Message #2041

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 4D-interactive puzzles in MagicTile
Date: Mon, 05 Mar 2012 00:24:14 -0600

Hi Ed,

It is 4D, in that the puzzle only fits together in a truly regular way in 4
dimensions. But the puzzle faces are still 2D (the {4,4|4} has 16 2D
faces). This is unlike MC4D, which has 3D faces fitting together in 4D.
Still, you can see how some of the square faces of the {4,4|4} are being
warped due to the 4D -> 3D projection. As in MC4D, shift+left drag to do
some 4D rotations that will affect this warping.

None of the 16 faces are getting hidden. Whereas in MC4D, a 3D face can be
"back facing" or "front facing" relative to a 4D camera, these 2D faces
have no such orientation. How come? Use dimensional analogy to think of a
set of 1D edges embedded in 3D. Although a 2D polygon can be back facing
or front facing in 3D (relative to a 3D camera), a 1D segment can not - a
polygon will have a normal that can only point in two directions, but a
segment has an infinite number of normal directions. Similarly for a 2D
polygon living in 4-space. Hope that made sense, but in short, there is
not a way to hide the 2D puzzle faces based on their location in 4D.

However, I should mention that skew polyhedra divide space into two halves
(the IRPs divide 3D space in two, the 4D skews divide the hypersphere
surface into two 3D partitions). It is therefore possible to consider one
of these halves the "inside" and the other the "outside". If we then
interpreted the puzzle as a solid 3D object which was the inside half, we
could hide faces based on that. I chose not to do this though, in keeping
with the MagicTile abstraction of representing puzzles as 2D surfaces
alone. In MagicTile, the Rubik’s Cube is not a solid cube, only a tiling
of squares on a 2D surface.

Hope this helps clarify some.

All the best,
Roice

On Sun, Mar 4, 2012 at 2:20 PM, Eduard Baumann <baumann@mcnet.ch> wrote:

> **
>
>
> Awesome !!!
>
> I like the new forms.
>
> Question:
> example { 4 4|4 }
> Is this 3D or 4D. I see only 4 cells.
> If 4D: are there hidden cells (as in tesseract the exterior 8th cell)?
>
> Kind regards
> Ed
>
>
>