# Message #1981

From: schuma <mananself@gmail.com>

Subject: Re: hemi-puzzles!

Date: Sat, 24 Dec 2011 17:35:17 -0000

Roice,

Thank you for such a special Xmas gift. After seeing your description of the topology, I solved this puzzle. The solution is quite traditional, solving the pieces type by type. For each type I solved the "special" polygons first to make sure nothing weird happens. For the pieces related to special polygons, I had to carefully select the moves so that they don’t intersect in an unexpected way. Aside from this there’s no more difficulty.

But I have to say your analysis is very very helpful and illuminating. Without the planar graph I won’t understand how it works.

Maybe I’ll apply your method to analyze and try the other asymmetric puzzles in the future. Before today I don’t really know how to deal with them.

The other hemi-polyhedral puzzles are pretty straightforward though.

Merry Christmas and happy new year to everyone!

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> Here’s a little more on the {3,5} 8-Color. These puzzles with asymmetrical

> colorings are strange, but they arise naturally from the math that

> identifies cells with each other, so I wanted to understand things a little

> better.

>

> To do that, I made a graph of the 4 vertices, 10 edges, and 8 faces all

> "rolled up". By that I mean each of these features is only shown once,

> rather than shown multiple times (with hand waving that "this face is

> identified with that one", as the MagicTile presentation requires). An

> image of my graph with default MagicTile colors is

> here<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/1602345595/view>,

> and some observations about it are:

>

> - The cyan and purple cells are

> henagons<http://en.wikipedia.org/wiki/Henagon> (polygons

> with one vertex and one edge). The face twisting of these two scrambles

> nothing.

> - The blue and orange cells are degenerate triangles. They have three

> sides, but only two vertices. I initially thought they were

> digons<http://en.wikipedia.org/wiki/Digon>,

> but they are more like a digon with a henagon subtracted out. I don’t know

> if there is a special name for this polygon.

> - The red, yellow, white, and green cells are proper triangles. (By the

> way, if you want to trace out the green and white triangles in the graph,

> note that the edge that goes off the top of the screen is the same edge

> that comes up from the bottom.)

> - Since there are different cell types, this puzzle represents a *

> non-regular* spherical polyhedron. It was cool to realize this could be

> done with MagicTile’s abstraction :)

> - Two of the vertices have 4 colors surrounding them, and two have 5

> colors surrounding them. Even so, the repeated color on a 4C vertex piece

> comes from different parts of a triangle, so the behavior is still 5C-like.

> - Since I was able to make this planar

> graph<http://en.wikipedia.org/wiki/Planar_graph>representation of the

> object, it was easier to see how it has the topology

> of the sphere.

>

> I haven’t tried to make sequences to solve it yet, but will. If anyone

> solves this puzzle, I’d love to hear about your experience with it!

>

> Roice

>

>

> On Fri, Dec 23, 2011 at 12:59 PM, Roice Nelson <roice3@…> wrote:

>

> > Hi all,

> >

> > I added some hemi-puzzles, all ones we haven’t seen before. The

> > hemi-dodecahedron and hemi-cube are not new, but I made them vertex turning

> > this time. There are also hemi-octahedron and hemi-icosahedron puzzles

> > now. All of these have the topology of the projective plane.

> >

> > I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not

> > symmetrical (like the {8,3} 10-Color and some of the other hyperbolic

> > puzzles). It turns out to have 8 faces, 10 edges, and 4 vertices, so the Euler

> > Characteristic <http://en.wikipedia.org/wiki/Euler_characteristic> shows

> > it has the topology of a sphere. I’ll try to write a little more about

> > this 8C puzzle soon.

> >

> > You can download the latest by clicking here<http://www.gravitation3d.com/magictile/downloads/MagicTile_v2_Preview.zip>

> > .

> >

> > Happy Holidays,

> > Roice

> >

>