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This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 2a and 2b 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. In this group, there is an R inscribed on the front of model 2a and an I inscribed on the front of model 2b.
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 models 2a and 2b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 2a is related to the level curves on model 2b. Similarly, the placement of the gradient curves model 2b is related to the level curves on the model 2a.
The equation involving complex variables on which models 2a and 2b are based is w2 = z4 - 1. That equation defines a surface in 4 dimensions. Models 2a and 2b are 3-dimensional models, each of which is defined by an equation using three real variables. Those, much more complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Model 2a is defined by u4 - (x4 - 6x2y2 + y4 - 1)u2 - 4x2y2(x2 - y2)2 = 0 while model 2b is defined by v4 + (x4 - 6x2y2 + y4 - 1)v2 - 4x2y2(x2 - y2)2 = 0.
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 or an I superimposed approximately where it appears on the corresponding model. While model 2b does not show it, the computer generated plot positioned to look directly along the plane y = x does show a significant hole and that same shaped hole can be seen by rotating the plot 90 degrees and looking directly along the plane y = -x.
L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71.
“Mathematische Modelle angefertigt im mathematischen Institut der k. technischen Hochschule in München unter Leitung von Professor Dr. Walther Dyck. Modelle zur Functionentheorie (Zu Serie XIV)” [“Mathematical Models Made in the Mathematical Institute of the Royal Technical College in Munich under the Direction of Professor Dr. Walther Dyck. Models for Function Theory (For Series 14)”], pp. 1-5 and fig. 2 plate I.
Currently not on view
Germany: Hesse, Darmstadt
plaster (overall material)
real part: 12.5 cm x 12.5 cm x 12.5 cm; 4 29/32 in x 4 29/32 in x 4 29/32 in
imaginary part: 11.4 cm x 12 cm x 11.7 cm; 4 1/2 in x 4 23/32 in x 4 19/32 in