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

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 A 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 upper sheet reads: “No. 415A (/) K. Riemann surface: the in-(/) verse transformation of (/) 415.” This means that the model is a Klein-Riemann surface for the same equations as for Baker’s Model 415 (MA*211257.076), i.e., v = y and 4u² = 1 − 4(x + y) (1− x) (1 − y), but with metal sheets above the wooden base that represent parts of the u,v-plane rather than the x,y-plane. This equation is found in Baker’s handwritten notes on Models 415 and 415a and differs from what appears in Baker’s 1931 catalog, “Mathematical Models Made y R. P. Baker,” and on the label for Model 415.

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 Riemann surface known as a Klein surface.

The actual surface representing the equations is in four dimensional space with real coordinates (u,v,x,y) but Model 415a is made up of a pair of two dimensional sheets and does not show the coordinates (x,y) that produce the points satisfying those equations. By using the equation v = y, one can reduce the actual number of coordinates to three, u, v, and x. Since the defining equations of the surface are at most quadratic in x, the number of points on the model for a given value of (u,v) 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 sheets, thus illustrating that the points that lie above one another represent a single curve. The two metal sheets above the wooden base represent the part of the u,v-plane where that number is one or two and there are two curves that represent the points where it is one. The curve defined by the endpoints of the vertical threads represents the boundary curve of the model. The defining equations imply that one of these curves is the line v = 1 and the other is a cubic curve, 4u² = v³ + v² − v, which has two components, one of which is closed and does not meet the line v = 1. The other runs to infinity.

This model may have been restrung since, as can be seen in image NMAH-AHB2019q017117, the current placement of vertical threads does not show exactly what is described above. There are now two threads that do not correspond to the actual boundary curves of the model. These threads connect the component of the cubic curve that is not closed to the line v = 1, making it appear as if the boundary curves defined by the threads include two closed components, the smaller of which includes part of the line v = 1. One of these extra threads is above the label (23) Point and the other is above (26) Point. In fact, the two “Point” labels are referring to the points (±½,1) where the solid lines u = ±½ meet the line of threads v = 1. The numbers in parentheses refer to vertical lines on the surface that are not visible on the model but represent the equations in three dimensional space with real coordinates (u,v,x). Because none of the metal outside of the correctly placed threads is part of the model of the surface, the points (±½,1) are the only points that connect the parts of the sheets with v≤1 to those with v≥ 1.

As this model is the inverse transformation of Model 415, the lines seen in this model are also related to the lines of a Clebsch Diagonal Surface as noted in the catalog entry for Model 415, i.e., for both the Clebsch Diagonal Surface, and the surface represented in Models 415 and 415a, all of the 27 lines on the surface are real and 3 of them are lines at infinity. However, while on Model 415 there are 24 lines shown on the model, on Model 415a there are only 22 lines shown since, as mentioned above, lines 23 and 26 represent vertical lines through the points (−½,1) and (½,1), respectively. The numbers that are omitted on Model 415 (13, 16, and 19) also do not appear on this model.

All the lines shown on Model 415a appear in pairs that are located above one another on the two sheets and, with the exception of the solid lines u = ±½ (lines 22 and 24 are u = ½, and lines 25 and 27 are u = −½), one line of the pair is represented as a solid line and the other as a dashed line. All the lines that appear on the model pass from one sheet to the other when they are tangent to a component of the cubic boundary curve and, except for u = ±½, this movement is seen as a switch between a solid and a dashed line. The points of tangency of u = ±½ occur when v = −1 and are exceptional because they are double points and, therefore, cause the movement between sheets to be done twice so the lines remain solid when they pass the point of tangency.

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.211257.077

accession number

211257

catalog number

211257.077

Object Name

geometric model

Physical Description

thread (overall material)

wood (overall material)

metal (overall material)

white (overall color)

red (overall color)

black (overall color)

copper (overall color)

bolted and threaded. (overall production method/technique)

Measurements

average spatial: 13.9 cm x 25.3 cm x 24.8 cm; 5 15/32 in x 9 31/32 in x 9 3/4 in

See more items in

Medicine and Science: Mathematics

Science & Mathematics

National Museum of American History

Subject

Mathematics

Record ID

nmah_1086171

Metadata Usage (text)

CC0

GUID (Link to Original Record)

https://n2t.net/ark:/65665/ng49ca746a9-5665-704b-e053-15f76fa0b4fa