Yanlin Yang, a Franklin & Marshall senior, had taken Professor of Mathematics Annalisa Crannell’s course in perspective geometry when she decided to embark on a project that has challenged human calculating since the advent of the ruler and compass.
“The research is about the spiral,” Yang said. “We tried to prove what can and cannot be drawn in order to prove that the spiral cannot be drawn with the classic mathematic tools, the ruler and compass.”
Crannell said perspective is about imagining a real three-dimensional world viewed through a two-dimensional window.
“What kind of things in the real world can you draw using the traditional Greek tools of ruler and compass?” Crannell said. “Lines and circles, obviously. A spiral is a circle that just keeps going up like a line, so you might think you can draw it with a ruler and a compass, but you can’t.”
Spirals cannot be drawn because Pi, the ratio of a circle’s circumference to its diameter, is needed in the process, Yang said. “You can’t draw Pi by the traditional Greek ruler and compass.”
One of the questions that “tortured Greeks and other geometers for centuries, even millennia, was, ‘If you have a square with a certain area, can you draw a circle with the same area?’” Crannell said. “This was such as big problem that instead of calling people ‘eggheads’ or ‘absent-minded professors,’ they called them ‘circle-squarers,’ for trying to do this impossible thing.”
Pi is a transcendental number, meaning it cannot solve a non-constant polynomial equation, which describes a form, Crannell said. “Because it’s a transcendental number, you can’t prove it in geometry,” she said. “And it turns out to be the same reason you can’t draw a spiral.”
For Yang, the research was enjoyable but challenging, sort of like being on a roller coaster, she said. “You finally figure something out one day and the next day you are disappointed,” Yang said. “But you never give up.”
Spirals can be drawn without ruler and compass, but those are only in approximations. “A computer can draw a really good approximation,” the professor said. “You can draw incredible approximations to spirals.”
Why is this important to the world? For now, that question is unknown.
“Most people don’t ask what use is a poem. They understand that it exists to illuminate, to be informative,” Crannell said. “The kind of math that we are doing is like that. It is wonderful to know the limits of logic, the limits of puzzle solving because it tells us something about the mathematical structure. And it might be useful someday.”