Euclid
In the Rainforest
“The noblest pleasure is
the joy of understanding.”
Leonardo da Vinci
Like Counting Rabbits
Imagine you start with one pair of baby rabbits. Each month, every grown-up pair has a new pair of babies. You get: 1, 1, 2, 3, 5, 8, 13, 21 … Each number is the two before it added together.
If you draw squares with those sizes and sweep a curve through their corners, you get the golden spiral — the same shape in seashells, hurricanes, and sunflower heads.
Convergence, φ, and Phyllotaxis
The Fibonacci recurrence F(n) = F(n-1) + F(n-2) produces a sequence whose consecutive ratios converge to φ = (1 + √5)/2 ≈ 1.618 — the positive root of x² = x + 1. Its continued fraction is all 1s, making it maximally irrational.
Phyllotaxis exploits this: each floret offset by 360°/φ² ≈ 137.5° guarantees no aligned rows and optimal packing. Spiral counts match consecutive Fibonacci numbers because F(n)/F(n+1) are the best rational approximations to 1/φ.
Huddy-Hint: switch to Phyllotaxis, then take control and drag while you move the Iterations slider.
The Pythagorean Tree
“Number is the ruler of forms and ideas.”
Squares on a Triangle
Take a right triangle. Stick a square on each side — the two smaller squares' areas add up to the big one exactly. a² + b² = c².
The tree stacks smaller right triangles on top of squares, over and over — the same law shrinking into the distance like a fractal.
Self-Similarity and the Theorem as Fractal Engine
From a base square of side s, erect an isosceles right triangle, attach two new squares to its legs. Each child has side s·cos(π/4). After n iterations: 2ⁿ terminal squares, total area bounded at 2× trunk area. The boundary is fractal with Hausdorff dimension > 1.
Huddy-Hint: switch to Organic, take control, then drag left and right.
Compass & Straightedge
“There is no royal road to geometry.” — Euclid
Only a Compass and a Ruler
Two tools: a compass and a straightedge (ruler with no numbers). No measuring — just circles and connecting points. Greeks built astonishingly precise shapes with just these moves.
Constructibility and Galois Theory
Compass-and-straightedge constructions = field extensions of ℚ via quadratic equations. A length is constructible iff its minimal polynomial has degree 2ⁿ. Hence: ∛2 (degree 3) can't be constructed, π (transcendental) can't square the circle. Regular n-gon constructible iff n = 2ᵏ · (distinct Fermat primes).
Tangle Field
“It from Bit.” — John Archibald Wheeler
Spooky Twins
Two magic coins. Flip one in New York — heads. Instantly, Tokyo — tails. Every time. No wire, no trick. That's entanglement.
Green and red dots are entangled pairs. Your pointer is an observer — move close and you collapse their world.
Bell States and Nonlocality
Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Measurement collapses the joint non-separable state. Bell's theorem (1964): no local hidden variables reproduce QM statistics. CHSH violations confirmed to ~100σ.
Concentric Numbers
“God does not play dice with the universe.” — Einstein
Mountain Maps and Ring Roads
The wavy lines are a hiking map. Every line traces one height. Lines packed close together mean a steep climb. Your pointer pushes a peak up under the sheet, and the rings rearrange around it.
Switch to Sacks Spiral and the map turns into numbers. Wind 1, 2, 3, 4 … outward in a spiral so every square number lands on the same straight line. Then light up only the primes. They refuse to scatter — they fall into curved lanes nobody put there.
Marching Squares & the Sacks Spiral
Contour and Relief share one scalar field: a sum of Gaussians plus a pointer-driven peak. The static term is baked once at grid resolution; only the pointer term is evaluated per frame. Isolines come from marching squares — a 4-bit corner index per cell, 16 cases, linear interpolation along crossed edges, batched into one path per level.
Sacks places n at polar (r = k√n, θ = 2π√n), so perfect squares fall on the positive x-axis and the turn spacing is uniform. Primes then concentrate along product curves — most visibly the Euler polynomial n² + n + 41, whose values trace a single arm. This is a picture of the same non-randomness that π(x) ~ x/ln(x) only averages over.
Huddy-Hint: take control in Contour and drag the peak into a saddle between two hills.
Bendy Curvy Spacetime
“Spacetime tells matter how to move; matter tells spacetime how to curve.”
The Bowling Ball on the Trampoline
Bedsheet plus bowling ball equals sag. Roll a marble across it and the marble curves. That is gravity: heavy things bend space, and everything else just follows the bend.
Add masses and drag them around. Light Paths shoots photons across the sheet — they change direction but never speed up. Orbits drops test particles in at the right speed to circle.
Einstein Field Equations and Geodesics
gμν encodes curvature; Gμν + Λgμν = (8πG/c⁴)Tμν relates it to stress-energy. Free fall follows geodesics of that metric.
What this canvas actually computes: displacement is a softened 1/(d + core) funnel clamped below d, which reads like a Flamm-paraboloid projection rather than a solution of the field equations. Light integrates direction under a Newtonian 1/r² pull and renormalizes speed each step, so rays deflect at constant c. Orbits are seeded at v = √(GM/r) and integrated with a softened potential. Real GR adds what a 2D Newtonian toy cannot: frame dragging (Kerr), Schwarzschild time dilation, and radiative loss (LIGO, 2015).
Huddy-Hint: take control in Orbits, drop a second mass, and watch the two-body scatter.