Message #2076

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: MagicTile, Topology of MT IRP {5,5} 8c F 0:0:0.85
Date: Tue, 24 Apr 2012 22:24:31 -0500

I’m glad to see this one getting attention - a nice excuse to take a closer
look :) Here are a couple thoughts to add to the mix:

• I think Melinda is right that this surface is orientable. The way I
  check is to do twists in the hyperbolic view and watch the copies. If you
  see any copies spin in an opposite sense, the surface is non-orientable.
  
• MagicTile internally calculates some aspects of the topology, and I
  can peek at the numbers in the debugger to get some free topology analysis
  (I should expose this). It told me the {5,5} 8C tiling has 8 faces, 20
  edges, and 8 vertices. It’s Euler Characteristic is therefore equal to 8 -
  20 + 8 = -4, corresponding to a genus 3 surface, also as Melinda pointed
  out.
  
• The faces are not pure pentagons on the abstract "rolled up" object.
  After identifications of the various elements, the faces have 5 edges, but
  only 4 vertices. (It’s easier to see which vertices are identified by
  looking at the vertex turning puzzle.)
  
• We could make a graph of the rolled up object to study connections
  between elements, but since the genus is 3, the graph won’t be planar like
  before <http://games.groups.yahoo.com/group/4D_Cubing/message/1980>. So
  we can’t draw it on a flat piece of paper without intersections.
  
• A nice shape to draw the graph on would be the "tetrus" (see image at
  the top of this
  paper<http://www.cs.berkeley.edu/~sequin/PAPERS/Bridges06_PatternsOnTetrus.pdf>).
  Each face would wrap around a tube of the tetrus so that two of the 5
  pentagon vertices would be doubled up into one.
  
• I haven’t carried this out yet, so it may be off track, but I suspect
  that the 4 pairs Nan describes could somehow be associated with the 4 hubs
  of the tetrus. My guess is there could be a nice graph drawing where each
  pair of pentagons lives near one of these hubs, one towards the inside of
  the tetrus and one towards the outside. The 8 vertices also beg to be
  placed near the hubs to make a nice graph. I’ll have to play with the
  thought more. It would be awesome to have a physical "whiteboard" tetrus,
  to draw and analyze these common genus-3 graphs :D

I’ve talked a little offline with Nan about macros on these puzzles with
asymmetric coloring. Setting up a macro in one location and applying it
elsewhere is definitely strange. We even discussed disallowing this, but I
decided to leave in the capability. The user has some extra responsibility
as a result, and must be aware of whether it makes sense to apply the macro
in a given position/orientation, based on where the macro was defined. As
Ed and Melinda found, it doesn’t always make much sense!

What MagicTile does to transform a macro is to take its pattern of clicks
on the underlying tiling (think of the uncolored, infinite tiling),
transform that to the new position/orientation on the tiling, then find
which new faces correspond to the transformed clicks and click them.
Macros behave much better on puzzles with symmetric colorings, where they
act the same no matter where applied.

I just had a thought…it would be extremely interesting if we could find a
macro on an asymmetrically colored puzzle that did two *different but useful
* things depending on where it was applied. Can you imagine a macro that
somehow could do both a 2-cycle and a 3-cycle. Now that would be cool!

seeya,
Roice

On Tue, Apr 24, 2012 at 7:25 PM, schuma <mananself@gmail.com> wrote:

> Hi,
>
> I just tried this puzzle. I agree that it’s not trivial at all. I’ve
> probably tried this puzzle a long time ago but I found it too asymmetric,
> so I gave up. But today I solved it.
>
> Around each vertex, "two" faces out of the "five" are identified, which
> breaks the symmetry. So the five "angles" around a vertex are not
> equivalent. For the same reason, the reference point can only be in a
> certain type of angle. The fact that different reference points lead to
> different results is not a bug. It’s just a consequence of asymmetry: if
> you hold the puzzle differently and apply the same sequence, the result is
> different.
>
> As I understand it, to flip an edge, the main idea is nothing but to let
> it go around a vertex. It’s a bit tricky not to affect other pieces. When I
> solved it I tried to apply [1,1] commutators intuitively and watched the
> orientation carefully at the same time. So I didn’t use macro for flipping
> edges. But I recorded a sequence which flips two edges in place, just to
> explain what I would do. It can be found here:
>
> http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/IRP55_flip.xml
>
> It’s just a save file, not a macro. You can use ctrl+z to go back to the
> starting point and use ctrl+y to play it. There are 14 moves. The first 8
> moves are the first 3-cycle: 2 setup moves + [1,1] commutator + undo setup
> moves. The next 6 moves are the second 3-cycle: 1 setup move + [1,1]
> commutator + undo setup move. The idea is just to take an edge and let it
> go around a vertex.
>
> It’s funny that the eight colors form four pairs: cyan+blue, green+orange,
> white+yellow, red+purple. The two colors in each pair have a special
> geometric relation, so that they intersect by two pieces. So their
> commutator is not a 3-cycle. To construct a 3-cycle using commutators, one
> should avoid using such pairs. Two colors from different pairs (for example
> red and white) intersect by one piece so their commutator is a 3-cycle.
>
> It seems like the paired pentagons have more stories in terms of
> geometry/topology. In the IRP view, they are co-planar. I’m not good at
> topology but I’d love it if any one explain it.
>
> ****
>
> By the way, a completely independent thing: here’s the wallpaper I’ve been
> using for a while.
>
> http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/Image 000.png
>
> It was generated by Magic Tile v2 by choosing a particular puzzle with
> proper parameters. No photoshop involved. Does anyone know what puzzle is
> it?
>
> Nan
>
>
>