Rectangular Hexahedrons as Fermat Bases of Quadrics - Čerin, Zvonk (Sarajevo Journal of Mathematics, 2024)
The concepts of a Fermat base of a quadric and of a Fermat locus of a rectangular hexahedron come from an old geometric problem by Pierre de Fermat about a semicircle on a side of a rectangle with ratio of adjacent sides equal to $\sqrt{2}$, which was resolved by synthetic methods first by Leonard Euler in 1750. An arbitrary rectangular hexahedron has a quadric as its Fermat locus. This quadric is either an ellipsoid, a rotational paraboloid or a hyperboloid with two sheets. Conversely, for every quadric from any of these three types one can ask to find all of its rectangular hexahedron Fermat bases which share a line of symmetry with the quadric. 2010 Mathematics Subject Classification. Primary 54H01