# Message #1979

From: Melinda Green <melinda@superliminal.com>

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.

-Melinda

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

> <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

> <mailto: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

>

>

>

>

>