Shape and space
1854
Curved space
Bernhard Riemann
What if space itself could be curved?
In a lecture in Göttingen, Riemann described geometry for spaces that bend, in any number of dimensions. On a curved surface, triangles’ angles need not add up to 180 degrees. Sixty years later, Einstein needed exactly this.
How they figured it out
To qualify as a lecturer at Göttingen in 1854, Bernhard Riemann had to give a trial lecture and offer three possible topics. Tradition said the first would be chosen. The 77-year-old Gauss chose the third, the one Riemann was least prepared for: the foundations of geometry.
Riemann described spaces where the rules for measuring distance change from place to place, in any number of dimensions, and said that whether our own space is curved is a question for physics, not mathematics. Gauss was reportedly deeply impressed. Riemann died at 39, long before Einstein showed he had been right.
The equation
Distance depends on where you are; the g tells you how the space stretches at each point.
See it in real life
A triangle with three right angles
Start at the North Pole, walk down to the equator, turn 90°, walk a quarter of the way around the Earth, turn 90° again and walk back to the pole. Your triangle has three right angles: 270°, not 180°. That is why long flights look curved on a flat map.
Where engineers use it
GPS and relativity
Riemann’s geometry describes the curved spacetime that GPS has to correct for.
Shape analysis
Curved-surface geometry is used in computer graphics, robotics and manufacturing curved panels.
Where it came from
- Euclid’s Elements c. 300 BC
Led to
Riemann asked what geometry looks like if Euclid’s flat-space assumptions are dropped.
What it led to
- General relativity November 1915
Built on
Einstein described gravity using Riemann’s geometry of curved spaces.