Thread: "Klein and Gnonomic projection models"

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If you have a moment, check out a couple new MagicTile options for
projection models. You can now switch to the
Klein model
for hyperbolic geometry, and to the analogous
Gnomonic 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},
{3,7},
Megaminx,
{3,5} Super Chop

...and the download
.

seeya,
Roice

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Thanks, Roice.=20

I simulated the effect of the Gnomonic Megaminx using Photo Booth as follow=
s:

https://picasaweb.google.com/lh/photo/BFAUmNHDi3snOmtjCSJWq313y-k7SArfvCdqr=
m7thV4

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

Nan

--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> If you have a moment, check out a couple new MagicTile options for
> projection models. You can now switch to the
> Klein model
> for hyperbolic geometry, and to the analogous
> Gnomonic model
> for spherical geometry. In both of these projections, geodesics appear
> straight. This means all tile edges will be straight line segments, rath=
er
> than the circular arcs of the default projections.
>=20
> In general, slices are not geodesics and will look curved. However, sinc=
e
> 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 straig=
ht
> lines! I found the 2x2x2 a nice one of these to pan around.
>=20
> 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 c=
an
> see more puzzle faces at once.
>=20
> On the hyperbolic side, the Klein model does feel like a pleasant way to
> interact with the {7,3}.
>=20
> Here are some pics...
> {7,3}/610107193/view>,
> {3,7}/1551284723/view>,
> Megaminxpic/1169195625/view>,
> {3,5} Super Chop4853720/pic/173024170/view>
>=20
> ...and the downloadcTile_v2.zip>
> .
>=20
> seeya,
> Roice
>