#
Women Mathematicians and NMAH CollectionsFrances Baker: Daughter of a Mathematical Model Maker

Image of Frances Baker, ca 1957. Gift of Frances E. Baker. (2006.3037.02) |

Frances Ellen Baker (1902–1995) is one of many women mathematicians who had a close relative who was also a mathematician. Her father was the England-born mathematician Richard Philip Baker, who is best known in the mathematical community for constructing mathematical models. Frances Baker spent most of her career teaching at women’s colleges, Mount Holyoke and Vassar. She was particularly involved with honor students, both by directing honor’s papers and through local chapters of the honor societies Phi Beta Kappa and Sigma Xi.

Questionnaires for both Bakers are preserved in the collection honoring American women in mathematics. In addition, the mathematics collections also include several objects donated by Frances Baker. Among these are the two doctoral hoods from the University of Chicago that she and her father received when they were awarded their PhD’s and a large number of mathematical models constructed by her father.

Riemann Surface. R. P. Baker Model #411z (w^{3}=z). Gift of Frances E. Baker. |

"Women Mathematicians and NMAH Collections - Frances Baker: Daughter of a Mathematical Model Maker" showing 3 items.

## Model of a Riemann Surface by Richard P. Baker, Baker #411z

- 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 100 of his models are in the museum collections.

- The mark 411 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. 411z (/) Riemann surface : w
^{3}= z. Model 411z is listed on page 17 of Baker’s 1931 catalogue of models as “w^{3}= z” under the heading*Riemann Surfaces*. The catalog description also notes that “411 is to serve as a first step to 412,” where Baker model 412z (MA*211157.075) is associated with a more complicated equation involving w^{3}.

- The model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w
^{3}= z where a complex number is 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 z after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex z-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 w-plane with the point w = 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 an angle of 30 degree. The front of the model is the edge on which 411 is inscribed so the two vertical rectangles lie above the polar axis, i.e. the ray emanating from the origin when the angle is 0 degrees, of the wooden base. This places every horizontal edge of the rectangles on a polar axis of a sheet.

- If z = 0, the equation w
^{3}= z is satisfied by only one value of w, i.e., w = 0. The point z = 0 is called a branch point of the model and for all other points on the z-plane the equation w^{3}= z is satisfied by three distinct values of w, each of which produces a different pair on the Riemann surface (if z = 1, the three distinct pairs on the Riemann surface are (1,1), and (1,(–1 ± √3 i)/2)). Thus there are three sheets representing the same disc in the z-plane and together they represent part of what is called a branched cover of the complex z-plane.

- Baker’s use of solid red circles, and dashed red and black circles indicates that each sheet is mapped continuously onto a different portion of the w-disk on the base. There are three radii of the disk on the base (the polar lines - rays emanating from the origin – for angles of 0, 120, and 240 degrees) that are the edges of sectors corresponding to quadrants on two different sheets. The order of the colors of the 30 degree sectors on the base starting at polar axis and proceeding counterclockwise correspond to the colors of the first through fourth quadrants of the top, middle, and then bottom sheets.

- The vertical rectangles mentioned above 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 the 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. While the defining equation determines branch points, branch cuts are not fixed by the equation. However, the single branch cut for any surface with only one branch point must run from that point out to infinity. The branch cut of this model is represented on each sheet by the horizontal edges of the vertical surface or surfaces meeting that sheet.

- Location
- Currently not on view

- maker
- Baker, Richard P.

- ID Number
- MA*211257.074

- accession number
- 211257

- catalog number
- 211257.074

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Academic Hood of Frances Ellen Baker

- Description
- This hood belonged to Frances Ellen Baker (1902–1995), who received her PhD in mathematics from the University of Chicago in 1934. The color of the velvet on the hood represents the type of doctorate awarded, with dark blue used for the degree of Doctor of Philosophy. The color of the reverse (interior) side of the hood, maroon, represents the school that awarded the degree, i.e., the University of Chicago whose colors are maroon and white. A matching hood (1985.0820.02) was awarded to Baker’s father, Richard Philip Baker (1866-1937), when he received his PhD in mathematics from Chicago in 1910.

- Frances Baker’s PhD dissertation,
*A Contribution to the Waring Problem for Cubic Functions*, was directed by L. E. Dickson, the first doctoral student of E. H. Moore, who directed her father’s PhD dissertation,*The Problem of the Angle-Bisectors*(1985.3145.01). Her younger sister, Gladys Elizabeth Baker (1908–2007) earned a doctorate in botany and mycology from Washington University in St. Louis in 1935.

- Frances Baker spent most of her career, 1942–68, teaching mathematics at Vassar College. She came to Vassar two years after her sister Gladys arrived there to teach botany. There Frances Baker directed several honors papers and served as an officer of the local chapters of the academic honor society, Phi Beta Kappa, and the science honor society, Sigma Xi.

- Location
- Currently not on view

- date made
- 1934

- date used
- 1934

- user
- Baker, Frances E.

- ID Number
- 1985.0820.03

- accession number
- 1985.0820

- catalog number
- 1985.0820.03

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Academic Hood of Richard P. Baker

- Description
- This hood belonged to to Richard Philip Baker (1866–1937) who received his PhD in mathematics from the University of Chicago in 1910. The color of the velvet on the hood represents the type of doctorate awarded, with dark blue used for the degree of Doctor of Philosophy. The color of the reverse (interior) side of the hood, maroon, represents the school that awarded the degree, i.e., the University of Chicago whose colors are maroon and white. A matching hood (1985.0820.03) was acquired by Baker’s daughter, Frances Ellen Baker (1902–1995), when she was awarded a PhD in mathematics from Chicago in 1934.

- R. P. Baker’s doctoral dissertation,
*The Problem of the Angle-Bisectors*(1985.3145.01), was directed by E. H. Moore, while his daughter Frances’s doctoral dissertation,*A Contribution to the Waring Problem for Cubic Functions*, was directed by L. E. Dickson, E. H. Moore’s first doctoral student. R. P. Baker’s younger daughter, Gladys Elizabeth Baker (1908-2007) earned a doctorate, in botany and mycology from Washington University in St. Louis in 1935.

- R. P. Baker is best known in the mathematical community for constructing mathematical models that he believed were necessary for the proper teaching of geometry. His 1931 catalog offered several hundred models. Several museum accessions include models made by Baker. See MA*211257.04 for a description of one of these models.

- Location
- Currently not on view

- date made
- 1910

- date used
- 1910

- user
- Baker, Richard P.

- ID Number
- 1985.0820.02

- accession number
- 1985.0820

- catalog number
- 1985.0820.02

- Data Source
- National Museum of American History, Kenneth E. Behring Center