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On Geometry

By Steve Humphrey

 

With Real Estate as one of the themes for the May issue, it made me think of houses, land, property lines – and geometry. For thousands of years, the Nile would flood annually, erasing all landmarks indicating property lines. Because of the flooding, the soil near the river was very fertile, and thus highly valued. To avoid fights between neighboring landowners, a method had to be devised to reestablish the old property lines, which Egyptian surveyors developed using back-of-the-papyrus, rough-and-ready rules. 

Centuries later, the Greek mathematician Euclid (323 BCE – 283 BCE) made these insights rigorous, and invented Geometry which literally translated means, “measure of the Earth.” He wrote 13 books which collectively came to be known as the “Elements,” which has been called the most successful and influential textbook ever written. In that treatise, he began with five intuitively obvious postulates, and, using only deductive logic, proved hundreds of theorems that had practical application in the study of planes and solids. For example, his first postulate can be stated as “two points define a straight line.” For reference, postulates, sometimes called “axioms,” are claims that are so obviously true that they don’t need proof or even evidence.

One postulate, the fifth, called the “Parallel Postulate,” can be expressed as follows: given a straight line and a point not on that line, exactly one straight line can be drawn through that point parallel to the given line. Picture a line drawn in the dirt in your garden and a plant near that line. How many lines can you draw through that plant (don’t kill it!) that are parallel to the line? For Euclid, the answer is “one.” This postulate is much more complicated than the others and was regarded as problematic almost immediately. Great effort was expended in attempting to show that it could be derived from the other postulates, thus making it a proved theorem, rather than an assumed axiom. These efforts were unsuccessful, so geometers in the 19th century, such as Riemann, Gauss and others, turned their attention to proving that it was independent of the other postulates. That is, could the first four postulates get along with different versions of the fifth, or would they fight? Two such contrary postulates were formulated. One, more than one line can be drawn parallel to the given line, and two, no such line can be drawn. It was shown that both resulting “Non-Euclidean” geometries were internally consistent.

The first is called “Hyperbolic Geometry” and can be illustrated by the surface of a horse’s saddle, on which the straightest lines possible are hyperbolas. The second is called “Spherical Geometry,” and is exemplified by the surface of a sphere, like the Earth. The “straight lines” on a sphere are the “great circles,” like the equator and the lines of longitude, and these are the shortest distance between any two points on the surface. This explains why a flight from Los Angeles to London goes over the North Pole, and not directly East. Think about the lines of longitude intersecting at both poles. In fact, any two great circles will meet twice. 

One consequence of the consistency of Non-Euclidean geometries is that determining the actual geometry of the Universe is a matter of empirical investigation, rather than simple mathematical calculation. One way to do this would be to measure the internal angles of a triangle, say as determined by three stars, or beacons on the peaks of three mountains. In Euclidean space, those angles add up to 180 degrees. In spherical geometry, they add up to more than 180, and in hyperbolic geometry, less than 180. As Einstein showed, the Universe we live in has a Non-Euclidean geometrical structure. 

Did you realize there was so much historical math and science involved in real estate? Beginning with the flooding of the Nile, we eventually get to two of the most significant theories about the physical world, thus explaining how current-day property lines came to be determined today.

Steve Humphrey has a Ph.D. in the history and philosophy of science, with a specialty in the philosophy of physics. He teaches courses in these subjects at the University of California, Santa Barbara and has taught them at the University of Louisville.