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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 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 exhibited at the German Educational Exhibit at the Columbian Exposition held in Chicago in 1893, where it was purchased by Wesleyan University.
This tan paper model of a hyperbolic paraboloid consists of thirteen portions of triangles, intersecting portions of another thirteen triangles. It is stored flat in a gray envelope which in turn is in a brown paper box with the other models in the series. A mark stamped on the model reads: Verlag von L. Brill in Darmstadt.
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.
Ludwig Brill, Catalog mathematischer Modelle..., Darmstadt: L. Brill, 1892, p. 1, 57.
Henry Burchard Fine and Henry Dallas Thompson, Coordinate Geometry, New York: Macmillan Company, 1931, p. 243-244.
Currently not on view
Germany: Hesse, Darmstadt
paper (overall material)
overall: 10 cm x 14 cm x 8 cm; 3 15/16 in x 5 1/2 in x 3 5/32 in