Model of a Riemann Surface by Richard P. Baker, Baker #405wn
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 w" 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 zn (/) Riemann surface : w<sup>2</sup> = z<sup>3</sup> - z. The label should have read "405 wn" and someone added a handwritten question mark after the "zn." Model 405wn is listed on page 17 of Baker’s 1931 catalog 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</sup> + y<sup>2</sup> = 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 wn after the number of the model indicates that the model is made up of spheres representing w-values. These spheres are called the sheets of the model. There is no part of this model in which values of z or pairs (z,w) are represented. However, it is possible that the coloring on this model is related to the painted part of the wooden base of one of three other Baker models of Riemann surfaces that are associated with the equation of this model.
label: Description
content: If w = ±<sup>4</sup>√ (4/27) or w = ±i<sup>4</sup>√ (4/27), the equation w<sup>2</sup> = z<sup>3</sup> - z is satisfied by two distinct values of z. For the two real values of w, ±<sup>4</sup>√ (4/27), z takes the values –1/√3 and 2/√3, and for the two imaginary values of w, ±i <sup>4</sup>√ (4/27), z takes the values 1/√3 and –2/√3. These four points together with the point z = ∞ are called branch points of the model and for all other points on the w-sphere the equation w<sup>2</sup> = z<sup>3</sup> - z is satisfied by three distinct values of z, each of which produces a different pair on the Riemann surface (if w = 0, the three distinct pairs on the Riemann surface are (0,0) and (±1,0)). Thus there are three sheets representing the complex w-sphere and together they represent what is called a branched cover of the complex w-sphere.
label: Description
content: On each of the sheets the equator is a thin circle and there are two great circles through the poles. On one of the great circles the values of w are purely imaginary while on the other they are real. Baker’s usual use of colors implies that the great circles facing the front and back represent imaginary numbers, while those facing the sides represent real numbers. Normally w = ∞ is at the north pole and w = 0 is at the south pole. However, as the four finite branch points of this model lie in the northern hemisphere, it appears that this model has that assignment of values reversed. The great circle facing the front and back has a thick white segment that connects the two imaginary branch points by way of w = ∞ at the south pole, while the other has a thick black segment connecting the two real branch points by way of w = 0 at the north pole. The parts of the great circles that connect two branch points are called branch cuts. This model has three, one is the black arc mentioned above and the others are the two halves of the white arc with ends at an imaginary branch point and infinity. 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 w values into the equation.
label: Description
content: There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, a model with Baker’s no. 405w (211257.068), has “405w” carved on the base Two others, Baker’s no. 405z (211257.070) and Baker’s no. 405zn (211257.071), both have the mark “405z“ carved on them. Baker carved a "w" or a "z" to indicate which variable is represented on the sheets of the model and added an "n" after the "w" or "z" 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.211257.069
label: accession number
content: 211257
label: catalog number
content: 211257.069
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: yellow (overall color)
label: Physical Description
content: blue (overall color)
label: Physical Description
content: green (overall color)
label: Physical Description
content: orange (overall color)
label: Physical Description
content: spheres bolted through base (overall production method/technique)
label: Measurements
content: average spatial: 13.2 cm x 34 cm x 10.4 cm; 5 3/16 in x 13 3/8 in x 4 3/32 in