Message #276

From: Roice Nelson <roice@gravitation3d.com>
Subject: Re: [MC4D] other polychora and polytope permutation puzzles
Date: Sat, 10 Jun 2006 02:09:18 -0500

Thanks for the cool comments Melinda and Don. I have just a couple inlines
below.

On 6/3/06, Melinda Green <melinda@superliminal.com> wrote:

> > The 4-simplex ( http://en.wikipedia.org/wiki/4-simplex) would be the
> > 4D analog of Pyraminx, and would be easier and less screen-crazy than
> > MC4D. That puzzle would also be interesting because as I imagine one
> > projection of it, there would be no center face. There would be 4
> > icosahedrons all pointing towards an empty center. The 5th
> > icosahedron would be the hidden face closest to the viewer (think of
> > the 3D to 2D case of projecting the icosahedron of a Pyraminx to 2D).
> Icosahedral pieces? Is that right? I’d definitely love to try my hand at
> solving this one since I can usually intuit my way to solving the 3D
> version fairly quickly.

Oops… After reading Don’s email today, I saw I had used the wrong term
here. I should have realized when Melinda questioned it. Replace all the
uses of "icosahedron" with "tetrahedron" to get what I was meaning by the 4D
analog of Pyraminx.

On 6/9/06, Don Hatch <hatch@plunk.org> wrote:
> So if you want to work with this R,T representation
> instead of matrices, you’d need to figure out three formulas:
> 1. How to apply the rotation R,T to arbitrary coords (x,y,z,w)
> 2. Given rotations R0,T0 and R1,T1, find their composition R,T
> 3. Given a rotation R,T, find its inverse R’,T’
> (i.e. R’,T’ such that R,T composed with R’,T’ is the identity).
> It would be interesting to see these formulas;
> I wonder if they collapse into something nice.

I don’t know if you have ever used the C++ boost library, but I found some
information there that might help answer these questions. They have a
quaternion library, whose main page is here:

http://www.boost.org/libs/math/quaternion/index.html

An example file using this library to convert 2 quaternions to a 4D rotation
matrix (which helps answer 1) is here:

http://www.boost.org/libs/math/quaternion/HSO4.hpp

From a wikipedia article (http://en.wikipedia.org/wiki/Quaternion_rotation),
I gathered the answer to 2 is that the composition of rotations is still
done by quaternion multiplication like you would do for 3D rotations. In
this case, I think you would just multiply all 4 quaternions in succession.
I bet the finding the inverse is straightforward for this reason too.

But I’d like to take the path of least resistance first in any playing I
might do with this stuff, and it sounds like that is probably using standard
rotational matrices.

I liked the prism suggestion btw. I hadn’t thought of that…

Roice