Ivars Peterson's MathTrek

September 2, 1996

# Beyond the Ellipse

There's a simple trick one can use to draw an ellipse. Tie the ends of a length of string to two pins (or thumbtacks) stuck in a sheet of paper on a drawing board. Then, keeping the string taut with the point of a pencil, allow the pencil to trace a path around the pins. The resulting curve is an ellipse, with the two pins, or fixed points, representing its foci.

Drawing an ellipse.

This drawing method takes advantage of the geometric fact that the sum of the distances from the foci of an ellipse is the same for all points on the curve. Thus, If A and B are the foci of an ellipse, the total distance (AP + PB) from the foci to any point P on the curve is constant.

One can also ask what curve results when the total distance from three given points is kept the same. For example, suppose that the pins are placed at the corners of an equilateral triangle. In this case, it's not possible to draw the figure using a pencil and string because the pins would end up getting in the way of the string. However, one can explore this possibility using a computer.

That's precisely what Bilge Demirkoz, a 16-year-old high school student in Istanbul,Turkey, did to investigate what happens for not only three but also four fixed points. She presented a guided tour of her findings in this neglected corner of mathematics one evening at the Seattle Mathcamp (see last week's MathLand article, Math Camp).

In the case of three points, her computer plots showed an oval figure that was obviously neither a circle (one fixed point) nor an ellipse (two fixed points). She called the result a trisoid. Its precise form depended on the given total length and the relative positions of the three given points. Four fixed points led to another curiously rounded shape.

Example of a trisoid.

When Demirkoz looked for the geometric form that results when the sum of the distances from two points minus the distance from a third point is kept constant, she found additional surprises. In this case, depending on the chosen constant distance, the figure has an outer boundary that looks somewhat like an ellipse (though it isn't) and sometimes has an inner boundary -- a hole -- that looks like a circle (but isn't).

These computer explorations represent just the first stage in a potentially rewarding mathematical investigation. They raise a variety of questions and prompt a number of conjectures about the characteristics and behavior of these curves.

Such results may even have relevance to physical systems, Demirkoz notes. For example, elliptical orbits arise when one body orbits another under the influence of gravity. It's possible that certain gravitational or electric fields could lead to these other kinds of orbits.

In Mathematics: Queen and Servant of Science, Eric Temple Bell writes: "A circle no doubt has a certain appealing simplicity at first glance, but one look at an ellipse should have convinced even the most mystical of astronomers that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has little to say. Possibly our own search for cosmic simplicities in the physical universe is of this circular kind -- a projection of our uncomplicated mentality on an infinitely intricate external world."

Perhaps the trisoid and its geometric cousins illuminate a small corner of the cosmic complexity.