Attractor · an interactive atlas
The structures reality wants to make.
106 live explorables at the frontier of mathematics, physics, information and life — primes and zeta zeros, black holes, quantum error correction, E8, the renormalization group. Poke every one of them; each tells you why it matters.
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Clifford attractor · move your cursor to bend it
Journeys
Guided stories through the atlas
Each stop explains why the next one matters. Step with ← →; every stop has its own link.
The Music of the Primes
From a doodle on a spiral to the Riemann Hypothesis: why the primes are a sum of pure tones.
Information Is Physical
From Shannon’s bit to black holes: how information became the deepest quantity in physics.
The Shape of Spacetime
Boosts, black holes and ripples in the fabric of the universe.
Quantum, Certified
Eight experiments that broke classical intuition — and how the classical world comes back.
How Complexity Emerges
Simple rules, rich worlds: from cellular automata to the renormalization group.
Hidden Integers
Symmetry and topology: the things that cannot change — and why they protect everything.
The Mathematics of Life
Spirals, cycles, strategies and drift — the math evolution discovered first.
Number
Primes, constants, and the hidden order of the integers.
Primes & Integers
The primes are the atoms of arithmetic. They look random — yet spirals, diagonals and comets keep appearing.
9Ulam Spiral
The diagonals Ulam saw, watched as they form. Add primes one at a time; toggle quadratic-polynomial overlays; pan the integer line.
Sacks Spiral Studio
Classic Sacks layout with controls: range, color by residue class, twin-prime highlighting, animated zoom along the radial axis.
Prime Heartbeat
Each prime fires a Gaussian wavepacket from its Sacks position; the interference field accumulates into the prime's spectral signature.
Resonance Loom
Sweep the angular step α from 0 to 1 and watch which values lock primes into petals. A spectrum of the fertile angles.
Moiré Studio
Three layered prime fields with full transform + composite-mode controls. Built for finding interference patterns by eye.
Goldbach Comet
For each even n, plot the count of ways to write it as the sum of two primes. The "comet" pattern reveals deep modular structure.
Heegner Numbers
e^(π√163) ≈ 262537412640768744 with twelve nines after the decimal. The j-invariant, class number 1, and Euler's prime polynomial n²+n+41.
Erdős Primitive Sets
The 1935 conjecture, proven by Lichtman in 2022 with von Mangoldt weighting. Build your own primitive set; watch your sum chase the prime bound.
Collatz Conjecture
3n+1. Watch any starting integer's path to 1. The conjecture is verified to 2⁶⁸ but unproven. Erdős said mathematics wasn't ready for it.
The Music of the Primes
One function, ζ(s), encodes every prime. Its zeros are the notes; the primes are the song.
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Zeta on the Critical Strip
|ζ(s)| as a heatmap across the critical strip 0 < Re(s) < 1. The zeros punch dark holes through the surface.
Riemann Zeros
Non-trivial zeros of ζ(s) construct ψ(x) as a sum of cosine waves. Add zeros one at a time; watch the staircase emerge from the smooth.
The Explicit Formula
The prime staircase is a smooth line plus one wave per zeta zero. Add zeros one at a time and watch steps appear exactly at the primes. The music of the primes.
Prime Sonification
Listen to ψ(x): each Riemann zero is a frequency in the prime distribution. Audio sweep from γ₁ outward.
Wigner Semicircle
Eigenvalues of random symmetric matrices distribute into a semicircle. Universal. Same statistics control nuclear energy levels and Riemann zeta zeros.
Where Constants Come From
π, e and their cousins aren’t arbitrary — they fall out of processes: needles, limits, spirals, chaos.
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e as a Limit
Watch (1 + 1/n)^n climb toward e as n grows, side-by-side with π(x) climbing toward x/ln(x). Same e, two different jobs.
Compound Pulse
Compound interest as e in disguise. Slide the compounding frequency from yearly to continuous and watch e fall out of the math.
Euler’s Identity
e^(iθ) traced through θ ∈ [0, 2π] in the complex plane. At θ = π, the trace reads -1, and 1 + e^(iπ) = 0 falls out as a geometric fact.
π Three Ways
Vieta 1593 (nested radicals), Wallis 1656 (infinite product), Euler 1734 (Basel sum). Three formulas, one constant.
Buffon's Needle
Drop random needles. Count line-crossings. π falls out of the ratio. 1733 — the first Monte Carlo experiment in history.
The Gaussian
Why does ∫e^(−x²) dx = √π? Poisson's trick: square the integral and rotate. π appears because the 2D bell is rotationally symmetric.
Khinchin's Constant
Geometric mean of continued-fraction coefficients converges to K ≈ 2.685 for almost every real number. The CF algorithm has a universal statistic.
Constants Compass
Type a number; the compass searches for expressions in e, π, φ, √2 that land near it. Identity hunter for amateurs and Ramanujans alike.
Approximation & Modularity
How well can a fraction imitate an irrational — and why does that question lead to the deepest symmetries in mathematics?
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Continued Fractions
Type a number; see its continued fraction expansion and the convergents that approximate it. Watch e's repeating pattern emerge.
Stern-Brocot Tree
Binary tree of all positive rationals in lowest terms. Trace any irrational; the denominators along the path are the convergents of its continued fraction.
Diophantine Map
For any irrational α, plot the best rational p/q approximations and their errors. The "spectrum" of how well α is approximable.
Modular Forms
SL(2,ℤ) acting on the upper half-plane. j-invariant. Monstrous moonshine — coefficient 196884 = 196883 + 1 = dimension of Monster's first rep.
Elliptic Curve Group Law
Draw a line through two points on a cubic, find the third, reflect — that's "addition." It secures Bitcoin and TLS, and it cracked Fermat's Last Theorem.
Shape
Fractals, strange spaces, knots — geometry beyond Euclid.
Fractals & Chaos
Iterate a simple rule and you get infinite detail — and futures no one can predict.
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Mandelbrot & Julia
z² + c iteration. Hover the Mandelbrot to pick c and the Julia set draws live. Click to zoom infinitely into the fractal boundary.
Mandelbulb
3D analog of the Mandelbrot. Spherical-coords cheating gives a cathedral-like fractal with no relation to nature, yet looks intensely organic.
Cantor Set
Remove middle thirds forever. Total length zero, uncountably many points, fractal dimension log 2 / log 3 ≈ 0.631. The first fractal.
Apollonian Gasket
Three tangent circles + Descartes' theorem = an infinite fractal of integer-curvature circles. The most arithmetically loaded fractal in math.
Bifurcation & Feigenbaum δ
The logistic map's path to chaos. Period doubles at ratios that converge to δ = 4.669 — same for any peaked 1D map. Universal.
Lorenz Attractor
Three coupled ODEs. The original "butterfly effect" — two near-identical starts diverge into different lobes within a few hundred steps.
Double Pendulum
Two coupled pendulums = chaos. Add a twin at 10⁻⁴ offset, watch them diverge in seconds. The most accessible deterministic chaos.
Strange Spaces
Negative curvature, a fourth dimension, surfaces with no inside, order without repetition.
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Hyperbolic Tilings
Regular {p,q} tilings of the hyperbolic plane in the Poincaré disk model. The "third geometry" — what Escher drew, what AdS/CFT uses.
Riemann Sphere
Stereographic projection between sphere and complex plane. Infinity becomes a point. Conformal, circle-preserving, the foundation of complex analysis.
Tesseract
4D hypercube rotating through dimensions our brains never built. XW/YW/ZW rotations turn the cube inside out — vertices pass through faces.
Hopf Fibration
The 3-sphere as a bundle of linked circles, projected to 3D. The first non-trivial fiber bundle in history; foundation for gauge theory.
Klein Bottle
Non-orientable surfaces — Möbius, Klein, cross-cap. Walk along one and "left" becomes "right." Cannot embed in 3D without self-intersection.
Penrose Tiling
Two tile shapes, never periodic, 5-fold symmetry forbidden in crystals. Shechtman's quasicrystals (Nobel 2011) made these physical in matter.
Voronoi & Delaunay
Voronoi cells (nearest-neighbor regions) and their dual Delaunay triangulation. Honeycomb, Brillouin zones, basalt columns, image stippling.
Knots & Symmetry
When is a tangle truly tangled — and what is the most symmetric object that can exist?
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Knot Theory
Classify knots by their Jones polynomial. Witten 1989: knot polynomials are partition functions of Chern-Simons theory.
Borromean Rings
Three rings locked together, no two linked pairwise. Cut any one — the others fall free. Higher-order topology: Brunnian link, Efimov nuclei.
The E8 Root System
240 roots in 8 dimensions, projected live onto the Coxeter plane into eight perfect rings of 30. Often called the most beautiful object in mathematics.
Matter & Spacetime
Gravity, action, and the statistical physics of everything.
Gravity & Spacetime
Mass tells spacetime how to curve; spacetime tells mass how to move.
9Lorentz Transformations
Spacetime diagram with shearing axes. Light cones at 45° invariant. Click events; watch their boosted-frame coordinates. Special relativity made geometric.
Schwarzschild Orbits
Geodesics around a black hole. ISCO at 6M, photon sphere at 3M, event horizon at 2M. GR's first triumph: Mercury's precession.
Kerr Black Hole
A spinning BH drags spacetime around it. Inside the ergosphere even light must corotate. Penrose process: up to 29% of mass-energy is extractable.
Gravitational Lensing
Drag a black hole through a star field. Watch Einstein rings form. The dark shadow + photon ring is what EHT actually imaged.
Penrose Diagram
Maximally extended Schwarzschild as a finite diamond. Four regions, two universes, light cones at 45°. Once past the horizon, the singularity is in your future.
Gravitational Waves
Two black holes spiral in and shake spacetime. The strain h(t) is a chirp — rising tone, rising amplitude, merger, ringdown. What LIGO heard in 2015.
N-Body Gravity
200 particles attracting via Newton. Watch clusters form, galaxies collide, the chenciner-Montgomery figure-8 orbit, gravitational scattering.
The CMB Power Spectrum
Acoustic peaks in the cosmic microwave background. Slide Ω_b, Ω_cdm, Ω_k and watch peak positions shift. The most-constrained data in cosmology.
Cherenkov Cone
Why a nuclear reactor pool glows blue: charged particles outpacing light-in-medium leave a bow-shock cone of photons behind them.
Action, Fields & Gauge
Nature takes the path of stationary action — and hides its forces inside symmetries.
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Brachistochrone
Curve of fastest descent. Cycloid beats line, parabola, and arc. Newton solved Bernoulli's overnight challenge in 1696 — "ex ungue leonem."
The Path Integral
A particle takes every path at once; each contributes a phasor e^(iS/ℏ). Wild paths cancel, the classical path survives. Watch least action emerge as ℏ→0.
The Higgs Potential
Spontaneous symmetry breaking made visible. V(φ) potential with continuous vacuum ring. Drop a ball — it rolls into one of infinitely many vacua.
Berry Phase
Parallel-transport a vector around a closed loop on a sphere. It returns rotated by the enclosed solid angle. Foucault pendulum, Berry phase, gauge holonomy.
Aharonov-Bohm
Electrons pick up a phase from a magnetic field they never touch. The vector potential A is more real than the field B. Gauge theory's first signature.
Statistical Physics
Trillions of particles, a few simple laws: how order, disorder and criticality emerge.
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Maxwell-Boltzmann
Hard-sphere gas thermalizes from a delta-function speed distribution to the MB distribution. Boltzmann's H-theorem in action.
Galton Board
Balls bounce off pegs; the histogram approaches the Gaussian bell curve. The Central Limit Theorem in physical form. Bias the pegs and watch the peak shift.
Bose, Fermi & Boltzmann
Three distributions side-by-side. Bosons crowd, fermions exclude, classical particles distribute exponentially. The reason matter is solid.
Ising Model
Magnetic spins on a lattice; Metropolis algorithm. Above T_c = 2.27 J/k_B, randomness; below, spontaneous order. Onsager's exact 1944 critical exponents.
Percolation
Occupy each site with probability p. At p_c ≈ 0.5927 a spanning cluster suddenly appears. The simplest phase transition.
Sandpile Criticality
Bak-Tang-Wiesenfeld. Drop grains; cells topple at 4; avalanches form. Power-law size distribution emerges without tuning — self-organized criticality.
Renormalization Group
Zoom out by majority-voting blocks of spins. Away from T_c detail washes out; at criticality the system looks the same at every scale. Why universality exists.
Quantum
Superposition, entanglement, topology — and perhaps the origin of space.
Quantum Foundations
Superposition, measurement, entanglement — the experiments that broke classical intuition, one by one.
8Schrödinger Wavepacket
Gaussian wavepacket launched at a potential. Watch tunneling, spreading, and reflection. The Schrödinger equation propagated numerically.
Stern–Gerlach Chain
Sequential spin measurements. cos²(θ/2) transmission. The middle SG destroys information set by the first — non-commuting observables.
Quantum Zeno
Measure a qubit often enough and it stops evolving. The watched pot really doesn't boil — used in quantum error correction.
Wigner Function
Phase-space representation of a quantum state. The negative regions are non-classicality made visible — Schrödinger cats glow with them.
Quantum Walk
Quantum walker spreads ballistically (∝ N) — classical walker diffusively (∝ √N). Bimodal distribution with peaks at ±N/√2.
CHSH Bell Game
Alice and Bob cooperate against a referee. Classical max: 75%. With entangled qubits: 85.36%. Quantum violates local realism in a measurable game.
GHZ Paradox
Three-party entanglement where quantum predicts +1 with certainty and every local hidden-variable theory predicts −1. No probability — direct contradiction.
Quantum Teleportation
Step through the 3-qubit BBCJPW circuit. Alice measures, sends 2 classical bits, Bob applies correction. The qubit appears on Bob's end.
Quantum Matter & Holography
Topology protects quantum states — and entanglement may be what spacetime is built from.
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Hofstadter Butterfly
Fractal energy spectrum of electrons in a magnetic field. Avila Fields Medal 2014 (Ten Martini Problem). Quantum Hall conductances are the gap topologies.
The Chern Number
An integer counts how many times a quantum state wraps a sphere over momentum space. It can't change smoothly — so edge currents are topologically protected.
Anyon Braiding
In 2D, exchanging identical particles can give any phase e^(iθ) — not just ±1. Non-abelian variants are how topological quantum computers store qubits.
The Page Curve
Black hole evaporation. Hawking says information is destroyed; Page says it must come back; 2019 island formula proves Page is right from gravity alone.
Ryu-Takayanagi
Holographic entanglement entropy: pick a region on the boundary, the minimal bulk geodesic computes the entanglement. Geometry IS information.
MERA Tensor Network
A hierarchical tree of tensors encodes a critical quantum state. The geometry is hyperbolic — Swingle: this is AdS/CFT. Spacetime from entanglement.
Life & Complexity
Emergence, evolution and strategy: how simple rules build living worlds.
Emergence
Local rules, global structure: computation, spirals and fractal growth from almost nothing.
5Wolfram’s Cellular Automata
All 256 1D elementary cellular automata. Rule 30 (chaos, used as Mathematica's RNG), Rule 90 (Sierpinski), Rule 110 (Turing-complete).
Game of Life
Conway's two-rule universe. Gliders, glider guns, R-pentominos. Turing-complete. The canonical demonstration that simple rules → universal computation.
Turing Patterns
Two chemicals, one activator + one inhibitor, diffusing at different rates. Zebra stripes, leopard spots — biology emerging from chemistry, 1952.
Belousov-Zhabotinsky
Chemical clock. Homogeneous solution spontaneously breaks symmetry into spiral waves. Same math describes cardiac fibrillation.
Diffusion-Limited Aggregation
Lightning, mineral dendrites, viscous fingering, bacterial colonies. Fractal dimension ~1.71. Same random-walk + stick recipe everywhere.
Living Systems
The mathematics evolution found first: golden spirals, predator cycles, drift, foraging, prediction.
6Phyllotaxis
Fibonacci spirals in plants. Move the angle 0.2° off the golden — packing breaks. The unique optimal seed-placement angle, made visible.
Lotka-Volterra
Predator-prey cycles. Closed orbits in (rabbits, foxes) phase space. Click to start from any initial condition — Hudson's Bay lynx-hare data fits perfectly.
Lévy Flights
Brownian vs heavy-tailed walks side by side. Rare giant leaps explore sparse worlds far faster than diffusion — sharks and T-cells seem to search this way.
Genetic Drift
Wright-Fisher model: in finite populations, alleles fix or vanish by chance alone. Kimura's neutral theory — most evolution is noise.
DNA Topology
Linking = Twist + Writhe. The conservation law that makes topoisomerase drugs work. 3D rotatable double helix with supercoiling.
Free Energy Principle
A brain is a prediction machine. Bayesian observer minimizes surprise about the world. Friston: all persistent self-organizing systems do this.
Games & Mechanisms
Strategy, cooperation, fairness — and how to design rules that make honesty the best policy.
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Hawk–Dove & the ESS
Replicator dynamics finds the evolutionarily stable strategy p* = V/C. Selection without rational agents — the bridge between Darwin and game theory.
Axelrod Tournament
Iterated Prisoner's Dilemma. 10 strategies face off in round-robin. Tit-for-Tat wins. Try with noise to see why generous variants take over.
Ultimatum Game
Rational theory says accept any offer > 0. Humans reject below 30%, even at cost to themselves. Behavioral economics in one game.
Nim & Sprague–Grundy
Three piles of stones. The XOR of pile sizes tells you whether you've already lost. Play against the optimal AI; understand the XOR strategy.
VCG Auction
Second-price auction where truthful bidding is dominant strategy. Vickrey 1996 Nobel. Foundation of modern algorithmic mechanism design.
Gale-Shapley
Stable matching via propose-and-reject. Step through the algorithm. NRMP, NYC school choice, kidney exchange — all run on this.
Voting Paradoxes
Plurality, Borda, Condorcet — three methods, sometimes three winners. The Condorcet cycle, Arrow's impossibility, no perfect democracy.
Information
Bits are physical. Entropy, computation, and the machines that learn.
Information & Computation
A bit is physical. It has entropy, costs energy to erase — and now it runs the world.
7Shannon Information
Entropy as the irreducible bits in a message; channel capacity C = 1 − H(f) as a hard speed limit. The 1948 theorems that built the digital age.
Maxwell's Demon
A demon sorts molecules for free — until you count the cost of its memory. Erasing one bit costs kT ln2 (Landauer). Information is physical.
Fourier Transform
Draw a signal; see its spectrum decompose into sine waves. JPEG, MP3, MRI, radar — every digital technology uses this.
PageRank
Random surfer on a graph; long-run probability = dominant eigenvector. The algorithm that built Google. Brin & Page 1998.
Shor's Algorithm
Quantum factoring turns the hardness behind RSA into easy period-finding. Watch aˣ mod N cycle, the QFT peak into a comb, and the factors fall out.
The Surface Code
Protect a qubit you're not allowed to look at. Click to inject errors; stabilizers light up as a syndrome. The path to fault-tolerant quantum computing.
Self-Attention
The one operation inside every LLM: softmax(Q·Kᵀ/√d)·V. Each word looks at every other and pulls in a weighted blend. Hover a row to see what it attends to.
Dog Physics
The author’s notebook: why observers agree on a shared world — and a conjecture about where objectivity comes from.
2Quantum Darwinism
Zurek's R_δ: how a quantum fact becomes classical when it's redundantly broadcast across many environment fragments. The plateau is objective reality.
REG Bridge
Schematic of the proposed bridge from Zurek's environmental redundancy to Jacobson's thermodynamic derivation of Einstein's equations.
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