Geometric Model, L. Brill No. 62. Ser. 5 No. 16d, Cyclide (Parabolic Spindle Cyclide)

Description:

As an undergraduate at the École polytechnique in Paris, French naval engineer and mathematician Charles Dupin (1784-1873) described a set of surfaces based on inversions of the torus – surfaces that came to be called Dupin cyclides. All the lines of curvature on such surfaces are circles or straight lines. Dupin published a description of cyclides in 1822. Between 1879 and 1885, Alexander Brill and his students and colleagues would publish several different models of Dupin cyclides. This is one of them, sold by the German firm of Ludwig Brill from 1879. It was exhibited at the Columbia Exposition held in Chicago in 1893 and then acquired by Wesleyan University.

This model is of a parabolic spindle cyclide. It is formed by the inversion of a spindle torus (a torus formed when a circle rotates about an axis that passes through a chord of the circle). It is one of four models of cyclides designed by Peter Vogel (1856-1915), a student in Munich who would go one to teach mathematics at the war academy in that city.

The plaster model has a cusp-shaped piece that arches over the base. Various circular arcs, representing lines of curvature, are on the surface. A paper number tag reads: 62. Another paper tag reads: Dupin'sche Cyclide (/) Verl. v. L. Brill. 5 Ser. Nr. XVId.

References:

L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 11, 66.

G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 28-30.

Ulf Hashagen, “ Die Mathematik und ihre Assistenten an der TH Munchen (1868-1918), Mathematics and Theoretical Physics, Symposia Gaussiana, ed. M. Behara, R. Fritsch, and R.G. Lintz, 1995, pp. 136-140.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea, 1952, pp. 217-219.