Axiom.

Mathematics, drawn. No lecture, no prerequisites — a straightedge, a compass, and four results that demonstrate themselves. The chalk moves when you do.

Proposition I

Proposition I · Rearrangement

In a right triangle, the square on the hypotenuse equals the two squares on the legs, taken together.

One square of side a + b. Four copies of the same right triangle. Keep your eye on the space they don't cover.

Three triangles slide and turn; none is stretched, none overlaps. The uncovered area — two squares before, one square after — never changed. It only changed shape. The pale outlines remember where a² and b² used to live.

a²+b²=c²

Attributed to Pythagoras of Samos, c. 530 BC. No algebra required: area is conserved, so the equation is forced.

Proposition II · Locus

The points that sit exactly twice as far from A as from B form a perfect circle.

Walk wherever you like, provided your distance to A stays double your distance to B. Your path closes behind you.

Every spoke is measured: yellow legs run to A, pink legs to B, and yellow is always exactly twice pink — 2ℓ against ℓ. Apollonius of Perga proved the path closes, four centuries before coordinates existed.

Proposition III · Growth

Cut the largest square from a golden rectangle, and a golden rectangle remains. Forever.

The first nine cuts of infinitely many. Each square hosts a quarter of a circle; beneath them coils the true spiral.

White: the classical compass construction — one quarter-circle per square. Gold: the true logarithmic spiral r = φ^2θ/π, which grows by φ every quarter turn and meets each square only at its corner. Arc and spiral disagree by less than a chalk-width. But they disagree.

φ = (1 + √5) / 2 ≈ 1.618 033 988…

Proposition IV · Chance

Drop ten thousand accidents through ten rows of pegs, and a law appears.

Each ball turns left or right with even odds, ten times. Nobody steers. The pile signs its own theorem.

Balls dropped: 0
Sample mean: —
Sample σ: —
Predicted: μ 5 · σ 1.581

The gold curve is the normal law N(5, 2.5) that de Moivre extracted from the binomial in 1733; the pink ticks are the exact expectation C(10,k)·n/2¹⁰ for each bin. The more accidents, the better they agree.

A proof without words is still a proof. The board asks only that you watch closely.

AXIOM is a small press for visual mathematics — prints, folios, and a slow journal of proofs that draw themselves. New propositions are chalked each equinox. Bring nothing.