I am trying to cover a sphere (surface of a ball in 3D) with identical unilateral triangles. At first step an octahedron is inscribed into a (unit) sphere. Its faces make the first generation of 8 triangles, obviously unilateral. Next generation triangles are created as follows: centers of edges of selected "old" triangle are found and projected onto the sphere. This replaces an old triangle with 4 new, smaller ones. The procedure is repeated for all "old" triangles, producing in effect the new
generation of triangles. Unexpectedly, new triangles are no longer unilateral, only 2 edges are of equal length (after, say, 5th generation is created). The observed length differences, of order of 6-10%, seem far too large to blame rounding errors as their source. So, what am I doing wrong?
All calculations are done exclusively in Cartesian coordinates.
Well, my procedure shouldn't be directly related to spherical geometry. The sphere with radius=1 is centered at (0,0,0) and the projection of each newly created points onto the sphere is nothing else but normalization of its own radius vector, rAB, to unit length. Here rAB = (rA + rB)/2 (rAB, rA, rB are familiar 3D vectors).
I'm sorry, I do not understand your answer. You see no problem, since offspring triangles are necessarily equilateral (as I think) - or just the opposite?
Are you trying to construct a Platonic solid with more than 20 triangles? This is provably impossible :) (http://mathworld.wolfram.com/PlatonicSolid.html)
So, as usually: "if you look into your program and don't see a bug then it probably isn't there".
I appreciate a very concise explanation given by Karl Fabian, right to the point, as well as the other given by Dirk Lorentz - less formal, but giving the idea WHY and HOW this effect occurs at all. Great thanks to both of you!
On the margin: I hope this short discussion will be enlightening and valuable for others. I know lot of papers describing similar construction, i.e. partitioning a sphere into triangles. None of them ever mentions the triangles are not equilateral, leaving the (false!) impression they are.
I decided to ask this question here, on ResearchGate, only after unsuccessful conversations with my colleagues. Long live ReseachGate!
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