Thread: "MagicTile Considerations"

In the theorems of Astrelin and Baumann we have the formulation =84turning =
the puzzle as a whole arround a face center". Meant is to do this in the po=
incare disc 2D view. Doing the corresponding manipulation is not easy. You =
often have the impression that the effect goes in the opposite direction th=
an you want. If you switch to the 3D skew view you have the normal 3D manip=
ulations plus some other (ctrl) weird manipulations remembering the 4D aspe=
ct (movements with selfintersections).=20
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.=20

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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"
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
chapterof
"The
Eightfold Way ", 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 wrote:

> In the theorems of Astrelin and Baumann we have the formulation =84turnin=
g
> 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 tur=
n
> and the corresponding 3D skew "turn"? In the case of the runcinated 5-cel=
l
> you fix two "opposite" prisme sides.
>

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That is an interesting question.