Model of a Riemann Surface by Richard P. Baker, Baker #405zn
date: label: date made
content: ca 1915-1935
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 "405 z" 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. 405 wn (/) Riemann surface : w<sup>2</sup> = z<sup>3</sup> - z. The label is incorrect and should read "405 zn". Model 405zn is listed on page 17 of Baker’s 1931 catalogue of models as “w<sup>2</sup> = z<sup>3</sup> - z” 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>2</sup> = z<sup>3</sup> - z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. The spheres of the model are called complex, or Riemann, spheres. They are formed with north and south poles representing ∞ and 0, respectively. The equator represents the points x + yi with x<sup>2 + y</sup>2 = 1. Thus the points ±1 and ±i are equally spaced along the equator. Riemann surfaces and spheres are named after the 19th-century German mathematician Bernhard Riemann.
label: Description
content: Baker explains in his catalog that the zn after the number of the model indicates that the model is made up of spheres representing z-values. These spheres are called the sheets of the model. It appears as if painted part of the wooden base of the model represents the Riemann surface as a torus, i.e., a donut, formed by pasting together the ends of the stripes to form a cylinder and then joining the ends of the cylinder.
label: Description
content: If z = 0 or z = ±1, the equation w<sup>2</sup> = z<sup>3</sup> - z is satisfied by only one value of w, i.e., w = 0. These three points together with the point z = ∞ are called branch points of the model and for all other points on the z-sphere the equation w<sup>2</sup> = z<sup>3</sup> - 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, ±√6)). Thus there are two sheets representing the complex z-sphere and together they represent what is called a branched cover of the complex z-sphere. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base into which it is mapped.
label: Description
content: On each of the sheets the equator is colored red and there are great circles through the poles that are colored yellow and black. The points on the yellow great circle are purely imaginary while those on the black great circle are real. Thus the real non-zero branch points, z = ±1, lie on the equator and on the black great circle, while the other two branch points are at the north and south poles. The darkened parts of the black great circle are called branch cuts. Assuming the pair (1,0) lies on the Riemann surface along edge shared by the center (yellow and green) stripes on the base and that the pair (–1,0) lies along the edges of the outer stripes on the base, one of the branch cuts runs between join z = 0 and z = 1 and other between z = –1 and z = ∞. These branch cuts are 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. Thus one can construct the Riemann surface as a torus by cutting the spheres along the branch cut and sewing the two spheres together along those cuts while matching the four branch points.
label: Description
content: There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, with Baker's number 405z (MA.211257.070) has "405z" carved on the base. Two others, Baker's number 405 w (MA.211257.068) and Baker's number 405wn (MA.211257.069) have the mark "405w" on the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" to indicate that the sheets of the model are spheres.
label: Location
content: Currently not on view
place: label: place made
content: United States: Iowa, Iowa City
topic: label: Subject
content: Mathematics
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
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content: Mathematical Association of America Objects
publisher: label: Related Publication
content: Baker, Richard P.. Mathematical Models
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.211257.071
label: accession number
content: 211257
label: catalog number
content: 211257.071
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: red (overall color)
label: Physical Description
content: yellow (overall color)
label: Physical Description
content: orange (overall color)
label: Physical Description
content: green (overall color)
label: Physical Description
content: spheres bolted through base (overall production method/technique)
label: Measurements
content: average spatial: 13.4 cm x 25.2 cm x 25.2 cm; 5 9/32 in x 9 29/32 in x 9 29/32 in