4 - Non-Euclidean Geometry

Ancient Greeks

Introduced abstraction, rigor, and proofs; math studied for its own sake using general principles and formal proofs.

Thales of Miletus

First use of deductive reasoning in geometry; calculated pyramid heights and ship distances; discovered 5 geometric theorems.

Pythagoras of Samos

Pythagorean Theorem: (a^2 + b^2 = c^2) for right triangles.

Plato of Athens

Founded the Academy; inscription: “Let no one enter, who is ignorant of Geometry.”

Eratosthenes of Cyrene

Calculated the Earth’s circumference (~24,900 miles, only 100 miles off).

Archimedes of Syracuse

Used Method of Exhaustion to calculate circle areas; showed (\pi) is between (3 \frac{1}{7}) and (3 \frac{10}{71}).

Euclid’s Undefined Terms

Point, line, plane, incidence.

Euclid’s Common Notions

Things equal to the same thing are equal; adding/subtracting equals gives equals; coinciding things are equal; whole > part.

Euclid’s Postulates

1. Line between any two points. 2. Extend a line segment into a line. 3. Circle with any center and radius. 4. All right angles equal. 5. Parallel Postulate: If interior angles < 180°, lines meet.

Playfair’s Axiom

Through a point not on a line, exactly one line is parallel to the given line.

Non-Euclidean Geometry

Retains first 4 Euclidean postulates; changes the fifth postulate.

Hyperbolic Geometry

At least two parallel lines through a point; triangle angles < 180°; no rectangles/squares.

Hyperbolic Geometry Mathematicians

Bolyai, Lobachevsky, Gauss.

Poincare Disk Model

Points = inside unit circle; lines = diameters or perpendicular arcs; hyperbolic triangle area = (\pi - (\alpha + \beta + \gamma)).

Half-Plane Model

Lines = verticals or semicircles with diameters on the real axis.

Elliptic Geometry

No parallel lines; triangle angles > 180°; lines = great circles; similar triangles are congruent.

Riemann’s Spherical Model

Points = on unit sphere; lines = great circles; triangle area = ((\alpha + \beta + \gamma) - \pi).

Globe Terms

Longitude = east-west position; Meridian = constant longitude; Prime Meridian = 0° through Greenwich; Antimeridian = 180° (date change line).

Pentagramma Mirificum

Star polygon on a sphere; composed of arcs of 5 great circles; internal angles = right angles.

Einstein (1915)

Predicted space is curved; mass and energy bend space-time; gravity = movement along curved space-time.