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I have found a photo of a interesting geometric lampshade, but I don't recognise what it's underlying geometry is. It looks as if it could be constructed from regular pentagons, hexagons and triangles.

enter image description here

Ross Millikan
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John Spence
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  • have you checked https://en.wikipedia.org/wiki/List_of_polygons,_polyhedra_and_polytopes ? – JMP May 06 '17 at 13:15
  • Theres also a rhombus. No wait the rhombi are one third of the hexagonal shape. – mathreadler May 06 '17 at 13:17
  • Kind of reminds me of the classical football shape with alternating pentagons and hexagons, before they started with the round ball thing. I think the hexagon is shrunk just enough to fit exactly one more of those third rhombi in between. – mathreadler May 06 '17 at 13:25
  • Geodesic sphere / some variant of a rhombicosidodecahedron – Marcus Andrews May 06 '17 at 13:49
  • It looks like it could be an Archimedean solid, but it's not. Maybe it can only be constructed if some of the polygons are not regular. – John Spence May 06 '17 at 14:28
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    It looks like it's derived similarly to a "snub dodecahedron". In the snub, two neighboring pentagons are separated by a pair of equilateral triangles. Joining a vertex of the pentagon to an edge-midpt on the other subdivides these triangles, giving a quad formed of two right triangles. In your figure, it looks like the corresponding quad is formed of two equilateral triangles. It appears that appropriate scaling and rotating of the pentagons would have the desired effect, while also doing the right thing along the axes of three-fold symmetry. – Blue May 06 '17 at 16:57
  • @Blue I did notice that scaling and rotating the pentagons on a snub dodecahedron would produce this shape, but it would be nice to know a precise geometric way of doing this. I would like to draw it in a 3D CAD program. – John Spence May 07 '17 at 08:48

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