A Four-Point Theorem: Yet Another Variation on an Old Theme A Four-Point Theorem: Yet Another Variation

The subject of this article belongs to a “neighborhood” of the four-vertex theorem, which in its simplest form, states that the curvature of a plane oval (a smooth closed curve with positive curvature) has at least four critical points. Since its publication by Syamadas Mukhopadhyaya in 1909, this r...

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Bibliographic Details
Published inThe Mathematical intelligencer Vol. 47; no. 2; pp. 171 - 175
Main Author Tabachnikov, Serge
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2025
Subjects
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Gems and Curiosities
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Simulation
Theoretical
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Summary:The subject of this article belongs to a “neighborhood” of the four-vertex theorem, which in its simplest form, states that the curvature of a plane oval (a smooth closed curve with positive curvature) has at least four critical points. Since its publication by Syamadas Mukhopadhyaya in 1909, this result and its ramifications have generated a vast literature. We give but one reference: [ 5 , Lecture 10]. In what follows, we freely use basic facts of elementary differential geometry of the sphere and the hyperbolic plane, and we omit references to numerous textbooks on the subject.
ISSN:0343-6993
1866-7414
DOI:10.1007/s00283-024-10399-2