Message #1897

From: schuma <>
Subject: Re: Infinite Regular Polyhedra Puzzles
Date: Sun, 06 Nov 2011 06:51:08 -0000

Hi all,

I am comparing the {3,7} in the Poincare Disk view, and the IRP {3,7}. As Roice said, the patterns of connections between the polygons are different. I try to see how it changes the puzzle in terms of solving.

The difference affects more about the global properties rather than the local properties. Thus I focus on the orbits of a certain type of 1C pieces in the edge-turning {3,7}, because they have to do with the global topology. The orbits are important because a solver needs to deduce the global orientation from the orbits, if there’s no additional aids like piece-finding.

First let’s look at the {3,7} in the Poincare Disk view.
<> (I hope this link works)
The pieces labeled by the red color are in an orbit. There are eight pieces in an orbit. These eight pieces stay in the orbit forever. Each edge turn swaps two pieces connected by a red line segment. There are 168/8 = 21 orbits.

If we move on to the IRP {3,7}, and trace an orbit, we can find 24 pieces in each orbit and there are 168/24 = 7 orbits. It can be verified by focusing on one such piece, and keeping moving it. After 24 moves it comes back to its original place. Since it’s hard to make a clear illustration as the previous one, I saved the 24 twists in this save file:

The sizes and structures of orbits are different for these two versions of {3,7}. I think deducing the global orientation in IRP {3,7} is harder. So at least in the edge turning {3,7}, in one stage of the solution, the Poincare disk version and the IRP version are quite different. For the vertex turning one, it seems like there’s no this kind of difference. I’m not sure about the face turning one.


— In, Roice Nelson <roice3@…> wrote:
> Hi all,
> I’m excited to share some new puzzles based on Melinda’s "infinite regular
> polyhedra" (IRP) work. Please visit her
> page<>on
> these generalized polyhedra for background. The new puzzles will show
> in
> the puzzle menu in the latest build of MagicTile (click to
> download<>
> ).
> Each puzzle has both an IRP and a Poincare Disk view (which share the same
> topology), and you can toggle between the two views at any time via a new
> setting. One lovely thing about the IRP puzzles is that they feel more like
> working with a traditional 3D puzzle, though they are still by no means
> traditional! {3,7}, {4,6}, and {5,5} variants are available to start with,
> and I’ve placed a few screenshots
> here<>
> .
> If you study the new {3,7} enough, you’ll discover it is slightly different
> than the Klein Quartic version. It has 56 colors, but in order to map to
> the IRP, the pattern of connections between faces is new. However, it will
> still be just as difficult a permutation puzzle, so if you’d prefer to work
> with the IRP representation, my offer still stands to send a crystal cube to
> the first solver of all three {3,7} puzzles. It doesn’t matter if you
> choose to use the KQ or the IRP versions.
> I want to say thanks to Melinda for exposing me to these objects, for
> studying them and creating accessible data files in the first place, and for
> helping me get past various hurdles when trying to incorporate them into
> MagicTile.
> More IRP puzzles will surely trickle out over time too :)
> Cheers,
> Roice