Paper Model of a Hyperbolic Paraboloid, M. Schilling No. 35, Carton Ser. No. 6

In the nineteenth and early twentieth century, students studying technical subjects often learned about the representation of surfaces by equations in courses in solid analytic geometry. Schools in Europe, the United States, and Japan sometimes purchased models to illustrate such surfaces. The firm of Ludwig Brill in Darmstadt and its successor Martin Schilling in Leipzig published this one as part of a series of paper models (the “Carton” series) designed by Alexander Brill and first issued in 1874. This example was published by Schilling.

This tan paper model of a hyperbolic paraboloid consists of thirteen portions of triangles, intersecting potions of another thirteen triangles. It is stored flat in a light green envelope which in turn is in a brown paper box with the other models in the series. A tag on the model reads: Karton-Modelle (/) von Flachen zweiter Ordnung. (/) Nr. 6. (/) Hyperbolisches Paraboloid. (/) Verlag von Martin Schilling in Halle a. S. A further mark on the envelope reads: Leipzig.

The saddle-shaped surface is represented by the equation: + y²/ b² - x²/a² = -2z. Sections by any plane where x = c or y=c (c being an arbitrary constant) are parabolas. Sections parallel to the plane z = 0 are hyperbolas. The top edges of the pieces are straight lines, illustrating that the hyperbolic paraboloid is a ruled surface.

Compare MA.304723.19 to 1985.0112.021.

References:

Ludwig Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, p. 1, 57.

M. Schilling, Catalog mathematischer Modelle. . ., Leipzig, 1911, p.1-2, 114.

Henry Burchard Fine and Henry Dallas Thompson, Coordinate Geometry, New York: Macmillan Company, 1931, p. 243-244.

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