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He was immediately drawn to this geometry in 2003 when he saw a small, 19th-century mathematical model in a classroom at Tokyo University, designed to illustrate “a surface of revolution with a constant negative curvature,” also known as a pseudosphere.

Hiroshi Sugimoto, Mathematical Model 009, Surface of revolution with constant negative curvature, 2006.

This equation, which gives the pressure in terms of the principal radii of curvature, though here proved only in the case of a surface of revolution, must be true of all surfaces.

The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.

We know that the radius of curvature of a surface of revolution in the plane normal to the meridian plane is the portion of the normal intercepted by the axis of revolution.