String Model of a Riemann Surface by Richard P. Baker, Baker #310

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

The typed part of a paper tag on the underside of the base reads: No. 310 (/) Riemann surface. It is listed as a thread (or string) model with title Riemann Surface and further described as “Showing connection of sheets at branch cuts. Two cycles of two and one of three visible.” The model is listed on page 17 of Baker’s 1931 catalog of models not far above the section titled Riemann Surfaces and in which all the models listed, including nine in the museum collections (MA.211257.067 thru MA.211257.075), are made using sheet metal, rather than thread. Unlike the other nine models, neither the catalog entry for nor the paper tag on this model indicates the equation that defines the Riemann surface.

However, Baker’s notes on this model indicate that it is defined by the equation, zw³ - zw + 2 = 0 (Richard P. Baker Papers, University of Iowa Libraries, Iowa City, Iowa) so this model is intended to help understand the Riemann surface defined by that equation. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. A complex plane is like the usual real Cartesian plane but with the horizontal axis representing the real part of the number and the vertical axis representing the imaginary part of the number. Riemann surfaces are named after the 19th-century German mathematician Bernhard Riemann.

For most values of z, the equation zw³ - zw + 2 = 0 is satisfied by three distinct values of w. There are three exceptions to this; they are called the branch points of the Riemann surface. When z = 0, only w = infinity satisfies the defining equation. When z = 3√3 the defining equation is satisfied by two distinct values of w, √3/3, which is a double root, and -2√3/3. Similarly, when z = -3√3 the equation is also satisfied by the two distinct values of w, the double root -√3/3 and 2√3/3. Thus, 0 and ±3√3 are the branch points of this Riemann surface.

This model is made using thread that represents movement along straight lines on the three sheets of the Riemann surface. All of the thread, some of which is missing, runs in a direction perpendicular to the real axes of the sheets that represent z planes. One can think of the pair of long bolts that support the thread as together representing a copy of the real axis of a z plane. These pairs of bolts are separated in order to show that movement along a thread can produce movement from one sheet to another. Sets of 3 pieces of thread nearest each end of a bolt remain on the same sheets (7 sets nearest one side of the frame and 8 sets nearest the opposite side of the frame). For the remaining thread there are two different patterns each containing 12 sets of 3 pieces of thread for which 2 pieces start on one sheet of the model and end on another sheet.

The branch cuts are the curves along which the switch from one sheet to another takes place. In this model there are two branch cuts, each of which is a line segment that runs from z = 0 to one of the two non-zero branch points. One branch cut is represented by the 12 points where the thread on either side of the bolts supporting the top sheet meet as they move to the middle sheet and the other by the 12 points where the thread on either side of the bolts supporting the middle sheet meet as they move to the bottom sheet. The pieces of thread that move from one sheet to another are analogous to the vertical surfaces on Baker’s other Riemann surface models.

The cycles referred to in the catalogue description refer to the movement between the sheets while following a path of the inputs of values into the equation as the path goes around a branch point. After three turns following a path around the branch point that is the endpoint of both branch cuts, i.e., z = 0, the path will have gone through all three sheets and ended where it started, thus making Baker’s “cycle of three” visible. After two turns following a path around either branch points that is the endpoint of just one a branch cut, i.e., z = ±3√3, the path will have gone through the middle sheet and one of the other sheets and ended where it started, thus making both of Baker’s “two cycles of two” visible.

Currently not on view