This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 3 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 the front of model 3, indicating that the vertical axis can represent either u or v.

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 1/2 cm. on model 3. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 3 is related to the level curves on that model.

The equation involving complex variables on which model 3 is based is w^{4} = 1 - z^{2}. That equation defines a surface in four dimensions. Model 3 is a three-dimensional model and is defined by two almost identical equations, each using three real variables. Those, very complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that one includes u’s and the other v’s and they define the same model. The polynomial equations that define model 3 start with a term with a coefficient 256 and exponent of the variable 16.

Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. Computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. 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 approximately where they appear on model 3. While model 3 does not show it, the computer generated plot positioned to look directly at the front shows a hole with a complicated boundary.

References:

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 125, p.122, and vol. 2 (*Commentary*), pp. 71-72, 75-76.

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