Model of a Riemann Surface by Richard P. Baker, Baker #411z
 Description

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 100 of his models are in the museum collections.

The mark 411 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. 411z (/) Riemann surface : w^{3} = z. Model 411z is listed on page 17 of Baker’s 1931 catalogue of models as “w^{3} = z” under the heading Riemann Surfaces. The catalog description also notes that “411 is to serve as a first step to 412,” where Baker model 412z (MA*211157.075) is associated with a more complicated equation involving w^{3}.

The model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{3} = z where a complex number is 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 19thcentury 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 zplane. 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 wplane with the point w = 0 at its center. The disk is divided into twelve sectors, piepieceshaped parts of a circle centered at 0, each of which has an angle of 30 degree. The front of the model is the edge on which 411 is inscribed so the two 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 the rectangles on a polar axis of a sheet.

If z = 0, the equation w^{3} = z is satisfied by only one value of w, i.e., w = 0. The point z = 0 is called a branch point of the model and for all other points on the zplane the equation w^{3} = z is satisfied by three distinct values of w, each of which produces a different pair on the Riemann surface (if z = 1, the three distinct pairs on the Riemann surface are (1,1), and (1,(–1 ± √3 i)/2)). Thus there are three sheets representing the same disc in the zplane and together they represent part of what is called a branched cover of the complex zplane.

Baker’s use of solid red circles, and dashed red and black circles indicates that each sheet is mapped continuously onto a different portion of the wdisk on the base. There are three radii of the disk on the base (the polar lines  rays emanating from the origin – for angles of 0, 120, and 240 degrees) that are the edges of sectors corresponding to quadrants on two different sheets. The order of the colors of the 30 degree sectors on the base starting at polar axis and proceeding counterclockwise correspond to the colors of the first through fourth quadrants of the top, middle, and then bottom sheets.

The vertical rectangles mentioned above 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 the 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 branch points, branch cuts are not fixed by the equation. However, the single branch cut for any surface with only one branch point must run from that point out to infinity. The branch cut of this model is represented on each sheet by the horizontal edges of the vertical surface or surfaces meeting that sheet.
 Location

Currently not on view
 Object Name

geometric model
 maker

Baker, Richard P.
 Physical Description

wood (overall material)

metal (overall material)

other, cardboard (overall material)

red (overall color)

green (overall color)

yellow (overall color)

blue (overall color)

bolted and soldered. (overall production method/technique)
 Measurements

average spatial: 24.6 cm x 24.7 cm x 25.4 cm; 9 11/16 in x 9 3/4 in x 10 in
 place made

United States: Iowa, Iowa City
 ID Number

MA*211257.074
 accession number

211257
 catalog number

211257.074
 subject

Mathematics
 See more items in

Medicine and Science: Mathematics

Science & Mathematics

Women Mathematicians
 Data Source

National Museum of American History, Kenneth E. Behring Center