The Geometric Viewpoint

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Pythagoras in the hyperbolic plane

The Pythagorean theorem is surely the most famous of all mathematical theorems. The simplicity of its statement (that a right triangle with sides of length a and b and hypotenuse of length c satisfies a^2 + b^2 = c^2) and the multiplicity of beautiful proofs (like the one shown at right from Byrne’s edition of Euclid’s Elements) contribute to its memorability.

Byrne's version of Euclid's proof of the Pythagorean theorem.

Byrne’s version of Euclid’s proof of the Pythagorean theorem.

The fame of Pythagoras’ theorem in Euclidean geometry makes it natural to ask if it holds in other geometries. For instance, is the Pythagorean theorem true in hyperbolic geometry, that radical challenger to Euclid’s throne?

The naive answer is “No, the Pythagorean theorem does not hold in hyperbolic geometry”, as it is logically equivalent to Euclid’s 5th postulate (which is the defining difference between Euclidean and hyperbolic geometry). However, there is a theorem in hyperbolic geometry which is analogous to Pythagoras theorem:

The hyperbolic Pythagorean theorem
If a right triangle in the hyperbolic plane has sides of length a and b and a hypotenuse of length c, then \cosh(a)\cosh(b) = \cosh c.
Right triangle in the hyperbolic plane

Right triangle in the hyperbolic plane

As in the figure at left, the edges of the triangle are hyperbolic geodesics (we’ll review what those are below), with the sides of length a and b adjacent to the right angle and the hypotenuse (of length c) is the edge across from the right angle. The function \cosh x = \frac{e^x + e^{-x}}{2} is hyperbolic cosine.

The hyperbolic plane

The hyperbolic plane was discovered by Bolyai and Lobachevsky when they investigated the effect of replacing Euclid’s parallel postulate with an alternative. They operated purely deductively: they had no graphical representation of hyperbolic geometry to work with. Later mathematicians, such as Klein and Poincaré, discovered ways of representing hyperbolic geometry inside of Euclidean geometry by giving new meanings to terms such as “line”. Using calculus, we can give a succinct description (called the Poincaré disc model) of the hyperbolic plane as follows.

The entirety of hyperbolic geometry will take place inside the open unit disc (the blue disc at left) in the plane \mathbb R^2. The unit circle (the boundary of the disc) is not part of the world in which we do hyperbolic geometry. We refer to it as the infinity circle. A path \gamma(t) = (x(t), y(t)) for t_0 \leq t \leq t_1 in the disc has length defined by an integral similar to the integral defining path length in euclidean geometry. The length of \gamma in euclidean geometry is given by \int_{t_0}^{t_1} ||\gamma'(t)||\thinspace dt, where ||v|| denotes the magnitude of the vector v. The length of \gamma in hyperbolic geometry on the other hand is given by the integral \int_{t_0}^{t_1} \frac{||\gamma'(t)||}{1 - ||\gamma(t)||^2} \thinspace dt. A path between points A and B is a geodesic if it has length no greater than the length of any other path between A and B. Geodesics in hyperbolic geometry are the analogue of straight lines in euclidean geometry. If there were light in the hyperbolic plane, it would travel along geodesics.

Right triangle drawn with geodesics

A triangle is formed by three geodesics intersecting pairwise.

In the Poincaré disc model of hyperbolic geometry it turns out that the geodesics are segments of diameters of the disc and portions of circles in \mathbb R^2 which intersect the infinity circle at right angles. If three geodesics intersect in three points, not all lying on the same geodesic, then the three geodesics define a triangle. The image on the right shows our triangle arising from three geodesics. It may seem as though the triangle we’ve drawn is somewhat special in that two of the sides lie on diameters of the disc. However using hyperbolic isometries (the analogue of euclidean translations, rotations, and reflections) we may move (without changing lengths or angles) any hyperbolic triangle so that two of its sides lie on diameters, as we have indicated.

Proving the hyperbolic Pythagorean theorem

Here is a sketch of the proof of the hyperbolic Pythagorean theorem. It is an abbreviated version of the proof given by Martin Greenberg in his excellent text Euclidean and non-Euclidean Geometries.

The triangle ABC

The triangle ABC

Let \triangle ABC be a right triangle in the hyperbolic plane with C the right angle. Without loss of generality, we may assume that the vertex A is the origin and that two of the edges, one of which is the hypotenuse, are portions of diameters, as in our picture. Let d(AB) be the hyperbolic distance from point A to point B and \overline{AB} the euclidean distance.

We have already defined the hyperbolic cosine function \cosh. The hyperbolic sine function is defined similarly: \sinh(x) = (e^x - e^{-x})/2 and the hyperbolic tangent function is simply \text{tanh }(x) = \sinh(x)/\cosh(x).

Using our path length formula, it is straightforward to verify that \text{tanh}(d(AB)) = \frac{2\overline{AB}}{1+\overline{AB}^2} and \text{tanh}(d(AC)) = \frac{2\overline{AC}}{1+\overline{AC}^2}.

We begin by showing:

Lemma: \sin A = \frac{\sinh a}{\sinh c} and \cos A = \frac{\text{tanh } a}{\text{tanh } c}

Once we have those equations, the hyperbolic Pythagorean theorem can be derived from the equality \sin^2 A + \cos^2 A = 1 by applying identities for \sinh and \cosh analogous to the identities involving \sin and \cos. We leave it as a pleasant challenge to the reader to work out those details.

Extending geodesics

Extend the geodesics into the plane outside the disc

We now set about proving the Lemma, by considering the image in \mathbb R^2 at right. Extend the geodesics making up the sides of \triangle ABC into \mathbb R^2. This means that the geodesic a is now part of a circle H centered at a point O.

The circle H intersects the infinity circle in two points. Join those two points by a line (drawn in red in the figure) and let P and Q be the points where that line intersects the extensions of the other two edges of \triangle ABC, as in the picture. (The points P and Q have a special relationship to the points B and Q: they are the images of P and Q under the conversion from the Poincaré disc model of the hyperbolic plane, to the Klein model of the hyperbolic plane. But that is a story for a different day.)

The key to the whole business is to apply the definition of cosine to the euclidean triangle \triangle PAQ. Doing so, we obtain: \cos A = \overline{AQ}/\overline{AP}. Converting to hyperbolic distances, we arrive at

(*)     \cos A = \text{tanh }b/\text{tanh }c.

Let R be the point, other than B, where the line AB intersects the circle H. The point R is the result of applying the inversion (x,y) \to \frac{1}{x^2 + y^2}(x,y) to the point B. This implies that \overline{AR} = 1/\overline{AB}. Hence,

(**)      \overline{BR} = \overline{AR} - \overline{AB} = 2/\sinh d(AB).

Recalling that c = d(AB), we have \overline{BR} = 2/c. Letting S be the other intersection point between the line AC and the circle H, we also have \overline{CS} = 2/\sinh b.

The angle BOT

The angle BOT is equal to the angle at B of triangle ABC.

Finally, as in the figure at left, let T be the orthogonal projection of O', the center of H onto the line AB. Some rather easy arithmetic, using the fact that the angles of a euclidean triangle sum to \pi, shows that the angle BOT is equal to the angle B. Combining this fact with (*) and (**), we conclude that \sin B = \sinh b/ \sinh c. Interchanging the roles of A and B in the preceding argument, concludes the proof of the lemma.

Final Thoughts

The euclidean and hyperbolic planes are certainly the most important of the two-dimensional geometries. The third most important geometry is spherical geometry. There is also a version of the Pythagorean theorem for triangles on the sphere. Thurston, in his famous book Three-dimensional Geometry and Topology, sketches a strategy for giving a combined proof of the law of cosines in the hyperbolic plane and in the sphere. The corresponding Pythagorean theorems follow from that. For the hyperbolic law of cosines, Thurston uses the hyperboloid model of the hyperbolic plane, which gives a unification of the Poincaré disc model and the Klein model alluded to earlier.

Finally, many of the beautiful proofs of the Pythagorean theorem make use of dissections of a square and the fact that a^2 is the area of a square with sides of length a. Is there a dissection proof of the hyperbolic Pythagorean theorem?

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