Basic question about geodesic spheres

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Eebster the Great
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Basic question about geodesic spheres

Postby Eebster the Great » Wed Jun 08, 2016 4:51 am UTC

So I know very little about geodesic spheres, but the way they seem to be explained online makes no sense. For instance, they seem to imply that there are many vertices at which three congruent regular hexagons meet. That's clearly impossible, since then the dihedral angle would be pi. So when I see pictures of these spheres, what exactly is going on? Are the hexagons not perfectly regular? Are they not perfectly congruent? Are they not perfectly flat? Or do they not meet perfectly at the vertex?

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jaap
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Re: Basic question about geodesic spheres

Postby jaap » Wed Jun 08, 2016 8:07 am UTC

They are not regular and not all congruent, unless you have the simplest case of the soccer ball with 20 regular hexagons between 12 pentagons. They are all flat though (i.e. the 6 vertex points of each hexagon lie in a plane).

The wikipedia page has a decent description of geodesic domes made with triangles, where you start with an icosahedron, subdivide its regular triangle faces into smaller regular triangles, and then distort them by projecting them out onto the circumsphere. The triangles are then all no longer regular (except for the 20 triangles in the centre of each subdivided icosahedron face, if they have central triangles rather than a central vertex).

A dome/sphere made with hexagons and pentagons is the dual polyhedron of one of those made with triangles. Its hexagons will therefore also be irregular, just like the underlying triangles were. The hexagons will be slightly larger the further away they are from the pentagons, because that is where the triangles were projected out furthest and were largest. There may be 20 regular hexagons on the sphere, each lying centrally between three pentagons, but all the other hexagons will be irregular.The sphere will still have icosahedral symmetry, so every irregular hexagon can occur up to 60 times, or 120 if you count mirror images.

By the way, with Euler's formula you can prove that if you make a sphere with just hexagons and pentagons, 3 coming together at each vertex, that you will need exactly 12 pentagons.

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Eebster the Great
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Re: Basic question about geodesic spheres

Postby Eebster the Great » Sat Jun 18, 2016 5:32 pm UTC

Cool thanks, that makes sense. Sorry I didn't respond, the "View Your Posts" button is broken and it took me a while to find this thread again.


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