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

Object Details

Baker, Richard P.

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 one hundred of his models are in the museum collections.

The mark 410 w 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. 410w (/) Riemann surface : (/) w² = z⁵ - z (/) 2 models. The 2 models refers to this model and model No. 410z (211257.073) 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 w after the number of this model indicates that the metal disks above the wooden base represent copies of a disk in the complex w-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 z-plane with the point z = 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 a central angle of 30 degrees.

There are eight values of w for which the equation w² = z⁵ - z is satisfied by only four values of z. These eight points all lie on a circle centered at w =0 with radius slightly less than 3/4 (the exact value is 2 divided by the 8th root of 5⁵). Two points have real values, two have purely imaginary values, and the remaining four lie on the circle, half way between a real and a purely imaginary point. These eight points on the w-plane are called branch points of the model and for all other points on the w-plane the equation w² = z⁵ - z is satisfied by five distinct values of z, each of which produces a different pair on the Riemann surface (if w = 0, the five distinct pairs on the Riemann surface are the origin and the points (0,±1) and (0,±i).). Thus there are five sheets representing the complex w-plane and together they represent part of what is called a branched cover of the complex w-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.

On all but the middle sheet the same pattern appears, a curve that looks like an ellipse and two curves that look like apples. Each half of the outer curve maps onto the same portion of the outermost of the five circles appearing on the base, while each of the apple-like curves maps onto the same portion of the next smaller circle on the base. In addition, each of the eight branch points lies inside one of the eight apple-like curves. The middle sheet, which is much more complicated, has five wavy closed curves that map onto various portions of the five circles on the base. The line segments that appear on all five sheets are mapped to radii or diameters on the base.

On upper sheet the two dark points mark the approximate locations of two of the branch points of 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 w 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 and are represented by the horizontal edges of the vertical surfaces. In this model the movement is always between the middle sheet and one of the other four sheets.

Location

Currently not on view

Credit Line

Gift of Frances E. Baker

ca 1906-1935

ID Number

MA.211257.072

accession number

211257

catalog number

211257.072

Object Name

geometric model

Physical Description

wood (overall material)

metal (overall material)

orange (overall color)

green (overall color)

yellow (overall color)

blue (overall color)

bolted and soldered. (overall production method/technique)

Measurements

average spatial: 27.8 cm x 25.4 cm x 24.6 cm; 10 15/16 in x 10 in x 9 11/16 in

See more items in

Medicine and Science: Mathematics

Science & Mathematics

National Museum of American History

Subject

Mathematics

Record ID

nmah_1086110

Metadata Usage (text)

CC0

GUID (Link to Original Record)

https://n2t.net/ark:/65665/ng49ca746a9-3bf0-704b-e053-15f76fa0b4fa