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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 9a and 9b 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 a vertical face of model 9a and an I inscribed on a vertical face of model 9b. However, since there is no vertical face of either model that is parallel to the x axis, the R and I are inscribed on faces that meet the x axis in what appears to be a 30 degree angle.
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 models 9a and 9b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 9a is related to the level curves on model 9b. Similarly, the placement of the gradient curves model 9b is related to the level curves on the model 9a.
Models 9a and 9b are based on a Weierstrass P-function. These complex valued functions are named after the nineteenth century German mathematician, Karl Weierstrass and 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 Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. One tiling associated with the Weierstrass P-function defining models 9a and 9b is made up of rhombuses whose angles are 60 and 120 degrees and with one pair of sides parallel to the x axis. One can see one such rhombus in models 9a and 9b by joining the four points where the tops of each pair of cropped spires meet. The sides of the rhombus parallel to the x axis pass through the center of the cropped spires in model 9a and pass between a cropped spire and a hole in model 9b.
The tiling by rhombuses leads to an alternate tiling by regular hexagons with one pair of sides parallel to the y axis and with the structures that were centered at each vertex of the tiling by rhombuses sitting at the center of each hexagon of the tiling. Models 9a and 9b are made up of four such hexagons that have been slightly trimmed. This tiling is difficult to see directly on the model but can be easily seen in two dimensional plots mimicking bird’s-eye views of the surfaces that were produced using the program Mathematica. These plots have been superimposed with an outline in red of four of the tiling hexagons and thicker black lines that show the models’ footprints. In addition, +’s and –‘s have been placed to indicate the location of the cropped spires (+’s) and the tapering holes (-‘s). These two-dimensional plots also show why the models do not have rectangular footprints and why the R and I could not be placed on faces parallel to the x axis.
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. The computer generated versions show only the surfaces, and 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. For models 9a and 9b, the computer generated versions show that the spires are hollow and the pairs of tapering holes are downward pointing versions of the pairs of upward pointing hollow spires. These versions are produced by taking x and y values from a rectangle grid so extra portions of the surfaces are seen, including parts of single spires in the left front and right rear corners.
L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-73.
“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. 1-3, 7-8, 11-13 and fig. 6 plate III.
Currently not on view
Germany: Hesse, Darmstadt
plaster (overall material)
real part: 16 cm x 21 cm x 15 cm; 6 5/16 in x 8 9/32 in x 5 29/32 in
imaginary part: 16 cm x 21 cm x 15 cm; 6 5/16 in x 8 9/32 in x 5 29/32 in