Model of a Riemann Surface by Richard P. Baker, Baker #410Z

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 410 z is inscribed on an edge of the wooden base of this model and the typed part of a paper tag on the base reads: No. 410z, w (/) Riemann surface : (/) w² = z⁵ - z (/) 2 models. The w is crossed but the 2 models refers to this model and model No. 410w (MA.211257.072) that are associated with the same equation. Both models are listed on page 17 of Baker’s 1931 catalog of models as w² = z⁵ - z under the heading Riemann Surfaces. This means that both models represent a Riemann surface consisting of pairs of complex numbers, (z, w), for which w² = z⁵ - 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 catalog that the z after the number of this 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 disk on the wooden base of the model represents a disk in the complex w-plane with the point w = 0 at its center. The disk is divided into sixteen sectors, pie-piece-shaped parts of a circle centered at 0, each of which has a central angle of 22.5 degrees.

If z = 0, ±1, or ±i, the equation w² = z⁵ - z is satisfied by only one value w, i.e., w = 0. These five points on the z-plane are called branch points of the model and for all other points on the z-plane the equation w² = z⁵ - 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, ±√30)). Thus there are two sheets representing 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 a sector or sectors on the base into which it is mapped.

The five dark blue points on the upper sheet of the model are marking the approximate locations of the five branch points of the model. The five branch points appear again on the lower sheet and the four colors on each sheet are represented in the regions surrounding the branch points on that sheet. 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 all of the branch cuts run out to infinity although two of them meet at a point that is not a branch point. One of the two branch cuts that meet runs from z = 0; the two branch cuts meet on the upper sheet at the point where two green regions meet two pink (possibly once purple) regions. All the branch cuts are represented by the horizontal edges of the vertical surfaces.

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