[Fluxus] commits: icosphere

[Kassen]

Claude  wrote:

(What follows is a bit off-topic, sorry..)
>
>
Looks completely topical to me, really.


> Good question, and the answer (perhaps surprisingly) is yes.
>
>
I'm not that surprised, I imagine that if we *couldn't* then we wouldn't be
sure we knew about all of the platonic ones, for example.... but maybe I'm
taking shortcuts now. Of course nothing guarantees that it would be easy :-)


> If you're interested in this kind of stuff, the book "Regular Polytopes" by
> HSM Coxeter is the one to get - it's got a *lot* of information in it, and
> not too pricy.
>
>
Sounds good!


> I have some Haskell code that follows Coxeter's equations for 3D and 4D,
> but it's quite a long process:
> 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?


> 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?


> 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?


>
> 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!

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.
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