Euclid
In the Rainforest
"The noblest pleasure is
the joy of understanding."
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: Toggle Phllyotaxis then adjust iterations + drag
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: Toggle Organic then drag left/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 mouse 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 Magic Numbers
Wavy lines like a hiking map — each ring = same "height." Close lines = steep climb. Your mouse pushes up a peak like a bedsheet.
"Prime Sieve" lights up primes: numbers divisible only by 1 and themselves.
Marching Squares and Eratosthenes
Contour: marching squares algorithm — 4-bit index per cell, 16 cases, linear interpolation on edges. Elevation = sum of Gaussians + mouse peak.
Sieve: iterate i from 2 to √n, mark multiples. O(n log log n). Prime distribution governed by π(x) ~ x/ln(x), connected to Riemann Hypothesis via ζ(s) zeros.
Bendy Curvy Spacetime
"Spacetime tells matter how to move; matter tells spacetime how to curve."
The Bowling Ball on the Trampoline
Bedsheet + bowling ball = sag. Roll a marble — it curves. That's gravity. Heavy things bend space.
Click to place masses. "Light Paths" = photons bending. "Orbits" = tiny moons.
Einstein Field Equations and Geodesics
gμν encodes curvature. Gμν + Λgμν = (8πG/c⁴)Tμν. Free-fall = geodesics. This sim approximates Newtonian M/r² with visual warping; real GR: frame-dragging (Kerr), time dilation (Schwarzschild dτ²), gravitational waves (LIGO 2015).