The last chapter for **Solid Geometry and Higher Dimensions** is on **Non-Euclidean Geometry**. Gardner starts by talking about **Euclidean Geometry**, and how Euclid was smart enough to recognize that his fifth postulate was different from the first four. That is, the statement “through a point on a plane, not on a given straight line, only one line is parallel to the given line” can not be proven without relying on some other approach that is derived from the parallel postulate. That is, if you assume that the sum of the angles of every triangle equals two right angles, you can not prove this assumption without using the parallel postulate as well.

Gardner goes on to talk about Hungarian mathematician Farkas Bolyai, who, in his attempt to prove the parallel postulate in the early 1800’s, not only realized it was independent of the first 4 axioms, he went on to show that a consistent geometry could be created assuming that through a point you could have an infinite number of parallel lines. Unfortunately, Farkas waited too long to publish his work, and Nikolai Lobachevski beat him to it. Even Farkas’s friend, Gauss, claimed to have worked out the same findings several years earlier and never bothered to publish them. Then there’s Italian Jesuit Giralamo Saccheri, who worked out both forms of non-Euclidean geometry in a Latin book published in 1733, titled “**Euclid Cleared of All Blemish**.” The problem was that claiming that non-Euclidean geometry was as true as the Euclidean form would have been very dangerous at that time, so he published the book while simultaneously denouncing his own findings

If the parallel postulate is altered to have infinite parallel lines through a point, you get a hyperbolic space where all triangles have sums of their angles that add up to less than 180 degrees, and the sum decreases as the triangle gets bigger. The circumference of a circle is greater than pi times the diameter, and the measure of curvature of a plane is negative. The other type of geometry, called “elliptic,” was simultaneously developed by Ludwig Schlafli and Bernhard Riemann (yes, the zeta function Riemann). In this geometry, NO parallel line can be drawn through a point to the given line. Here, the sums of the angles of the triangle add up to more than 180 degrees, and the circumference of a circle is less than pi times the diameter.

One of the last mathematicians to doubt non-Euclidean geometry was Lewis Carroll. H.S.M. Coxeter is quoted as saying “It is a strange paradox that he, whose **Alice in Wonderland** could alter her size by eating a little cake, was unable to accept that the area of a triangle could remain finite when its sides tend to infinity.” Gardner presents M. C. Escher’s **Circle Limit III** woodcut as an example of what Coxeter was talking about for hyperbolic space. In fact, Escher based **Circle Limit III** on a 1957 paper on crystal symmetry that Coxeter wrote and sent him.

(**Circle Limit III**, from the wiki entry. All rights belong to their owners. Images used here for review purposes only.)

Note that in this woodcut, the lines running through the fish are perpendicular to the circle’s edge. If you walked along one of these lines, every fish would be of exactly the same size. The thing is, you’d be getting smaller as you got closer to the edge of the circle, as would any measuring stick you used to check the size of the fish. You’d never get to the edge of the circle, because there is no “edge” from your perspective. Both you, and the fish, just keep getting infinitely smaller as you go (George Gamow talked about curved space in exactly this way in one of his books).

Elliptic geometry then can be modeled on the surface of a sphere. Euclidean straight lines become great circles, and no two are parallel. Einstein adopted a generalized form of this geometry, formulated by Riemann, to show that the curvature of physical space “varies from point to point depending on the influence of matter.” According to Gardner, one of the greatest revolutions in physics that came about from the General Theory of Relativity was that physics can be simplified by assuming physical space to have an elliptic structure.

The rest of the chapter is dedicated to “cranks” that were convinced that Einstein was wrong, that you can’t have non-Euclidean space, and that they were sure they’d proved Euclid’s parallel postulate.

**Challenge:** Prove the parallel postulate.

It’s ok, I’m willing to wait.