Model of a Riemann Surface by Richard P. Baker, Baker #412Z

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 412 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. 412z (/) Riemann surface : (/) z(1 + w³)² = (1 – w³)². This model is listed on page 17 of Baker’s 1931 catalog of models as z(1 + w³)² = (1 – w³)² under the heading Riemann Surfaces. According to the catalog Baker No. 411z (MA.211257.074), “411 is to serve as a first step to 412. ”That model is associated with the equation w³ = z.

This model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which z(1 + w³)² = (1 – w³)². 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 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 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 twelve sectors, pie-piece-shaped parts of a circle centered at 0, each of which has an angle of 30 degrees. The front of the model is the edge on which 412 is inscribed, so the vertical rectangles lie above the polar axis, i.e. the ray emanating from the origin when the angle is 0 degrees, of the wooden base. This places every horizontal edge of a vertical rectangle on the polar axis of one or two sheets.

If z = 0, the equation z(1 + w³)² = (1 – w³)² is satisfied by only three values of w, i.e., the three complex roots of w³ = 1, 1 and (–1 ± √3 i) / 2, and (–1 + √3 i) / 2. If z = 1, the equation z(1 + w³)² = (1 – w³)² is satisfied by only one finite value of w, w = 0. Both z = 0 and z = 1 are called branch points of the model and for all other finite points on the z-plane the equation z(1 + w³)² = (1 – w³)² is satisfied by six distinct values of w, each of which produces a different pair on the Riemann surface (if z = –1, the six values of w are the six complex roots of w⁶ = –1). Thus there are six 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. Baker’s use of red and black radii dividing the sheets into quadrants and white unit circles on all of the sheets indicated the continuous mapping of each sheet onto a 60 degree quadrant on the base. The top sheet is mapped into the sector defined by the polar lines (rays emanating from the origin) at 0 and 60 degrees. The subsequent sheets are defined by the five other 60 degree sectors defined by the polar lines at 60, 120, 180, 240 300, and 0 degrees.

On each sheet there is a circle of radius one drawn in white. These six circles are mapped to the six rays on the base emanated from the origin on which the three cube roots of i of –i lie. The circle on the upper sheet is mapped onto the ray with defining angle of 30 degrees. The subsequent circles are mapped onto rays at 90 (the positive imaginary axis), 150, 210, 270 (the negative imaginary axis), and 330 degrees.

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. In this model there are branch cuts along the positive real axis from z = 0 to z = 1 and from z = 1 to infinity. The movement produced when meeting an infinite branch cut is between the pair of sheets that lie above and below the vertical edges that define the branch cut, i.e., between sheets 1 and 2, 3 and 4, and 5 and 6. For branch cuts running from z=0 to z=1, the movement produced is also between consecutive sheets but is between sheets 1 and 6, 2 and 3, and 4 and 5.

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