Message #1979

From: Melinda Green <>
Subject: Re: [MC4D] Re: hemi-puzzles!
Date: Sat, 24 Dec 2011 02:15:05 -0800

What a beautiful graph!
I have to fool around with this strange little puzzle.


On 12/23/2011 10:47 PM, Roice Nelson 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
> <>,
> and some observations about it are:
> * The cyan and purple cells are henagons
> <> (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 <>, 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
> <> 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 <
> <>> 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
> <> 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
> <>.
> Happy Holidays,
> Roice