help someone
come up with a better idea) is the Schlafli in combination with a number for
the puzzle type, say the number you get by counting the sequence of
possibilities as slicing circle size increases. So the descriptive name
would be something like "p.q.n", and the Skewb Diamond in this scheme would
be 3.4.6 (I think). What I don’t like about it is the last number, since it
is difficult to even count these, and though algorithmically producible,
isn’t descriptive in a friendly way.

seeya,
Roice

On Sat, Feb 26, 2011 at 2:03 AM, schuma <mananself@gmail.com> wrote:

> Hi,
>
> After playing it for a while, I found it a great tool to explore puzzles.
> Thank you for sharing it. It’s similar in spirit to Jaap’s applet:
>
> http://www.jaapsch.net/puzzles/sphere.htm
>
> But Roice’s handles more general geometries other than only the sphere.
>
> (1) In {4,4}, when the size of circles is right, and only two circles are
> allowed to rotate, you get a real puzzle: Rashkey
> http://www.jaapsch.net/puzzles/rashkey.htm
> which is a neat and hard puzzle to solve.
>
> (2) Roice said:
> > - On the {3,6}, if you make the circles larger than the parent cell, you
> can
> > slice into 3-per-side by making the slicing circle go 2/3rds the way
> across
> > some adjacent cell edges (which simultaneously puts it 1/3rd the way
> across
> > some incident cell edges). This feels like a nice puzzle to me, with a
> > pretty star pattern in the middle of each cell. You can do the same
> thing
> > on the {3,5} icosahedron, but in that case, the cuts are not evenly
> spaced
> > along an edge and the star patten is not quite as regular.
>
> This way of slicing {3,5} should be precisely Gelatinbrain’s (2.1.4):
> http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f1.gif
>
> (3) In {5,5}, when I increase the size of circles close to the maximum
> value, suddenly all the cuts jump to the outside of the hyperbolic
> plane…… Is there a particular reason or just a bug?
>
> (4) Roice said:
> > - There is a very cool midpoint slicing of the {3,4}. The doubled-up
> > slicing circles form a cuboctahedron, so this is a case where things do
> fit
> > together quite nicely.
>
> This puzzle is the Skewb Diamond (a shape mod of Skewb). It’s interesting
> that it can be viewed as a cuboctahedron.
>
> (5) A {6,3} puzzle with large circles is simulated by Gelatinbrain (7.1.1,
> 7.1.2, 7.1.3, with different repeating patterns).
>
> In general, I’d like to see some of the puzzles of this kind, especially in
> the hyperbolic plane. Since there is no macro function, I don’t think I can
> solve puzzles with too many small pieces. Brandon, I’m not that crazy…
>
> Nan
>
>
>