What Is the Shape of a Cupola?

Abstract

This article examines the shape of a surface obtained by a hanging flexible, inelastic material with prescribed area and boundary curve. The shape of this surface, after being turned upside down, is a model for cupolas (or domes) under the simple hypothesis of compression. Investigating the rotational examples, we provide and illustrate a novel design for a roof that has the extraordinary property that its shape, although natural, is modeled by a surface of revolution whose axis of rotation is horizontal.

MSC::

• 53A10
• 49J10

Acknowledgments

The author wishes to thank the anonymous referees for a careful reading of the manuscript, providing many useful remarks and corrections. These suggestions greatly helped to improve the final version. In particular, one of the referees pointed out the reference [21] of Nitsche for the Example 1. This work has been partially supported by the grant no. PID2020-117868GB-I00 Ministerio de Ciencia e Innovación.

Notes

1 Actually Hooke considered the problem of the hanging cable writing an anagram in Latin that deciphers to “ut pendet continuum flexile, sic stabit contiguum rigidum inversum” which translates as “as hangs the flexible line, so but inverted will stand the rigid arch” ([14, p. 31]).

Additional information

Notes on contributors

Rafael López

Rafael López works in classical differential geometry, in particular, surfaces with prescribed mean curvature. He is a Professor of Mathematics at the University of Granada where he received his Ph.D. in 1996. Rafael enjoys performing mathematics outreach activities in schools using soap bubbles and, in his spare time, he likes trekking and cycling in Sierra Nevada.