Geometry
What if the parallel postulate is false?
For two thousand years the fifth postulate was thought to be a theorem waiting for a proof. It is not, and what happens when you deny it is not a contradiction but a second geometry, equally consistent and describing the universe rather better.
Euclid's axioms
Which of the five is not like the others?
Two thousand years of trying to prove a postulate
The parallel postulate is longer and less obvious than the other four, and the history of attempts to derive it is the history of nearly discovering something else.
What Euclid assumed without saying
Several steps in the Elements use facts about betweenness that are never stated. Hilbert's repair is the document's second half.
Non-Euclidean space
Can a consistent geometry disagree with the world?
A triangle whose angles do not sum to 180
On a sphere they sum to more, on a saddle to less, and neither is an error. The film builds both and checks the other four postulates still hold.
Curvature
Can a surface know it is curved?
Measuring bending from inside
Gauss proved that a surface's curvature can be determined without leaving it, which is why the idea survives into general relativity.
Straight lines that are not straight
A geodesic is the straightest available path, which on a curved surface is not straight. The document treats the variational argument in full.
Geometry, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume XI ↓