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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 horn cyclide. It is formed by the inversion of a horn torus (a torus with no hole – the surface formed when a circle rotates about an axis that is tangent to it). 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 model consists of two cusp-shaped plaster pieces held together by an iron crossbar. Several circles (lines of curvature) are indicated on the surface. A paper number tag reads: 60. Another paper tag reads: Dupin'sche Cyclide (/) Verl. v. L. Brill. 5. Ser. Nr. XVIb.
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.
Website of the Mathematics Institute at Oxford University, accessed September, 2017..
Currently not on view
Cyclide (Spindle Cyclide)
plaster (overall material)
iron (overall material)
overall: 5 cm x 15 cm x 18.5 cm; 1 31/32 in x 5 29/32 in x 7 9/32 in