Women are slowly becoming more represented in the area of mathematics. Women are also writing, publishing, and being awarded more for their contributions to the study of mathematics. Carolyn Gordon is an example of a successful female mathematician who has made strides in the field of geometry.
Carolyn Gordon was born on December 26, 1950. She received her Ph.D. from Washington University in 1979. Her main area of interest concerning research is Riemannian Geometry. Carolyn has published over 45 papers and served in many professional organizations such as the American Mathematical Society (Curriculum Vitae). Carolyn is also married to a fellow mathematician and researcher, David L. Webb (ams.org 1).
Carolyn is very well known for working on a question first posed by Mark Kac, can you hear the shape of a drum?" A drum essentially makes sound because of a membrane (the part one beats on) that is stretched across some frame. When one beats on a drum, the membrane vibrates causing a spectrum of frequencies. The spectrum of frequencies can also be called the set of pure tones or the normal-modes (maa.org 1).
"Physicists and mathematicians have long recognized
that the shape of a boundary enclosing a membrane plays a crucial role in determining the membrane’s spectrum of normal-mode vibrations" (maa.org 1)
Kac began wandering whether the normal-mode vibrations was sufficient enough to determine a drum’s shape, hence the question can one hear the shape of a drum. It had been known that one could "hear" the area and perimeter of a drum and in 1991, Carolyn Gordon, David Webb, and Scott Wolpert proved that the answer to Kac’s question is now and different shaped drums can make the same sound.
Carolyn was one of three that helped find a pair of different geometric shapes that have the same normal-mode frequencies when though of as drums. Gordon was at Washington Univeristy when the discovery was made and the mathematics department had T-shirts printed with the proof showing that the drums sound alike on them. While the proof was simple, finding the drums was not. Carolyn and her colleagues had to make use of many fields of mathematics like partial differential equations and geometrical analysis (ams.org 2).
Carolyn’s drums are "mathematical drums." A mathematical drum is a shape in a plane that has an interior bounded by something like a square, polygon, or circle. The sounds produced by the drum are solutions to a wave equation, which is used to describe any that that moves in waves like sound. The interior of the drum vibrates while the boundary, or shape the membrane is stretched across, determines the actual frequencies given off from the drum. Since the frequencies determine the sound of a drum and the boundary determines the frequencies, the shape determines the sound (ams.org 2).
Gordon and her colleagues proved that while shape determines sound, sound does not determine the shape of a drum. The drums that Gordon and her colleagues found are, in their most simple shape, eight sided polygons (amath 1).
The proof demonstrating that these drums sound alike is demonstrated by Figure 3 (picture from www.ams.org/new-in-math/hap-drum/fig16.html).
"Figure 3 shows how to recombine pieces A–G of a standing wave on the seven half-crosses of the first drum into a standing wave on the second drum" (ams.org 3). In other words, when A–G are rearranged, they make a solution to the wave equation that would be the same as on the second drum. By rearranging the pieces of the polygon, the drums can be made so that they give identical solutions to the wave equation, meaning their frequencies are the same.
Carolyn Gordon’s work on whether sound determines the shape of a drum has given rise to lots different questions that mathematicians are now working on like are there certain properties that make drums sound alike (ams.org 2). Carolyn Gordon is further proof that women contribute to the study of mathematics and can be an inspiration to other female mathematicians.