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Russian Football

Vincent Pantaloni posted a fascinating picture of an unusually designed Russian football, and wondered if it could be a polyhedron, or if it required curved sides. I think this sketch answers yes it's a polyhedron, but the pentagons are only near regular. (So it's not a Johnson solid.) Here are links for Vincent's applet and original tweet. Great conversation developed with Daniel Ruiz Aguilera, Simon Gregg, Martin Holtham, Steve Phelps and more. I did algebra to figure out how to get the edge between octagons to be congruent to the edge of the octagon (after several tries!) and then intersected planes to get the fifth point of the pentagon.