A Model for Elliptic Geometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

In a famous lecture in 1854, Bernhard Riemann constructed an infinite family of non-Euclidean geometries, giving a formula for a set of metrics on the unit ball in Euclidean space. The simplest of Riemann’s metrics gives rise to elliptic geometry.

Pleading the Fifth

Euclid’s fifth postulate, in the form known as Playfair’s axiom, states that, for any given line ${\ell}$ in a plane and any point ${P}$ not on ${\ell}$ , there is exactly one line through ${P}$ that does not intersect ${\ell}$ . In hyperbolic geometry, there are infinitely many lines through ${P}$ not intersecting ${\ell}$ , while in elliptic geometry, all lines through ${P}$ intersect ${\ell}$ .

The distinction can be illustrated by considering two straight lines in a two-dimensional plane that are both perpendicular to a third line (see Figure above). In Euclid’s geometry, the two lines remain equidistant from each other (they are parallel). In hyperbolic geometry, they veer away from each other, while in elliptic geometry, they curve toward each other and intersect.

The fifth postulate is more complicated — and more cumbersome — than Euclid’s first four postulates:

To draw a straight line from any point to any point.
To extend a finite line segment to a straight line.
To describe a circle with any centre and radius.
That all right angles are equal to one another.

Euclid appears to have been uncomfortable with the fifth postulate. The first 28 propositions of the Elements do not use the parallel postulate or anything equivalent to it. Geometry based on only the first four postulates is described as absolute geometry.

Models

Planar Euclidean geometry arises from our concept of a flat plane. We are so familiar with the points and lines in this plane, that we rarely think of it as a ‘model’ of Euclidean geometry. But with non-Euclidean geometries, which clearly do not apply to the usual points and lines in a plane, it is far from obvious that there is any circumstance in which such a geometry might apply.

The first model for hyperbolic geometry was provided in 1868 by Eugenio Beltrami, who showed that the surface of a pseudosphere has the required negative curvature to model a portion of hyperbolic space.The obvious model for elliptic geometry is a sphere on which great circles are the “lines”. However, it is clear that two great circles intersect in two antipodal points: just think of the equator and the great circle comprising the Greenwich meridian and the date-line. Again, thinking geographically, we see that all meridians pass through both the North and South poles. In Euclidean geometry, there is always a unique line passing through any two distinct points.

If we want a system that respects Euclid’s first four postulates, two ‘lines’ must intersect in a single ‘point’. This can be accomplished by identifying pairs of antipodal points; so, a ‘point’ in this model is actually a pair of points. Since each such pair of points lie on a radial line through the origin, we can associate the set of ‘points’ with the set of lines through the origin. This set of lines is known as the real projective plane ${\mathbb{P}^2}$ . However, there is no metric defined for the projective plane while in elliptic geometry a metric is defined enabling lengths and angles to be measured.

The elliptic distance between two points ${A}$ and ${B}$ is the angle ${AOB}$ where ${O}$ is the centre of the sphere. In Euclid’s plane, two triangles can have equal angles but different areas: they are similar but not conjugate. On the sphere, this is no longer true. Indeed, the area is determined by the sum of the angles. And this sum is always in excess of two right angles.

So long, Pythagoras

Image: http://www.quora.com

The triangle with vertices at the geographic locations ${(0^\circ \mathrm{N},0^\circ \mathrm{E})}$ , ${(0^\circ \mathrm{N},90^\circ \mathrm{E})}$ and ${(90^\circ \mathrm{N},0^\circ \mathrm{E})}$ has three right angles. All sides are of the same length, and it is clear that the Pythagorean theorem cannot hold. If we denote the sides by ${a}$ , ${b}$ and ${c}$ , we have

$\displaystyle \cos c = \cos a \cos b$

The result ${c^2 = a^2 + b^2}$ is recovered in the limit of small triangles.

We note that the spherical triangle shown in the figure is only “half the story”. In elliptic geometry, it is identified with the antipodal triangle, having vertices diametrically opposite those of the triangle shown.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

>> Review in The Irish Times <<

* * * * *

ThatsMaths