# Message #2599

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] MagicTile Considerations

Date: Thu, 10 Jan 2013 18:07:32 -0600

That is an interesting question.

The effects of rotating the 4D skew polyhedron will show up in the 2D view,

and vice versa (since the disk is the "universal

cover<http://mathworld.wolfram.com/UniversalCover.html>"

of the polyhedron). So given a rotation, the same two faces will remain

fixed in both the 4D view and the 2D view. Consider a rotation about the

white face, in which case the other fixed face is ‘Color 30’. For a 180

degree rotation of the disk, notice how the ‘Color 30’ face ends up in the

same place after the rotation as well, and also rotated 180 degrees!

The permutation restrictions you guys found seem to make sense when you

look at the 4D objects. On the runcinated 5-cell, check out how a 90

degree rotation is not a symmetry of the object, whereas a 180 degree

rotation is a symmetry.

Related to this topic, I recommend the Thurston

chapter<http://library.msri.org/books/Book35/files/thurston.pdf>of

"The

Eightfold Way <http://library.msri.org/books/Book35/contents.html>", which

discusses the {7,3} and the unique effects that happen there when you

rotate that tiling in the disk model.

Roice

On Thu, Jan 3, 2013 at 4:28 AM, Eduard <ed.baumann@bluewin.ch> wrote:

> In the theorems of Astrelin and Baumann we have the formulation „turning

> the puzzle as a whole arround a face center". Meant is to do this in the

> poincare disc 2D view. Doing the corresponding manipulation is not easy.

> You often have the impression that the effect goes in the opposite

> direction than you want. If you switch to the 3D skew view you have the

> normal 3D manipulations plus some other (ctrl) weird manipulations

> remembering the 4D aspect (movements with selfintersections).

> Question: what is the exact translation between the poincare disc 2D turn

> and the corresponding 3D skew "turn"? In the case of the runcinated 5-cell

> you fix two "opposite" prisme sides.

>