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# GEOMETRIES OTHER THAN EUCLID’S PART 2

YESTERDAY, WE left geometry in a quandary. Mathematicians were trying to accommodate Euclid’s Fifth Postulate, the one about parallel lines, into 19th-century goals of mathematical rigor. Could it follow from the other assumptions? Could this postulate be modified or even left out entirely?

In The Mathematical Experience, authors Philip J. Davis and Reuben Hersh note the pitfalls of seeking a non-Euclidean geometry: “It seemed to be on the edge of madness. The birth pains were severe.”

The Mathematical Experience, by Philip J. Davis and Reuben Hersh, Birkhäuser, 1995, reprinted 2002.

“It was found,” they write, “that there are not one but two non-Euclidian geometries. They currently go by the names of Lobachevskian (or hyperbolic) geometry and Riemannian (or elliptical) geometry.”

Lobachevsky’s Fifth. Russian mathematician Nikolai Ivanovich Lobachevsky replaced Euclid’s Fifth by allowing at least two parallels through that given point off that given line. He announced the logical consistency of this possibility in 1826. His fundamental work, Geometrical Researches on the Theory of Parallels was published in 1840.

Nikolai Ivanovich Lobachevsky, 1793–1856, Russian mathematician, one of the founders of non-Euclidian geometries.

Lobachevsky is perhaps familiar from satirist Tom Lehrer’s wonderful Николай Иванович Лобачевский Ой!.

Riemann’s Fifth. German mathematician Bernhard Riemann took the Fifth in the opposite direction: Given a line, he posited there are no parallels through that point off that line. In 1854, Riemann presented his paper On the Hypotheses which Underlie Geometry at the University of Göttingen.

Georg Friedrich Bernhard Riemann, German mathematician, one of the founders of non-Euclidean geometries.

Riemann’s discoveries in geometry, algebraic geometry, and complex manifold theory are still being applied in innovative ways in mathematical physics. The Riemann curvature tensor describes curvature of space, a fundamental aspect of Einstein’s Theory of Relativity.

A lattice analogy of the deformation of spacetime caused by a planetary mass. Image by mysid.

Three Possible Worlds: Euclidean, Lobachevskian, and Riemannian. Davis and Hersh assemble a helpful table describing the three world models portrayed by these three different geometries.

If you’re fooling in our traditional world as perceived by Newton, the Euclidean model is just fine. If you’re into Einstein’s relativistic view, the Riemannian model is appropriate. If you’re savoring geometry purely as an intellectual endeavor, try Lobachevsky. Or at least enjoy Tom Lehrer’s satiric tribute. Ой! ds