Dive into the math and mystery of the gomboc—a convex, homogeneous 3D shape with a single stable and a single unstable equilibrium that 'writes' itself back upright. We trace its origins from Arnold’s 1995 question to Damokos and Varkoni’s 2006 construction, explore the links to self-righting tortoises, explain why a 2D gomboc can’t exist due to the four-vertex theorem, and follow the polyhedral chase—plus surprising cameo appearances in nature, art, and design.

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