Hyperbolic Geometry

Chapter 8: Models of Geometry

Learning objectives

Describe the Poincare disk and upper half-plane models of the hyperbolic plane
Identify hyperbolic lines (geodesics) in each model and compute angles at intersection points
Use the angular-defect formula to compute hyperbolic triangle area
Connect hyperbolic geometry to negative Gaussian curvature and to deep-learning Poincare embeddings

What happens if you replace "exactly one parallel" with "infinitely many parallels"? The discovery of Bolyai, Lobachevsky, and Gauss in the 1820s was that the answer is a perfectly consistent geometry — hyperbolic geometry — with its own theorems, formulas, and applications. It is the $K = -1$ end of the curvature classification, the natural home of saddle surfaces, and (in a twenty-first-century twist) the geometric substrate of modern word embeddings in deep learning. This section meets two standard models, the rules for drawing lines in each, and the central paradox: triangles have angle sums strictly less than $\pi$ , and the deficit is the area.

The Poincare disk model

The Poincare disk $\mathbb{D} = {(x,y) : x^2 + y^2 < 1}$ models the whole hyperbolic plane as the open unit disk, equipped with the metric:

ds^2 = \dfrac{4(dx^2 + dy^2)}{(1 - x^2 - y^2)^2}

The metric blows up near the boundary, so the boundary circle is at infinite hyperbolic distance — "points at infinity" from inside. Geodesics (hyperbolic straight lines) are: (a) diameters of the disk, and (b) arcs of Euclidean circles that meet the boundary at right angles. Angles between geodesics agree with Euclidean angles (the metric is conformal), but lengths and areas do not.

The upper half-plane model

An equivalent model is the upper half-plane $\mathbb{H} = {(x,y) : y > 0}$ with metric:

ds^2 = \dfrac{dx^2 + dy^2}{y^2}

Geodesics are: (a) vertical Euclidean rays $x = c$ , and (b) Euclidean semicircles with centre on the $x$ -axis. The $x$ -axis itself is "at infinity," analogous to the boundary circle in the disk model. The two models are related by a Mobius transformation that maps disk to half-plane, and either is equally good for calculation — pick whichever makes a given problem easiest.

The angular defect equals the area

The central formula of hyperbolic geometry is the angular defect theorem. For a hyperbolic triangle with interior angles $\alpha, \beta, \gamma$ on a surface with constant Gaussian curvature $K = -1$ :

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

The angle sum is strictly less than $\pi$ , and the smaller the sum the bigger the triangle. The maximum possible area is $\pi$ , reached when all three angles approach zero — an ideal triangle with vertices on the boundary at infinity.

Striking corollary: there are no similar triangles in hyperbolic geometry (besides congruent ones). Two triangles with the same angles must have the same area, hence the same side lengths. Hyperbolic geometry is rigid.

Circumference grows exponentially

The circumference of a hyperbolic circle of radius $r$ (on a surface with $K = -1$ ) is $2\pi\sinh r$ , not $2\pi r$ . For small $r$ the two agree, but $\sinh r \sim \dfrac{1}{2}e^r$ for large $r$ : the boundary grows exponentially with radius. There is "more room" in hyperbolic space — one reason it appears naturally in problems with exponential branching, like phylogenetic trees and social networks.

Where this shows up

- **Poincare disk embeddings in deep learning:** Recent NLP architectures (Nickel-Kiela 2017, Facebook AI Research) embed hierarchical data — WordNet, ontologies, taxonomies — into the Poincare disk because hyperbolic space can fit exponentially many nodes with low distortion, matching the branching structure. Standard Euclidean embeddings need very high dimension to do the same job. - **Escher's art:** M. C. Escher's *Circle Limit* series of woodcuts (1958-1960) are hyperbolic tessellations of the Poincare disk: triangles, fish, or angels of fixed hyperbolic area appearing smaller and smaller toward the boundary while staying the same size to a hyperbolic observer. Escher learned the geometry from H. S. M. Coxeter's diagrams. - **Cosmology and open universes:** The Friedmann equations of general relativity admit solutions where the spatial sections have constant negative curvature — an "open" universe modelled on the hyperbolic three-space

\mathbb{H}^3

. The Planck mission's measurements constrain the curvature parameter

\Omega_K

to be very close to zero, but a slightly hyperbolic universe is still allowed. - **Computer networks and routing:** Routing algorithms on large internet topologies exploit hyperbolic embeddings of the autonomous-system graph to find near-optimal paths in constant time per query. The exponential boundary growth of hyperbolic space matches the scale-free structure of the internet. - **Cell biology:** The wavy edges of lettuce leaves and the ruffled boundaries of certain corals exhibit constant negative Gaussian curvature — they are physical realisations of the hyperbolic plane. The exponential boundary growth is the geometric reason these tissues have so much surface relative to their interior.

Pause and think: In the Poincare disk model, look at two intersecting "circular-arc geodesics." Why is the angle between them at the intersection the same as the Euclidean angle between the two arcs at the same point? (Hint: the hyperbolic metric is conformal to the Euclidean one.)

Try it

Predict first: a hyperbolic triangle has angles $\pi/3, \pi/4, \pi/4$ on a surface with $K = -1$ . Compute its area.
An ideal triangle in $\mathbb{H}^2$ has all three vertices on the boundary. What are its angles, and what is its area on a $K = -1$ surface?
True or false: in hyperbolic geometry, the perpendicular distance from a fixed line is not a constant along a parallel line. (Hint: parallel lines in $\mathbb{H}$ diverge.)
Compute the hyperbolic circumference of a circle of radius $r = 2$ on a $K = -1$ surface and compare with the Euclidean value $2\pi(2) \approx 12.57$ .

A trap to watch for

The Poincare disk is not a piece of the Euclidean plane with funny labelling — it is a genuine model of a geometry where the parallel postulate fails. Two "lines" can be parallel (never meet inside the disk) while sharing a single point on the boundary, or they can diverge with no closest-approach point. The temptation to import Euclidean intuition through the visual surface is strong; resist it. Always compute with the hyperbolic metric, not with Euclidean ruler-and-protractor measurements on the disk.

What you now know

You can describe the Poincare disk and upper half-plane models, identify geodesics in each, compute hyperbolic triangle areas via angular defect, and locate hyperbolic geometry in the $K < 0$ slot of the curvature classification. You can also point to its modern reappearance in machine learning Poincare embeddings. The next section completes the trichotomy with the $K > 0$ case: elliptic geometry, where there are no parallel lines at all.

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 8.
Greenberg, M. J. (2008). Euclidean and Non-Euclidean Geometries (4th ed.). W. H. Freeman, ch. 7-10.
Stillwell, J. (2008). Mathematics and Its History (3rd ed.). Springer, ch. 18 (Bolyai, Lobachevsky, Beltrami).
Coxeter, H. S. M. (1969). Introduction to Geometry (2nd ed.). Wiley, ch. 16 (hyperbolic tessellations and Escher).
Penrose, R. (2004). The Road to Reality. Knopf, ch. 2 (hyperbolic geometry as a model of physical space).