A geodesic dome - Math Central

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Question from Roger:

In my Science-Fiction series, I have a Dyson's Sphere tiled with regular hexagons. The number of hexagons is over 300,000 and the radius of the Sphere is roughly 80,000,000 miles. The actual size of the Sphere and hexagons have been left flexible until I can come up with a definite number of hexagons that would fit. My problem is the pattern of hexagons which would fit within the sphere without leaving gaps or overlapping.

My best guess has been to use four equilateral triangles composed of 78606 hexagons, (396 per edge) arranged around the sphere with six 'zippers' to connect them and four 'caps' at the points, for a total of 316804 hexagons.
Given the fact that each Hex is the same size, does this seem plausible? Is there some pattern formula I can use to play with these figures? Simple divsion of areas will not work if the number derived will not fit into the pattern to leave a perfectly tiled surface. Thank you.

Roger,

As I interpret your question, what you want is a shell composed of hexagons attached along their edges. In such a case, what you probably want is not a tetrahedron (composed of 4 equilateral triangles), but a geodesic dome -- this figure is based on the icosahedron. It will have 12 vertices symmetrically placed about the sphere at which 5 hexagons meet, and a large number of vertices where 6 hexagons come together. Details of their construction (including how many hexagons are possible) can be found on the internet.

Chris

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