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This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. It was manufactured by the Darmstadt publishing company of Ludwig Brill and is model number 4 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.
The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later.
Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on model 4, but on adjacent vertical faces, with the I on what looks like the back of the model.
On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on model 4. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 4 is related to the level curves.
The equation involving complex variables on which model 4 is based is w = 1/z. That equation defines a surface in 4 dimensions. Model 4 is a three-dimensional model and is defined by an equation using three real variables. That equation is found by replacing w by u + vi and z by x + yi, to get u + vi = (x – yi)/(x2 + y2), which is equivalent to the two equations u = x/(x2 + y2) and v = –y/(x2 + y2).. Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that switching from u to v, changes x to –y and y to –x. Therefore, these equations define the same model but with the x and y axes rotated 90 degrees.
Only points on the curved surface of the solid model satisfy the equation that defines it; points in the solid plaster that supports that surface do not satisfy that equation. Computer generated versions show only the surface so are able to show details of the model lying below the complex z-plane. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R and an I superimposed to show the faces where they appear on model 4. The computer plots looking directly at the faces with the inscribed R and I show a hollow cropped spire and the same spire rotated 180 degrees around the origin to face downward.
L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71.
G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 126, p. 123, and vol. 2 (Commentary), pp. 71-73.
Currently not on view
Germany: Hesse, Darmstadt
plaster (overall material)
overall: 12 cm x 12.3 cm x 12 cm; 4 23/32 in x 4 27/32 in x 4 23/32 in