Geometry on a sphere and beyond
Tonight's idea
Euclid built geometry from five assumptions. Four are modest. The fifth is equivalent to this: through a point not on a line, there is exactly one line parallel to it. For two thousand years people tried to prove it from the other four. Bolyai and Lobachevsky saw why they failed: there are consistent geometries where it is false.
You are standing on one. On a sphere, the straightest path is a great circle, whose centre is the centre of the sphere: the equator, any meridian, the route a plane flies from London to Tokyo. Great circles are the sphere's "lines", and any two of them meet. There are no parallels at all.
So the theorems change. Start at the North Pole, walk down a meridian to the equator, turn 90°, walk a quarter of the way round the equator, turn 90° again and walk back to the pole. That triangle has three right angles. On a sphere the angles of a triangle always add to more than 180°, and the excess (angle sum minus 180°) is proportional to the area. That is Girard's theorem: for angles \(\alpha, \beta, \gamma\) in radians on a sphere of radius \(R\),
\(\text{Area} = R^2(\alpha + \beta + \gamma - \pi)\). In degrees: angle sum \(= 180° + 720° \times \dfrac{\text{area}}{\text{area of sphere}}\).
Check it: excess \(\frac{\pi}{2}\), area \(\frac{\pi}{2}R^2\), exactly one eighth of the sphere's \(4\pi R^2\). There is also a hyperbolic geometry with infinitely many parallels and triangles adding to less than 180°. Einstein's relativity says mass curves space: the fifth postulate is a question for physics.
Watch one: how big is a triangle with angles 90°, 90°, 45°?
Earth, \(R = 6371\) km. Vertices: the North Pole and the points on the equator at longitudes 0° and 45° E. The equator angles are right angles, and the angle at the pole is the longitude difference, 45°.
Excess \(= 90° + 90° + 45° - 180° = 45° = \frac{\pi}{4}\). Area \(= 6371^2 \times 0.7854 \approx 3.19 \times 10^7\) km². Check: the sphere's area is \(4\pi R^2 \approx 510\) million km², and \(\frac{\pi/4}{4\pi} = \frac{1}{16}\) of it is \(31.9\) million km².
Your turn
- Which of these are "straight lines" (great circles) on the Earth: the equator; the Arctic Circle; the Greenwich meridian; the Greenwich and 180° meridians together; the Tropic of Cancer? For each non-line, say why not.
- A triangle has vertices at the North Pole and at two points on the equator 120° of longitude apart. Find its three angles, its angle sum, the fraction of the Earth's surface it covers, and its area in km² with \(R = 6371\) km.
- Can a spherical triangle have angle sum 540°? 541°? Use Girard's theorem and the fact that a triangle fits inside a hemisphere (area less than \(2\pi R^2\)) to find the full range of possible angle sums.
- A lune is the region between two meridians, with angle \(\theta\) at each pole (an orange segment), \(\theta\) in radians. Explain why its area is \(\dfrac{\theta}{2\pi} \cdot 4\pi R^2 = 2\theta R^2\). What fraction of the sphere is the lune with \(\theta = 90°\)?
Stretch: rectangles
Split a quadrilateral along a diagonal into two triangles. Add the two triangles' angle sums to decide whether a quadrilateral with four right angles can exist on a sphere, and whether one can exist in the hyperbolic plane. What does that say about Euclid's fifth postulate?
Answers
- Great circles: the equator; the Greenwich meridian (half of the great circle it shares with the 180° meridian); the two meridians together form one complete great circle. Not great circles: the Arctic Circle and the Tropic of Cancer, whose centres are on the Earth's axis rather than at the Earth's centre; following one means constantly turning.
- Angles 90°, 90°, 120°; sum 300°; excess 120°. Fraction \(= 120/720 = \frac16\) of the sphere. Area \(= R^2 \cdot \frac{2\pi}{3} \approx 8.50 \times 10^7\) km², about 85 million km².
- Angle sum \(= 180° + 720° \times (\text{area}/4\pi R^2)\), and the area is strictly between 0 and \(2\pi R^2\), so the sum is strictly between 180° and 540°. 540° is the unreachable limit; 541° is impossible.
- A lune of angle \(\theta\) is the fraction \(\theta/2\pi\) of the full turn around the axis, so it takes that fraction of the sphere's area: \(2\theta R^2\). With \(\theta = \frac{\pi}{2}\): \(\pi R^2\), one quarter of the sphere.
Stretch. On a sphere each triangle has angle sum above 180°, so the quadrilateral's sum exceeds 360° and four right angles are impossible. In hyperbolic geometry each sum is below 180°, so the total is below 360°: also impossible. Rectangles exist only in Euclidean geometry, and their existence is equivalent to the fifth postulate.