Stories of Mathematics

Geometry of a Spherical City


by Corina Murg

Imagine a tiny, livable planet has just been discovered. The air is clean and the grass is greener. It will be colonized immediately and become humanity’s first City Planet. City planners are hired and space fares are being negotiated as we speak. YOU are appointed Chief Mathematician. You are given a week to construct a geometry system for your planet. After some measurements, you find out that the shape of the planet resembles a sphere with radius of approximately 5 miles. What type of geometry would best fit your new planet?

Do NOT panic. Bernhard Riemann designed the perfect geometry for your spherical city. You are free to take advantage of a concept that revolutionized mathematics: the non-Euclidean geometry.

The non-Euclidean geometry stemmed from the independent work of three mathematicians, Karl Friedrich Gauss, Nicholas I. Lobachevsky and Janos Bolyai. Ironically, each one of them had started their work on non-Euclidean geometry by trying to prove Euclid’s controversial Postulate 5:

“… if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

From high school geometry class, most of us are familiar with the version of Postulate 5 developed by John Playfair, known as Playfair’s Axiom: Through any point in the plane, there is only one straight line parallel to a given straight line.

The postulate was controversial among mathematicians because it referred to what happens at “infinity,” way beyond our physical space. In the end, Gauss, Lobachevsky and Bolyai reached the conclusion that the Euclidean parallel axiom cannot be proved. In fact, it is only one of three alternatives, the other alternatives being that there is no line, or that there are at least two lines which pass through any given point and are parallel to a given straight line.

These two latter alternatives are the foundations for the two main branches of non-Euclidean geometry: hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry (after mathematician Bernhard Riemann who developed it). In hyperbolic geometry, for a straight line L and any point N not on it, there are many other straight lines that pass through  N and do not intersect L.

The geometry system you are setting up for your planet should rely on the Riemannian geometry which states that there are NO parallel lines to a given line. The surface of your planet is curved, so there could be no straight lines. You might be wondering how one will find then the shortest distance between any two points A and B, as in Figure 1. The distance between two locations on your planet will be an arc. The shortest arc connecting A and B lies on what is called a great circle, a circle whose center is the center of your planet and passes through A and B.

Pic1

 It is obvious now why there will be no parallel lines on your new planet: any two “straight lines” (ie great circles) will intersect at two points, as in Figure 2 below. Of course, every “straight line” will have the same finite length, the length of a great circle’s circumference.

While in Euclidean geometry two points determine a unique line, in your planet’s geometry this will be possible only if the two points are NOT the opposite ends of a diameter, like U and V in Figure 2. If they are opposite ends of a diameter, like R and V, then there exists a multitude of “lines” (great circles) passing through the two points.

Pic2

Using your geometry system, your planet’s planners will have the freedom to envision neighborhoods and street plans of shapes and designs unheard of on planet Earth. Any triangular shape will be determined by the intersection of three arcs. However, the sum of angles of a triangle will always be greater than 1800. It will reach 1800 only as the area approaches zero. Take for example triangle NMP in the figure below. Two of the angles are right angles; therefore the sum of all three angles is at least 1800.

Pic3

An entire neighborhood could be the shape of a two-sided polygon called a lune, with streets running along each side of the neighborhood. In effect, this will create a grid type street plan, efficient for both pedestrians and drivers.

Pic4

Latitudinal roads could be added to divide neighborhoods into smaller blocks (Figure 5), and since all perpendiculars to a “line” (ie great circle) on the sphere meet in one point, all roads on your planet will lead to … either pole of your city.

Welcome to City Planet!

 

 

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