Message #2006

From: Roice Nelson <roice3@gmail.com>
Subject: Super Puzzle Sunday!
Date: Sun, 05 Feb 2012 18:26:09 -0600

Hi all,

When I wrote last, MagicTile had 63 puzzles configured. *Now it has over
450!* Here are some highlights:

• Tetrahedron puzzles: Projecting the tetrahedron out to the sphere
  results in large curvature of faces, so there are some genuinely new
  puzzles here (which have no corresponding GelatinBrain applet). Check out
  the ET puzzles in particular.
  
• Dodecahedron and Icosahedron: Added FT and VT hemispherically sliced
  puzzles (BigChop-like). These puzzles have "1:1:1" labels. I also added
  crazy versions mixing all three BigChops into one FEV SuperChop
  puzzle. Very pretty, but must be difficult!
  
• Hemispherically sliced puzzles for all the other platonic solids too
  (though the labels end up being different for the different shapes).
  
• {3,6} and {4,4} Klein Bottles<http://en.wikipedia.org/wiki/Klein_bottle>.
  And the great thing about Klein Bottles for this group is that they
  require 4 dimensions to represent the compact surface without self
  intersections :D
  
• Analogues of Pyraminx Crystal and other Dodecahedral slicings, but on
  the {6,3} tilings.
  
• Many new FT {7,3} puzzle cut depths.
  
• A class of mixed ET+VT puzzles that apply to various tilings. The
  edge turning circle size is set to the incircle of the tiling, and the
  vertex turning size is set to the circumcircle, making all slices intersect
  at tile centers. The labels of this class of puzzles are "E0:1:0 V0:0:1".
  On every single spherical puzzle, the initial projections have slices that
  appear as lines, and I found the Cube and Octahedron projections especially
  striking. These look nice on {p,q} tilings when q is larger than p, since
  the edge circles become comparatively smaller - besides the Octahedron, see
  the {3,5} and {3,7} for instance.
  
• A class of mixed FT+VT puzzles, "F0:1:0 V1:0:0". Slicing circles all
  intersect at edge midpoints, and it seems like these puzzles should
  generally be easy.
  
• Check out the {5,5} FEV puzzles. Since it is a self-dual tiling, the
  slicing around faces and vertices is congruent. Very cool looking, and I
  bet these are fun to solve.
  
• Three strange {8,4} colorings. It would be a fun mini-project to do
  analysis of the topological structure of these.
  
• *Just lots and lots and lots of new slicing in general, so have a look
  around.*

I tried to select puzzles that have reasonable looking slicing, though
perhaps pushed the boundaries of that in a some cases. Maybe some are too
difficult or too easy. If you find puzzles that you feel shouldn’t belong,
I’d appreciate feedback on that. I welcome suggestions for further slicing
too.

In case you are wondering, here is the meaning of the puzzle labels. I
wanted to have something auto-generated that gave insight into the depth of
the cuts. Hopefully it is terse enough. A slice is displayed with the
turning type, "F", "E", or "V", plus a cut diameter specified by three
numbers in the form "P:Q:R".

P is the number of tiling edge lengths<br>
Q is the number of incircles<br>
R is the number of circumcircles

These numbers are based on the side lengths of the fundamental p.q.2
triangle of the tiling. P is the side opposite the π/p angle, Q the side
opposite the π/q angle, and R the side opposite the π/2 angle (R is the
hypotenuse of fundamental triangle). The final slicing circle radius is
set to some linear combination of these three lengths. This approach
allows you to make much simpler looking expressions for cut depths. I
tried to avoid decimals when possible.

There is a new puzzle tree so you can more easily select puzzles, but I
still left in the onerous nested menus. This latest version also fixes
some bugs I’ve run across. I won’t detail those here, but I encourage you
to download the latest rather than use previous versions.

Some things I haven’t done, but would like to do: I should probably add a
description of the labels directly in the program, maybe even a visual one.
I’d also like to add preview pictures for all the puzzles, to make it
easier to see what they look like without actually having to build them.

One last thing: I have removed the "Preview" label :)

www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip

Cheers,
Roice