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You learned that the angles of a triangle add up to 180°. That is true on a flat page. Draw a triangle on a globe and the sum can be 270°. Bernhard Riemann asked what this means for the space we actually live in, and his answer became the language of Einstein's theory of gravity. This post tells his story and works through the globe triangle.
Who was Bernhard Riemann?
According to his MacTutor biography, Riemann was born on 17 September 1826 in Breselenz, in the Kingdom of Hanover, the son of a Lutheran minister. At school the director lent him Legendre's book on number theory, about 900 pages, and he read it in six days.
He entered the University of Göttingen in 1846 to study theology, as his father wished, but asked his father's permission to switch to mathematics, and got it. He spent two years in Berlin, where Dirichlet's style of clear, intuitive reasoning shaped his own, then returned to Göttingen. His 1851 doctoral thesis, supervised by Carl Friedrich Gauss, introduced what are now called Riemann surfaces. Gauss's report praised its "gloriously fertile originality."
What happened at the 1854 lecture?
To qualify as a lecturer, Riemann had to give a trial lecture. He offered three topics, two on electricity and one on geometry, and Gauss chose geometry, against Riemann's expectations. On 10 June 1854 he delivered "On the hypotheses which lie at the foundations of geometry."
The lecture did two things. First, it described how to set up a space of any number of dimensions with a rule for measuring distance at each point, what we now call a Riemannian space, and defined its curvature. Second, it asked what the dimension of real space is and which geometry actually describes it, questions MacTutor says were too far ahead of their time for most scientists to appreciate. MacTutor quotes an account that Gauss, returning to the faculty meeting, spoke about the lecture with rare enthusiasm. The text was printed only after Riemann's death; William Kingdon Clifford's English translation appeared in Nature in 1873, in two parts.
How can you tell a surface is curved from inside it?
This is the heart of the idea. The historian Hans Freudenthal, quoted by MacTutor, put it this way: beings living on a surface could discover its curvature from deviations from Pythagoras's theorem. You do not need to step outside the surface. Measuring triangles is enough.
Worked example: a triangle on a sphere. Take a sphere of radius R. Start at the north pole, walk straight down a line of longitude to the equator, turn and walk a quarter of the way round the equator, then turn and walk straight back up to the pole.
At each corner on the equator, the longitude line meets the equator at 90°.
At the pole, the two longitude lines are a quarter turn apart, so the angle is 90°.
Angle sum: 90° + 90° + 90° = 270°.
The excess over a flat triangle is 270° − 180° = 90°, or π/2 radians. Now find the area. The triangle covers a quarter of the northern hemisphere, which is one-eighth of the sphere: 4πR² ÷ 8 = πR²/2. Divide by R² and you get π/2, exactly the excess.
This is a general rule for spheres: angle excess = area ÷ R². Tiny triangles have tiny excess, which is why a football field looks flat even though the Earth is round. Riemann's insight was that curvature can be measured this way in any number of dimensions, and that it can change from place to place.
How did Riemann's geometry reach physics?
MacTutor notes that the lecture was not fully understood for about sixty years. Then Albert Einstein needed a way to describe gravity as the curving of space and time. In his 1916 paper, The foundation of the general theory of relativity, he built the theory on the geometry descended from Riemann's lecture. As Freudenthal wrote, quoted by MacTutor, in the mathematical apparatus developed from Riemann's address, "Einstein found the frame to fit his physical ideas."
What else did Riemann leave behind?
In 1859, elected to the Berlin Academy, Riemann sent a short report on the number of primes below a given size. In it he studied the zeta function and suggested that its non-obvious zeros all have real part one half. This is the Riemann hypothesis. The Clay Mathematics Institute lists it as one of its Millennium Prize Problems, noting that it has been checked for the first ten trillion solutions but never proved. School and college students also meet his name in the Riemann integral, his definition of when a function can be integrated.
His health was poor. He developed tuberculosis after 1862, spent long periods in Italy for the warmer climate, and died at Selasca on Lake Maggiore on 20 July 1866, aged 39.
What can a student take from Riemann's way of working?
Intuition first, then rigour. Riemann reasoned from pictures and ideas. When Weierstrass found a gap in one of his methods, Riemann accepted the criticism but held that his results were still correct. Hermann Schwarz later found other proofs, and in 1901 Hilbert gave the corrected form of the method that made Riemann's proofs rigorous.
Question what everyone assumes. Euclid's flat geometry was the geometry everyone used. Riemann asked whether real space has to follow it.
Small audiences are fine. MacTutor notes that his first lecture course had eight students, and he was completely happy.
Spherical triangles are not in the CBSE or JEE syllabus, but area, angles in radians and the idea of curvature are. If radians or the area of a sphere need revision, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can rebuild them one at a time. Read how Riemann's teacher worked in our post on Carl Friedrich Gauss, and the geometry Riemann went beyond in Euclid's Elements.
Take a ball and a marker today and test the 270° triangle for yourself.
