Message #2074

From: schuma <>
Subject: Re: MagicTile, Topology of MT IRP {5,5} 8c F 0:0:0.85
Date: Wed, 25 Apr 2012 00:25:18 -0000


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:

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. 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?


— In, Melinda Green <melinda@…> wrote:
> Hello Eduard,
> I tried your macro a13
> <>
> and it is interesting. I’ve not used the macros in TM before so I’m not
> sure how they are supposed to be used but when I click on a red face
> near the vertex adjacent to the white and blue face it indeed exchanges
> two edges but click it elsewhere makes a mess. This was done in the
> hyperbolic view (Show as Skew = false) because in IRP mode I get an
> error saying that macros are not available in IRP mode. It does seem to
> work anyway, so maybe it is fine for this model though I’m not certain
> that the behavior is the same as in the hyperbolic view. Regardless,
> this is certainly an excellent macro and probably all one would need to
> solve this odd, unlikely and lovely little puzzle.
> Strange as it may be, I am fairly certain that it is still an orientable
> surface. It has genus 3 which is common. I don’t know if there is
> anything odd about its topology and would be interested to learn
> anything more about it that people may discover.
> -Melinda
> On 4/24/2012 4:51 AM, Eduard wrote:
> > This puzzle is not trivial. I wanted to have a sequence (macro) which flipps to neighbouring edges. A little by luck I found such a sequence which got the name a13. Doing a walk which brings back an edge in flipped state takes a lot of place and it is not easy to find a 3-cycle in the complement. Applying this macro is not easy in this puzzle. Having the reference point inbetween the two flipped edges I had to try all 5 positions in the face to find the only one which works (in wrong positions there was no effect or completely other effects than a double flip). With a setup it was always possible to get arround this problem.
> > I think the problem is typical for the topology of this puzzle. Can anybody of you do a more profound analysis? Is this puzzle 1-sided? Is it orientable? We have only 8 colors what forces a non trivial "glueing" of edges (oriented edges?).
> > I uplaod my macrofile for this puzzle on the 4D cubing group. Try the macro a13.<br> >