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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 8 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 8. On adjacent faces of model 9 there is an R and an I inscribed, with the R on the face with the labels.
On each model in series 14 there are two sets of curves that act much like the lines on 2-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 8. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 8 is related to the level curves.
Model 8 is based on the derivative of the Weierstrass P-function on which Brill models 7a and b (1985.0112.139) are based. Complex valued Weierstrass P-functions are named after the nineteenth century German mathematician, Karl Weierstrass and the derivative of each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a, 7b, and 8 is made up of squares with sides parallel to the x and y axes and there are four such squares in each of the models. Model 8 is made up of four congruent sections each of which has a square base and three cropped spires alternating with three narrowing holes equally spaced around the center of each square.
Only points on the curved surfaces of model 8 satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The 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 surface produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Although each plot has both an R and an I superimposed approximately where it appears on the model, if there is an R or an I facing the front, the x axis is parallel to that face and the vertical axis is labeled u if R is on that face and is labeled v if I is on it. For model 8, as well as models 7a and 7b, the computer generated versions show four congruent sections. In model 8 each section includes three hollow spires alternating with three downward pointing versions of the hollow spires.
L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-72.
G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 131, p. 128, and vol. 2 (Commentary), pp. 71-72. 75-76.
Currently not on view
Germany: Hesse, Darmstadt
plaster (overall material)
overall: 16 cm x 17.2 cm x 16 cm; 6 5/16 in x 6 25/32 in x 6 5/16 in