Non-Euclidean Geometry

What does geometry look like if the parallel postulate is simply false?

5 chapters4 min read4 turning points 0 open problems

Branched from: Euclidean Geometry
Branched into: Geometric Topology + Group Theory + Riemannian Geometry
Figures: Carl Friedrich Gauss, Nikolai Lobachevsky, János Bolyai, Eugenio Beltrami, Felix Klein, Henri Poincaré

In brief

Non-Euclidean geometry is what you get by keeping all of Euclid's assumptions except the parallel postulate. Allow many parallels through a point and you get hyperbolic geometry, where space spreads out faster than flat space and a triangle's angles add up to less than 180°. Allow none and you get elliptic (spherical) geometry, where the angles add up to more. (Elliptic geometry also has to give up the assumption that a line can be extended forever without returning.)

Its discovery changed what an axiom is. Geometry's starting assumptions turned out to be choices, not self-evident truths, and choosing differently gave perfectly consistent worlds.

Key ideas

Hyperbolic geometryEnters 1829 – 1832: The geometry in which infinitely many lines through a point miss a given line. Space has constant negative curvature, triangle angles sum to less than $\pi$ , and the area of a disc grows exponentially with its radius.
Elliptic (spherical) geometry: The geometry with no parallels at all: every two "lines" (great circles on a sphere) meet. Curvature is positive and triangle angles sum to more than $\pi$ . Pilots and navigators work in it every day.
ModelEnters 1868: A concrete construction inside familiar mathematics in which a set of axioms all hold. A model proves the axioms are consistent, as long as the familiar mathematics is.
IndependenceEnters 1868: A statement is independent of a set of axioms if neither it nor its negation can be proved from them. The parallel postulate was the first famous case. The continuum hypothesis in set theory (Gödel 1940, Cohen 1963) is the most famous later one.
Transformation groupEnters 1872: Klein's organising idea: a geometry is determined by its group of allowed motions, and studies whatever those motions leave unchanged.

Chapter I

Denying the Postulate

Replace Euclid's fifth postulate with its opposite: through a point off a line, more than one parallel line can be drawn. Nothing breaks. You get a geometry, now called hyperbolic, that is internally coherent and strange in precise ways.

In it, the angles of a triangle always add up to less than $\pi$ . The shortfall is not random: it is proportional to the triangle's area. On a hyperbolic plane of curvature $-1/R^2$ ,

\text{Area} = R^2\,\bigl(\pi - (\alpha + \beta + \gamma)\bigr).

So there are no similar triangles of different sizes, because changing the size changes the angles. Every triangle's area is less than $\pi R^2$ . Circles grow exponentially with their radius. And at small scales, where the areas involved are tiny, all of this looks exactly like Euclid. That is why nobody had noticed.

Chapter II

Three Discoverers and a Silence

The breakthrough came from three places at once. Nikolai Lobachevsky lectured on it at Kazan in 1826 and published in 1829–30, in Russian, in a provincial journal few Western mathematicians read. János Bolyai, a Hungarian army officer, wrote it up as a 26-page appendix to his father's textbook in 1832.

Gauss, the most famous mathematician alive, answered Bolyai's father that to praise the work "would be to praise myself," because he had found the same results long before. He had written as much in private letters, but he had published none of it and never would. Some historians read this as caution about the controversy. Others doubt that his unpublished work ever amounted to a full geometry. Either way, the credit is contested, and this atlas marks it that way.

What none of them had was proof that the new geometry was consistent. Lobachevsky and Bolyai had pushed a long way without finding a contradiction, just as Saccheri had. But finding no contradiction is not the same as there being none.

Chapter III

From Fantasy to Model

Eugenio Beltrami closed the question in 1868. He showed that hyperbolic geometry already lives inside ordinary Euclidean space: on surfaces of constant negative curvature, like the trumpet-shaped pseudosphere, and in a disc whose chords play the role of straight lines. A contradiction in hyperbolic geometry would therefore be a contradiction in Euclidean geometry too. The parallel postulate is independent: it can neither be proved nor disproved from the other four.

Beltrami's surfaces came straight out of Gauss's theory of curvature, from the neighbouring differential-geometry branch. The two branches were already converging. The models standard today are the projective disc, the Poincaré disc and the upper half-plane. Beltrami had already described versions of all three. Felix Klein recast the first in projective terms in 1871, and from 1882 Henri Poincaré made the other two famous.

Chapter IV

A Closer Look: Triangles in the Poincaré Disc

Poincaré's disc model makes hyperbolic geometry visible. The whole infinite plane is drawn inside a circle. "Straight lines" are arcs of circles that meet the boundary at right angles, together with diameters. Angles are measured as they appear, but distances are distorted: the same step covers less and less of the drawing as you approach the edge, which is infinitely far away.

Draw three such arcs to make a triangle. Because the arcs bow inward, the corners look pinched, and the angles visibly add up to less than $180°$ . The shortfall is not arbitrary. For a plane of curvature $-1$ ,

\text{Area} = \pi - (\alpha + \beta + \gamma) .

Push the three corners out to the boundary circle and each angle shrinks to zero. The result, an ideal triangle, has angles summing to $0$ and area exactly $\pi$ , the largest any hyperbolic triangle can have, however long its sides. In Euclid's plane triangles can be arbitrarily large. In Lobachevsky's they cannot.

Parallels behave just as Lobachevsky said. Take a line and a point off it. Infinitely many arcs through the point never meet the line, and two of them, the limiting parallels, meet it only at the boundary, at infinity. Nothing in the picture contradicts Euclid's first four postulates. That is exactly Beltrami's point: the model sits inside ordinary geometry, so if hyperbolic geometry were inconsistent, so would Euclid's be.

Chapter V

Geometry as a Choice

Once several geometries were known to be consistent, "which one is true?" stopped being a mathematical question. Felix Klein's Erlangen Program (1872) organised the new situation. A geometry is defined by a group of transformations and studies whatever those transformations leave unchanged. Euclidean geometry keeps distances fixed under rigid motions. Projective geometry keeps only incidence and cross-ratio. Hyperbolic geometry has its own group of motions.

Whether physical space is Euclidean became a question for measurement. A popular story says Gauss tried to measure the angle sum of a triangle of mountain peaks for this reason. Historians doubt the story. What is certain is that the question later found an answer in general relativity.

Non-Euclidean geometry, as a separate subject, largely finished its own work. Its questions moved into its successors. Hyperbolic space turned out to be the geometry of most three-dimensional shapes, and it became a main character in geometric topology.

Turning points4

1829 – 1832REFORMULATIONContested
Hyperbolic geometry is published as a geometry in its own right
Nikolai Lobachevsky, in the Kazan Messenger (1829–30), and János Bolyai, in an appendix to his father's textbook (1832), each develop a full geometry in which many parallels pass through a point. Neither treats it as an absurdity waiting for a contradiction. It is a consistent-looking alternative to Euclid, complete with its own trigonometry.
What is disputed
Priority is disputed. Gauss wrote privately, for example to Taurinus in 1824 and to Bolyai's father in 1832, that he had reached the same conclusions years earlier, but he published nothing. Ferdinand Schweikart and Franz Taurinus also worked out related results in 1818–26. Historians generally credit Lobachevsky and Bolyai with independent first publication. How far Gauss's unpublished work went, and whether he ever fully accepted the new geometry, is still debated.
Carl Friedrich Gauss, Nikolai Lobachevsky, János Bolyai
› Sources (3)
- Lobachevsky, N. I. (1829–1830). O nachalakh geometrii [On the principles of geometry]. Kazanskii Vestnik.
- Bolyai, J. (1832). Appendix, scientiam spatii absolute veram exhibens. In F. Bolyai, Tentamen. Maros-Vásárhely.
- Gray, J. (2004). János Bolyai, Non-Euclidean Geometry, and the Nature of Space. Burndy Library / MIT Press.
1868PROOF
Beltrami shows hyperbolic geometry is as consistent as Euclid's
Eugenio Beltrami builds hyperbolic geometry out of ordinary Euclidean objects, on surfaces of constant negative curvature and in a disc model. Any contradiction in the new geometry would therefore be a contradiction in the old one. This settles the two-thousand-year question: the parallel postulate cannot be proved from the others.
Eugenio Beltrami
› Sources (1)
- Beltrami, E. (1868). Saggio di interpretazione della geometria non-euclidea. Giornale di Matematiche 6: 284–312.
1872REFORMULATION
Klein's Erlangen Program redefines what a geometry is
Felix Klein proposes that a geometry is the study of whatever stays unchanged under a chosen group of transformations. Euclidean, hyperbolic, elliptic, affine and projective geometry then become siblings, each defined by its own group, rather than rival claims about the one true space.
Felix Klein
› Sources (1)
- Klein, F. (1872). Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen.
1882REFORMULATION
Poincaré finds hyperbolic geometry inside complex analysis
Studying functions of a complex variable that are unchanged under certain groups of transformations, Henri Poincaré realises those transformations are exactly the rigid motions of the hyperbolic plane, drawn in a disc or a half-plane. Hyperbolic geometry stops being a curiosity about parallels and becomes a working tool in complex analysis and number theory.
Henri Poincaré
› Sources (2)
- Poincaré, H. (1882). Théorie des groupes fuchsiens. Acta Mathematica 1: 1–62.
- Gray, J. (2000). Linear Differential Equations and Group Theory from Riemann to Poincaré (2nd ed.). Birkhäuser.

Applications

Where it is used

Navigation
Great-circle routes
On the curved surface of the Earth the shortest path between two cities is an arc of a great circle, the "straight line" of spherical geometry. Long-haul flights and shipping routes follow them, which is why a flight from Europe to North America arcs far north on a flat map. Spherical trigonometry, where triangle angles exceed 180°, is the everyday arithmetic of navigation.
Special relativity↗ Physics · Special Relativity
The geometry of velocities
In special relativity, velocities do not simply add. Composing two boosts behaves like adding lengths in hyperbolic space, and the natural measure of speed (rapidity) is hyperbolic distance. Vladimir Varićak pointed this out in 1910. It is one of the cleanest places where non-Euclidean geometry is physically real.
Machine learning
Hyperbolic embeddings of hierarchies
Trees and hierarchies, like taxonomies and word hierarchies, grow exponentially, and so does hyperbolic space. Embedding such data in the Poincaré disc keeps distances far more faithful in a few dimensions than any flat embedding can. This is now a widely used technique in representation learning.
› Sources (1)
- Nickel, M. & Kiela, D. (2017). Poincaré embeddings for learning hierarchical representations. Advances in Neural Information Processing Systems 30.
Art
Escher's Circle Limit prints
After seeing a hyperbolic tiling in a paper by H. S. M. Coxeter in 1958, M. C. Escher made his Circle Limit woodcuts (1958–60). Fish and angels shrink toward the edge of a disc. In the Poincaré model's own measure they are all the same size.
› Sources (1)
- Coxeter, H. S. M. (1979). The non-Euclidean symmetry of Escher's picture 'Circle Limit III'. Leonardo 12(1): 19–25.

Open problems

Where the map runs out

No open problems are recorded here. This field's unanswered questions moved into its successors: Geometric Topology + Group Theory + Riemannian Geometry.

In brief

Key ideas

Denying the Postulate

Three Discoverers and a Silence

From Fantasy to Model

A Closer Look: Triangles in the Poincaré Disc

Geometry as a Choice

Where it is used

Great-circle routes

The geometry of velocities

Hyperbolic embeddings of hierarchies

Escher's Circle Limit prints

Where the map runs out

Further reading