Message #2033

From: schuma <mananself@gmail.com>
Subject: Re: Klein and Gnonomic projection models
Date: Wed, 22 Feb 2012 07:05:40 -0000

Thanks, Roice.

I simulated the effect of the Gnomonic Megaminx using Photo Booth as follows:

https://picasaweb.google.com/lh/photo/BFAUmNHDi3snOmtjCSJWq313y-k7SArfvCdqrm7thV4

For the Rubik’s cube, I can’t even do the cross in this view.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:
>
> 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
>