Message #1978

From: Roice Nelson <roice3@gmail.com>
Subject: Re: hemi-puzzles!
Date: Sat, 24 Dec 2011 00:47:39 -0600

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@gmail.com> 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
>