Message #2037

From: Roice Nelson <roice3@gmail.com>
Subject: 4D-interactive puzzles in MagicTile
Date: Sun, 04 Mar 2012 10:17:07 -0600

Recently, I learned that in addition to the regular skew polyhedra that are
infinite (Melinda’s IRPs), there are regular skew polyhedra which are
finite. They close back around on themselves in 4 dimensions. Here was an
awesome opportunity to make MagicTile genuinely applicable to the 4D_Cubing
group :D

To start, I’ve included 4D skew polyhedra associated with some duoprisms,
the runcinated 5-cell, and the bitruncated 5-cell, though there are additional
possibilities <http://en.wikipedia.org/wiki/Regular_skew_polyhedron> for
the future. These should be pleasant to solve in skew mode, and I’m sure
many interesting observations can be made of the shapes included so far.
For instance, check out how the surface of the {4,6|3} relates to 3
intersecting spheres.

I just uploaded some
pics<http://groups.yahoo.com/group/4D_Cubing/photos/album/597055459/pic/list>of…

• {4,4|4} duoprism, aka hypercube
  
• {4,4|5} duoprism
  
• {4,4|7} duoprism
  
• {4,6|3} runcinated 5-cell
  
• {6,4|3} bitruncated 5-cell

Although the duoprisms are easily seen as having genus 1, can you figure
out the genus of the other two? Not trivial to figure out by sight! I
won’t give away the answer, but will say both have the same genus.

I changed the 4D viewing controls slightly from MC4D, so that the 4D camera
distance could be altered with the mouse (I didn’t want to rely on a
slider). Things will mostly be familiar though, and details are in the
Help menu.

This feels like a great plateau, so now I’m going to take a break from
MagicTile for real :)

Enjoy! <http://www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip>
Roice