# Message #2032

From: Roice Nelson <roice3@gmail.com>

Subject: Klein and Gnonomic projection models

Date: Tue, 21 Feb 2012 20:57:30 -0600

If you have a moment, check out a couple new MagicTile options for

projection models. You can now switch to the

Klein<http://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model> model

for hyperbolic geometry, and to the analogous

Gnomonic<http://en.wikipedia.org/wiki/Gnomonic_projection> model

for spherical geometry. In both of these projections, geodesics appear

straight. This means all tile edges will be straight line segments, rather

than the circular arcs of the default projections.

In general, slices are not geodesics and will look curved. However, since

hemispherical slices on the sphere are geodesics, those will be

straight. Big Chop and Super Chop puzzles are therefore quite

interesting in the Gnomonic model, as literally everything becomes straight

lines! I found the 2x2x2 a nice one of these to pan around.

I went through a solve of the Rubik’s Cube using the Gnomonic projection,

and it is strange - a bit difficult honestly, mainly since only one

hemisphere makes it to the plane. The Megaminx may be easier since you can

see more puzzle faces at once.

On the hyperbolic side, the Klein model does feel like a pleasant way to

interact with the {7,3}.

Here are some pics…

{7,3}<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/610107193/view>,

{3,7}<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/1551284723/view>,

Megaminx<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/1169195625/view>,

{3,5} Super Chop<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/173024170/view>

…and the download<http://www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip>

.

seeya,

Roice