Tetrahedal Geometry from Areas

Abstract

We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter, or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles, and (an) edge length, the remaining parts being derivable by symmetry or by identities in the classic 1902 compendium of results on tetrahedral geometry by G. Richardson. We also provide an alternative proof using (bi)vectors of the result of Yetter that four times the sum of the squared areas of the medial parallelograms is equal to the sum of the squared areas of the faces. Despite the classical nature of the problem, it would not have been natural to consider had it not been suggested by recent work in quantum physics.

• MSC: Primary 51M25
• Secondary 51M04
• 51M20

Acknowledgments

The authors wish to thank the editors and referees of this Monthly for many helpful suggestions that improved the exposition of this article.

Additional information

Notes on contributors

Louis Crane

LOUIS CRANE earned his Ph.D. from the University of Chicago in 1985. He is best known for introducing categorification and the categorical ladder with Igor Frenkel, applying categorical state sums to general relativity with John Barrett and applying categorical methods to topological field theory with the second author.

David N. Yetter

DAVID N. YETTER earned his Ph.D. from the University of Pennsylvania in 1984. He is the author of the monograph Functorial Knot Theory: Categories of Tangles, Coherence, Categoral Deformations and Topological Invariants and more than 40 papers in areas ranging from topos theory to knot theory and from Euclidean geometry to operator algebra, though he is still best known as “the Y in HOMFLY-PT.”