Kassen wrote:
Claude wrote:
[snip]
1. Essentially in 3D you give the Schlaefli symbol {p,q} which means there
are q p-gons around each vertex (so {4,3} is a cube).
Got it.
2. From that you calculate a fundamental (spherical) triangle
Just to be clear; this is a triangle projected on a sphere, with all three
points at a given distance from what will be the origin of the shape?
Yes. It has its vertices at the center of a face, the center of an
edge, and the center of a vertex of the {p,q}.
This paper has a diagram of the "fundamental chamber in a cube" (figure
5 on page 7) which is essentially the same idea:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.127.6227&rep=rep1&type=pdf
Roe Goodman "Alice through Looking Glass after Looking Glass: The
Mathematics of Mirrors and Kaleidoscopes"
3. Then the whole symmetry group follows by reflections
Check, because we know all vertices and/or edges to be the same for the
regular shapes, right?
Yes - given that the {p,q} will form a regular solid, you can just
reflect over the edges of that fundamental triangle and it will cover
the whole sphere without gaps or overlaps.
4. Then you can pick a starting point inside the fundamental triangle
5. Then apply each of the group actions (as matrices)
You lost me at "group actions". Is that like the kind of operation where we
take a cube and make each face move away from the origin, then put a quad to
join it where edges were and triangles where there was a single vertex?
The kind of operation you describe is what I'd call truncation (it's
similar to slicing off corners and edges).
I'm probably misusing terminology, but how I meant "action" above is
that a group element in a symmetry group is an abstract symmetry
operation - like a reflection or a rotation (a rotation is a combination
of 2 reflections), and a group action would be a concrete representation
of a group element in a particular model (like a matrix transformation
or a permutation).
And you end up with a sort of truncated regular polyhedron, depending where
the point you picked is.
Points 1-3 can be done once for all time, the symmetry group never changes
and there are a finite number of "interesting" groups.
Check.
Some older info (need to update with newer info...):
http://claudiusmaximus.goto10.org/cm/2009-10-15_reflex_preview.html
http://claudiusmaximus.goto10.org/g/reflex/
And good pictures too!
Thanks! :)
When I have some spare time I'm going to try playing with this kind of
thing. I wonder to what degree we can say something sensible about
the symmetry in relation to pdata indices. That would be quite nice for
transformations... I quite liked the effect of stacking the different
iterations of Gabor's sphere.
Thanks for your explanation,
Kas.
Claude
--
http://claudiusmaximus.goto10.org
