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gptkb:Pappus–Guldinus_theorem
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surface area generated by revolving a plane curve equals the product of the length of the curve and the distance traveled by its centroid
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gptkb:Bieberbach_theorem
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A crystallographic group in n-dimensional Euclidean space has a free abelian subgroup of rank n of finite index.
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gptkb:Gödel's_Incompleteness_Theorems
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Any consistent formal system that is sufficiently expressive cannot be complete.
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gptkb:Gödel's_incompleteness_theorems
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Any consistent formal system that is capable of expressing elementary arithmetic cannot be both complete and consistent.
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