Group of Two Plaster Models for Function Theory by L. Brill, No. 179, Ser. 14
No. 7a and 7b
 Description

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 7a and 7b 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 zplane, 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 7a and an I inscribed on the front of model 7b.

On each model in series 14 there are two sets of curves that act much like the lines on twodimensional 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 7a and 7b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 7a is related to the level curves on model 7b. Similarly, the placement of the gradient curves model 7b is related to the level curves on the model 7a.

Models 7a and 7b are based on a Weierstrass Pfunction. 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 zplane by congruent parallelograms (like graph paper) so that the complex value of the Weierstrass Pfunction is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a and 7b is made up of squares with sides parallel to the x and y axes. There are four such squares in each of the models so models 7a and 7b are both made up of four congruent sections each of which has a square base and has at its center a pair of cropped spires and a pair of narrowing holes.

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 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 7a and 7b, the computer generated versions show the four congruent sections, each of which includes two spires that are hollow and two holes that are downward pointing versions of the hollow spires.

References:

L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 2930, 7072.

G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photos 129 (model 7a) and 130 (model 7b), pp. 126127. and vol. 2 (Commentary), pp. 7172, 7576.
 Location

Currently not on view
 Object Name

geometric model
 date made

1892
 maker

L. Brill
 Physical Description

plaster (overall material)
 Measurements

real part: 16.5 cm x 16.3 cm x 16 cm; 6 1/2 in x 6 13/32 in x 6 5/16 in

imaginary part: 16.5 cm x 16.3 cm x 16 cm; 6 1/2 in x 6 13/32 in x 6 5/16 in
 place made

Deutschland: Hessen, Darmstadt
 ID Number

1985.0112.139
 catalog number

1985.0112.139
 accession number

1985.0112
 subject

Mathematics
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Medicine and Science: Mathematics

Science & Mathematics

Geometric Models for Complex Analysis
 Data Source

National Museum of American History, Kenneth E. Behring Center