The Moebius strip and Viviani's windows
by
Renzo Caddeo
University of Cagliari, Italy
Coauthors: Stefano Montaldo (University of Cagliari), Paola Piu (University of Cagliari)

Poster

The Viviani's windows

The space curves called ``Viviani's windows'' are curves that solved a celebrated geometric puzzle: ``Aenigma Geometricum de miro opificio Testudinis Quadrabilis Hemisphaericae''. This is a (pseudo)-architectural problem proposed by Vincenzo Viviani, a disciple of Galileo, in 1692 (see , page 201, and [Roero1], [Roero2] for a complete and detailed treatment) formulated as follows: build on a hemispherical cupola four equal windows of such a size that the remaining surface can be exactly squared. Among several known solutions, the following was found by Viviani and by other eminent mathematicians of that time: the four windows are the intersections of a hemisphere of radius \alpha with two cylinders of radius \frac\alpha2 that have in common only a ruling containing a diameter of the hemisphere.

When we cut away four (or more) equal half-calottes from a hemisphere by means of four (or more) planes orthogonal to its boundary (an equator), we obtain a special dome vault. In italian this vault is called ``Volta a vela'' (``vela'' meaning ``sail''), because it resembles a sail filled by the wind. For this reason and for the fact that its area is independent of \pi (see [Loria]), Viviani gave to this surface the name ``Vela Quadrabile Fiorentina''. Here is a picture of a cardboard model of the supporting structure realized by the student Gregorio Franzoni (University of Cagliari) and the computer generated surface.

FIGURE

An example of a realization of such a cupola can be admired in the interior of the basilica of San Fedele in Milan.

FIGURE

Thus Viviani's windows give rise to special spherical curves. However, in general, by Viviani's window or curve it is meant a curve obtained intersecting a sphere with a round cylinder tangent to the sphere and to one of its diameters, a sort of spherical eight figure:

FIGURE

In the following rendering of the Osaka Maritime Museum (designed by Paul Andreu), in construction from 1997, and in the drawing below, the spheric shell is embellished by a metal grid generated by two families of curves (see ). We are not sure that these curves are really Viviani's curves, but certainly they could be.

FIGURE

A Möbius strip at the place of the cylinder

It may be interesting to remark that a Viviani's curve can also be obtained by intersecting a sphere with the non orientable analogue of the cylinder, that is, with a Möbius strip.

The picture below (realized with Geomview) shows four views of the intersection between the sphere, the cylinder and the Möbius strip that gives the Viviani's curve.

FIGURE

Bibliography

[Andreu] P. Andreu, Osaka Maritime Museum (Sotto e sopra il mare), L'ARCA, 133, L'ARCAedizioni, Milano, January 1999.

[Loria] G. Loria, Curve sghembe speciali, Ed. Zanichelli, Bologna, 1925.

[Roero1] C.S. Roero, L'intérêt international d'un problème proposé par Viviani, Actes de l'Univ. d'Été Hist. des Math., I.R.E.M. Toulouse, 1986.

[Roero2] C.S. Roero, The Italian challange to Leibnitzian calculus in 1692. Leibnitz and Viviani: a comparison of two epistemologies, V Int. Congress Leibnitz, Hannover, 1988.

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-99.