Euclid's Elements is a brilliant idea but its axioms and definitions do not meet modern standards of rigour. This project takes G. D. Birkhoff's four axioms (1932), follows the spine of the Elements, and rebuilds every theorem you meet in school from the ground up, so you can see where each theorem comes from. The full theorem atlas is now live — open any node below to watch its derivation.
Before the four axioms — what is a definition? what is an axiom? from Euclid through Hilbert to Birkhoff, why we start from here.
Points on a line are in bijection with the real numbers so that the distance between two points equals the absolute difference of their coordinates.
Any two points lie on a unique line.
Rays through a point are in bijection with [0, 2π) so that an angle equals the difference of the corresponding numbers.
If two triangles have two sides in proportion and the included angles equal, the triangles are similar.
These four are equivalent to Hilbert I + II + III + IV + V, but more direct: measurement is taken as primitive, and continuity is delegated to the real numbers.
Measurement convention: rectangle area = length × width, circle area πr², and circumference 2πr are accepted as base measure formulas — their rigorous proofs require limits and lie outside this atlas; every other area / arc-length formula is derived from these.
Below is the full backbone, all the way from the axioms to the area formulae. Every incoming edge marks a logical dependency on an axiom or an earlier theorem; open any node to watch its own derivation animation.