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I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture

enter image description here

Class III 8,11 floret planar net (source)

If anyone could point me in the right direction for an algorithm to calculate, and ideally group into petals and florets, the vertices, I would be most appreciative. Or otherwise if someone could help explain the maths behind it, so I can implement my own algorithm, that would be fantastic!

The model would be for a civilization or settelers of catan style tile-based game. I'm not fussed about the pentagons, as they can just be a special feature of the game.

Thanks!

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Start with an icosahedron, and replace each triangle with the interior of this picture, which is to say, pick any two vertices of degree six in that picture (the centers of the florets), complete the equilateral triangle, and map the interior points linearly onto the triangle of the icosahedron. As a final step, push all the new vertices out onto the bounding sphere to make it more spherical-looking. The process is very similar to making a geodesic dome model, except you are using a more interesting tiling than the standard map. If you find it easier, you can generate the floret tiling as the dual of the snub hexagonal tiling, which is to say, connect the centers of the faces of this tiling to get the floret tiling. The construction is closely related to the Pentagonal hexecontahedron, which can be viewed as the most basic version of this, where you choose the smallest possible triangle in the tiling and hence only get pentagonal pieces.

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    $\begingroup$ I did a quick mock up using blender and it seems to work! crystallinegreen.com/wp-content/uploads/2012/12/screenshot.png Going to translate that into an algorithm just to make sure before I mark question as answered $\endgroup$ – James Coote Dec 11 '12 at 13:15
  • $\begingroup$ Well done! By the way, if you map each larger triangle (with 4 sub-triangles) to a larger section of the picture, you can get more variety in the tesselations. The photo you have would use vertices 3 times up plus one in the up-right direction. $\endgroup$ – Mario Carneiro Dec 11 '12 at 22:57

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