Honors Geometry Unit 15 Review: Non-Euclidean Geometries

Hyperbolic geometry keeps Euclid's first four postulates and replaces the fifth. Through a point $P$ not on a line $l$, infinitely many lines pass through $P$ without ever meeting $l$. The surface has constant negative curvature, so picture a saddle that curves up in one direction and down in the other at every point.

You can't draw a perfect saddle on paper, so mathematicians use models. In the Poincaré disk model, the whole infinite plane fits inside a circle. Lines appear as diameters or as arcs that meet the boundary circle at right angles. Distances stretch as you approach the edge, so the boundary is infinitely far away.

Hyperbolic triangles always have angle sums less than $180°$. The shortfall, $180°$ minus the angle sum, is the angular defect, and area is proportional to it. On the standard hyperbolic plane (curvature $-1$), the area equals the defect measured in radians. This leads to a surprising result. Two triangles with the same three angles are congruent, so similar triangles of different sizes don't exist.

Worked example. A hyperbolic triangle has angles $40°$, $50°$, and $55°$. The sum is $40° + 50° + 55° = 145°$, which is under $180°$ as expected. The defect is $180° - 145° = 35°$. Converting to radians gives $35° \times \frac{\pi}{180°} = \frac{7\pi}{36} \approx 0.611$. On the standard hyperbolic plane, the area is about $0.611$ square units.

On a sphere, the job of a straight line is done by a great circle, which is a circle whose center is the center of the sphere. The equator is a great circle. A line of longitude and the opposite line of longitude together form a great circle. Lines of latitude other than the equator are small circles, so they don't count as lines. A shortest path between two points on a sphere follows an arc of a great circle. Antipodal points have infinitely many such paths. That's why a flight from New York to Tokyo looks curved toward Alaska on a flat map.

Any two great circles meet at two antipodal points, which are points directly opposite each other, like the North and South Poles. So spherical geometry has no parallel lines at all.

An ordinary spherical triangle is formed by three shorter great-circle arcs joining three noncollinear points, with no pair of points antipodal. Its angle sum is greater than $180°$ and less than $540°$. The amount over $180°$ is the spherical excess, and the triangle's area is proportional to it. With angles in radians, $A = (\alpha + \beta + \gamma - \pi)r^2$.

Worked example. Start at the North Pole and walk down a line of longitude to the equator. Turn $90°$, walk a quarter of the way around the equator, then turn $90°$ and head back north to the pole. All three angles are $90°$, so the sum is $270°$. The excess is $90°$, or $\frac{\pi}{2}$ radians. On a sphere with $r = 10$, the area is $\frac{\pi}{2}(10)^2 = 50\pi \approx 157.1$ square units. To check, notice that this triangle covers one-eighth of the sphere, and $\frac{1}{8}(4\pi \cdot 10^2) = 50\pi$.

A major difference among these geometries is their parallel-line rule. Playfair's axiom, a common form of Euclid's fifth postulate, says exactly one line through an outside point is parallel to a given line. Spherical geometry has no such lines, while hyperbolic geometry has infinitely many. Spherical and elliptic geometry are related but distinct.

The first four postulates describe drawing and extending segments, drawing circles, and the equality of right angles. Their usual interpretation needs changes on a sphere. Great circles loop back on themselves, antipodal points lie on infinitely many great circles, and geodesic circles cannot have arbitrarily large radii. Elliptic geometry identifies each pair of antipodal points as one point.

Curvature organizes everything. Zero curvature gives triangle sums of exactly $180°$. Positive curvature pushes the sum above $180°$, and negative curvature pulls it below. In both curved geometries, bigger triangles drift farther from $180°$. Only in Euclidean geometry are area and angle sum unrelated, which is why only Euclidean geometry has similar triangles of different sizes.

For about 2,000 years, mathematicians tried to prove the fifth postulate from the other four. In the early 1800s, Gauss, Bolyai, and Lobachevsky independently developed ideas of hyperbolic geometry. In 1868, Beltrami published models establishing its consistency relative to Euclidean geometry. Riemann described a broader framework for curved geometry in 1854, and Einstein later used curved geometry in general relativity to describe gravity.