Model of a Riemann Surface by Richard P. Baker, Baker #404Z
date: label: date made
content: ca 1906-1935
name: label: maker
content: Baker, Richard P.
notes: label: Description
content: 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.
label: Description
content: The mark 404 is inscribed on an edge of the wooden base of this model and the typed part of a paper tag on the bottom of the base reads: No. 404z (/) Riemann Surface: (/) w<sup>3</sup> + z<sup>3</sup> + 3wz = 0. This paper label is pasted on top of another paper label with the entry showing, No. 386, crossed out. Baker model 404z is listed on page 17 of his 1931 catalogue of models as w<sup>3</sup> + z<sup>3</sup> + 3wz = 0 under the heading <I>Riemann Surfaces</I>. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z, w), for which w<sup>3</sup> + z<sup>3</sup> +3wz = 0. 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.
label: Description
content: Baker explains in his catalogue 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 thirty-two sectors, pie-piece-shaped parts of a circle centered at 0, each of which has a central angle of 11.25 degrees. 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.
label: Description
content: If z = 0, the equation w<sup>3</sup> + z<sup>3</sup> + 3wz = 0 is satisfied by only one value of w, i.e., w = 0. However, if z is a complex cube root of -4, i.e., z = – <sup>3</sup>√4, <sup>3</sup>√4(1 – i√3) / 2, <sup>3</sup>√4(1 + i√3) / 2, then the defining equation is satisfied by two distinct values of w, one the negative of the double of the other, i.e., w = – <sup>3</sup>√2 and 2 <sup>3</sup>√2 , w = <sup>3</sup>√2 (1 + i√3) / 2 and – <sup>3</sup>√2 (1 + i√3), or w = <sup>3</sup>√2 (1 – i√3) / 2 and – <sup>3</sup>√2 (1 – i√3), respectively. These four points on the z-plane are called branch points of the model and for all non-branch points on the z-plane the equation w<sup>3</sup> + z<sup>3</sup> + 3wz = 0 is satisfied by three distinct values of w. Thus there are three sheets representing the complex z-plane and together they represent part of what is called a branched cover of the complex z-plane. Each of the non-zero branch points appears on only two of the sheets.
label: Description
content: A 120 degree sector on the upper sheet is colored darker than the other two 120 degree sectors of the sheet. The borders of this sector include two partial radii at the end of which there are small rods coming from below that mark the approximate locations of two of the non-zero branch points of the model. There are two non-zero branch points on each of the other sheets marked by similar configurations. Between the sheets there are vertical surfaces that 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. Each branch cut of this model runs between z = 0 and one of the three non-zero branch points and every branch cut is represented by the horizontal edges of a vertical surface
label: Description
content: It is clear that there is a branch cut on the middle sheet that is connected only to a branch point on the upper sheet by one of these vertical surfaces and another branch cut on the middle sheet that is connected only to a branch cut on the lower sheet. The coloring of the two vertical surfaces that run directly above one another shows that it neither of these surfaces is meant to mark a branch cut on the middle sheet. Instead, together the two vertical surfaces represent a movement from the branch cut on the upper sheet to branch cut on the bottom sheet. Thus for each pair of sheets, there is a branch cut representing movement between them.
label: Location
content: Currently not on view
topic: label: Subject
content: Mathematics
setName: label: See more items in
content: Medicine and Science: Mathematics
label: See more items in
content: Science & Mathematics
creditLine: label: Credit Line
content: Gift of Frances E. Baker
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: MA.211257.067
label: accession number
content: 211257
label: catalog number
content: 211257.067
objectType: label: Object Name
content: geometric model
physicalDescription: label: Physical Description
content: wood (overall material)
label: Physical Description
content: metal (overall material)
label: Physical Description
content: black (overall color)
label: Physical Description
content: purple (overall color)
label: Physical Description
content: blue (overall color)
label: Physical Description
content: yellow (overall color)
label: Physical Description
content: bolted and soldered. (overall production method/technique)
label: Measurements
content: average spatial: 20.8 cm x 25.2 cm x 24.8 cm; 8 3/16 in x 9 29/32 in x 9 3/4 in