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Le 25/10/2009 18:11, SharkD nous fit lire :
> I want to extend my library of fractals based on polyhedra.
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http://lib.povray.org/searchcollection/index2.php?objectName=FractalObjects&version=1.1&contributorTag=SharkD
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> What other types are there?
Answering after a long time...
The fractal-polyhedra shares two common properties:
1. the removed part is a reduction of the basic object.
2. the remaining parts can be splitted as a set of reduced basic objects.
Which can be transformed into: the basic object can be constructed as a
set of reduced ones; the holes are made by removing some of them, but to
keep the whole shape, it must not be at a vertex. (you would truncate
the shape)
in fact, only rule 2 is needed (as demonstrated by Sierpinski pyramid).
Practically: the cube makes the Menger sponge for a division of 3,
removal of central faces & cube. You might consider a division by more
than 3, or removing/leaving others coordinates (but keep the 8 corners).
(1 cube = 27 sub-cubes, removing 9 of them)
The Sierpinski Tetrahedron is a division of the tetrahedron in 4 corners.
The Sierpinski pyramid is a division of the pyramid in 5 corners.
Other pyramids-like ? Nope, as filling the base polygon is not possible
short of triangle (tetrahedron) or square (pyramid)
Other regular solid ?
Dodecahedron & icosahedron are not self-filling.
As Octahedron is the dual of cube, it might be interesting to
investigate... but in fact, Octahedron is just a double pyramid, so do
not expect a revolution. Try it anyway.
(octahedron = 6 corners)
Subdivision by 3 instead of 2 per face might provide a menger like
solid. (not really, it's not space filling)
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