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I want to make an origami of a sphere, so I planned to print some net of a pentakis icosahedron, but I have a image of another sphere with more polygons:


I would like to find the net of such model (I know it will be very fun to cut).

Do you know if it has a name ?

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So, a geodesic dome? – J. M. Dec 24 '10 at 13:03
thanks, that's it... now to lurk the web for a net... :) You should post it as an answer :p – jokoon Dec 24 '10 at 17:51
Another perspective on these polyhedra is that they are the duals of the fullerenes. A fullerene is a 3-valent convex polyhedron with exactly 12 pentagons, and some number of hexagons h (where all values of h except for 1 can occur). Fullerenes can be highly symmetrical or have only the identity group as their symmetry group. – Joseph Malkevitch Dec 24 '10 at 18:51
La Géode (éode) is a famous building in Paris (it is a movie theatre in a science museum). It would make a far more ambitious origami (apparently, it is made of 6433 triangles) but I can't find a net online. As it seems to be made with the classical icosahedron-decomposition method, I guess one can make one by hand. – LIE Jan 20 '11 at 21:38
btw, where can I buy those toys that expands which seems to be some kind of spheric molecule ? – jokoon Dec 26 '11 at 15:19
up vote 3 down vote accepted

This whitepaper on Geodesic Math may be helpful.

Probably less helpful is this Ruby Quiz I hosted on writing a program to calculate Geodesic spheres.

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Here is a net of a buckyball, from

alt text

It should be possible to turn this into the kind of net you're looking for by replacing the pentagons and hexagons with 5 and 6 isosceles triangles (the heights of the triangles determine the "elevation" of the center vertex from the original pentagonal/hexagonal faces and thus affect the sphericality of the result).

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There are many nets of the buckyball (buckminsterfullerene) but only one buckyball. There are many fullerenes with 20 hexagons, of which the buckyball is only one. There are many open questions about the extent to which one can realize dual fullerenes with equilateral and/or isosceles triangles. (The regular icosahedron is the only dual fullerene which can be realized with all equilateral triangles.) – Joseph Malkevitch Dec 24 '10 at 20:50
not triangle, but interesting enough :) – jokoon Dec 26 '11 at 15:18

Extensive information about the Polyhedra can be found at WolframAlpha. For example:

In Mathematica ( likely also in WA ) you can simply get the coordinates of the vertices of many polyhedra. For example with

 PolyhedronData["Dodecahedron", "VertexCoordinates"]
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