Model of a Klein-Riemann Surface by Richard P. Baker, Baker #415

Object Details

Baker, Richard P.

Description

This geometric model was constructed by Richard P. Baker in the early twentieth century 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 415 and the initials R. P. B. are carved into one edge of the wooden base of this model. The typed part of a paper label on the base reads: “No. 415 (/) Klein-Riemann surface: v = y (/) 4u² = 1 + 4(x + y) (1 − x) (1 − y).” Although Baker’s 1931 catalog, “Mathematical Models Made by R. P. Baker” lists the same equations, his handwritten notes on Model 415 and its inverse transformation, Model 415a (MA*211257.077), show that the actual equations that define these models are v = y and 4u² = 1 − 4(x + y) (1 − x) (1 − y). The metal sheets of 415 represent parts of the x,y-plane and those of 415a represent parts of the u,v-plane.

While Baker did not define a Klein-Riemann surface, and that term does not appear to have been used except by him, the identification of points on opposite ends of the threads suggests that he is referring to a generalization of a Riemann surface known as a Klein surface.

The actual surface representing the equations is in four dimensional space with real coordinates (x,y,u,v), but Model 415 is made up of a pair of two dimensional sheets and does not show the coordinates (u,v) that produce the points satisfying those equations. By using the equation v=y, one can reduce the actual number of coordinates to three, x, y, and u. Since the defining equations of the surface are at most quadratic in u, the number of points on the model for a given value of (x,y) can only be 0, 1 or 2.

Baker’s model illustrates the curves on the model where the number of points is 1 by using vertical threads to connect the two sheets, thus illustrating that the points that lie above one another represent a single curve. These curves represent the boundary curve of the model and, therefore, none of the metal outside of the threads is actually part of the model of the surface. While one of these components of the boundary is represented by a small circle, the other three are asymptotic to the solid lines that run alongside them so these boundary curves run to infinity.

.The entry for Model 415 in Baker’s 1931 catalog is followed by the explanation “This model is closely related to Clebsch’ [sic] diagonal surface, giving a concise method of drawing 24 lines of the surface, 3 being principal lines at infinity.” All of the 27 lines of the Clebsch diagonal surface are real and 3 of them are lines at infinity. Baker numbered the 24 lines shown on the model using numbers between 1 and 27, omitting the three numbers assigned to the lines at infinity (13, 16, and 19).

As seen in image NMAH-AHB2019q017118, the number of each of the visible lines is written next to the point where the line leaves the model. Of the 24 numbered lines, 21 appear on both sheets of the model and only six lines appear on only one sheet (22, 23, and 24 on the upper sheet and 25, 26, and 27 on the lower sheet). Each of these six lines is asymptotic to the two different infinite components of the boundary curve. Each of the lines numbered 1 through 6 pass through a point of tangency of these infinite components of the boundary curve and move from one sheet to the other at that point as indicated by switching the representation of the line between solid and dashed on both sheets with the solid and dashed portions appearing above one another. Similarly, the lines numbered 7 through 12, 14, 15, 17, 18, 20, and 21 switch between solid and dashed representations and switch sheets at six different points of tangency to the finite component of the boundary curve.

References:

Richard P. Baker Papers, University Archives, Special Collections, The University of Iowa Libraries.

Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 17.

Location

Currently not on view

Credit Line

Gift of Frances E. Baker

ca 1906-1935

ID Number

MA.211257.076

accession number

211257

catalog number

211257.076

Object Name

geometric model

Physical Description

thread (overall material)

wood (overall material)

metal (overall material)

white (overall color)

black (overall color)

red (overall color)

bolted and threaded. (overall production method/technique)

Measurements

average spatial: 13.6 cm x 25.4 cm x 25.4 cm; 5 11/32 in x 10 in x 10 in

See more items in

Medicine and Science: Mathematics

Science & Mathematics

National Museum of American History

Subject

Mathematics

Record ID

nmah_1086168

Metadata Usage (text)

CC0

GUID (Link to Original Record)

https://n2t.net/ark:/65665/ng49ca746a9-542c-704b-e053-15f76fa0b4fa