[Fluxus] commits: icosphere

[Claude Heiland-Allen]

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