Plaster Model for Function Theory by L. Brill, No. 182, Ser. 14 No. 10b
 Description

This model is one of a group of two plaster models that 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 10b of the group that also contains model 10a 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. There is an I inscribed on a vertical face of model 10b but, since there is no vertical face of the model that is parallel to the x axis, the I is inscribed on a face that meets 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 2dimensional 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 10b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 10ba is related to the level curves on model 10a, which is not in the museum collections.

Model 10b is based on the derivative of the Weierstrass Pfunction on which Brill models 9a and b (1985.0112.141) are based. Complex valued Weierstrass Pfunctions 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 zplane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass Pfunction is the same for corresponding points of the One can see one such rhombus in model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire. parallelograms of the tiling. The tiling associated with models 9a, 9b, 10a, and 10b 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 model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire.

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. Model 10b is 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 a two dimensional plot mimicking a bird’seye view of the surface that were produced using the program Mathematica. This plot has been superimposed with an outline in red of four of the tiling hexagons and thicker black lines that show the model’s footprint. In addition, +’s and –‘s have been placed to indicate the location of the cropped spires (+’s) and the tapering holes (‘s). These twodimensional 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 the 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 surface produced using the program Mathematica show scales to indicate the direction of at least two of the variables and each has an I superimposed approximately where it appears on the model. For model 10b, as well as models 9a and 9b, the computer generated versions show four congruent sections. In model 10b each section includes three hollow spires alternating with three downward pointing versions of the hollow spires. The plots are produced by taking x and y values from a rectangle grid so extra portions of the surface are seen, including parts of additional spires that are outside the footprint of the model. A version of this plot has been overlaid with the two sides of the rhombus that are parallel to the x axis and the long diagonal of the rhombus, which is parallel to the base of the vertical face with the inscribed I.

References:

L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 2930, 7073.

“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. 13, 78, 1113 and fig. 7 plate III.
 Location

Currently not on view
 Object Name

geometric model
 date made

1892
 maker

L. Brill
 Physical Description

plaster (overall material)
 Measurements

overall: 16.5 cm x 22.5 cm x 16.5 cm; 6 1/2 in x 8 27/32 in x 6 1/2 in
 place made

Deutschland: Hessen, Darmstadt
 ID Number

1985.0112.142
 catalog number

1985.0112.142
 accession number

1985.0112
 subject

Mathematics
 See more items in

Medicine and Science: Mathematics

Science & Mathematics

Geometric Models for Complex Analysis
 Data Source

National Museum of American History, Kenneth E. Behring Center