Model of a Riemann Surface by Richard P. Baker, Baker #405z

This geometric model was constructed by Richard P. Baker in about 1930 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 mark 405 z is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 w (/) Riemann surface : w² = z³ - z. Someone corrected the error on the label by hand, crossing out the w and inserting a z. Model 405z is listed on page 17 of Baker’s 1931 catalogue of models as “w² = z³ - z” under the heading Riemann Surfaces. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w² = z³ - z. 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.

Baker explains in his catalogue that the z after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex z-plane. These disks are called the sheets of the model. The painted part of the wooden base of the model represents a square in the complex w-plane with the point w = 0 at its center and the real axis along the line between the yellow and dark green stripes.

If z = 0 or z = ±1, the equation w² = z³ - z is satisfied by only one value of w, i.e., w = 0. These three points on the z-plane are called branch points of the model and for all other points on the z-plane the equation w² = z³ - z is satisfied by two distinct values of w, each of which produces a different pair on the Riemann surface (if z = 2, the two distinct pairs on the Riemann surface are (2, ±√6)). Thus there are two sheets representing the same disk in the complex z-plane and together they represent part of what is called a branched cover of the complex z-plane. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base in which the w coordinate lies.

For each sheet, the center of the disc is the point z = 0 and the solid black line through that point is the real axis. The branch points of this model all lie on the real axis. The point z = –1 is the point inside the green and yellow oval where the real axis meets the small red circle representing the unit circle with center z = 0. The point z = 1 is the other point where the real axis meets the small red circle; it is inside the oval that includes all eight colors used in the model.

The vertical surfaces between the two sheets are not part of the Riemann surface but call attention to what are called branch cuts of the model, i.e., curves on a sheet that produce movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of z values into the equation. While the defining equation determines the branch points, the branch cuts are not fixed by the equation but, normally, each branch cut goes through two of the surface’s branch points or runs out to infinity. In this model, one of the branch cuts connects z = 0 to z = 1 and the other runs from z = –1 to infinity; they are represented by the horizontal edges of the vertical surfaces.

There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, a model with Baker's number 405zn (MA.211257.071), has "405z" carved on the base. Two others, one with Baker's number 405w (MA.211257.068) and the other with Baker's number 405wn (MA.211257.069) have "405w" carved on the edge of the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" to indicate that the sheets of the model are spheres.

Currently not on view