arithmos
01Explore ↗Quadratic worlds
Mandelbrot: the parameter universe
One critical orbit maps an entire universe of quadratic rules.
THE MATHEMATICS OF WONDER
Small rules.
Extraordinary worlds.
Explore 40 remarkable shapes.
Then discover the maths that makes them.
START WITH A SIMPLE RULEThe world of z² + c
The collection
Explore. Understand. Build.40 explorations
01Explore ↗
02Explore ↗Quadratic worlds
Julia: one rule, infinite starting points
A single parameter transforms the fate of every starting point.
03Explore ↗Unexpected geometries
Gauss-sum curlicues
Add roots of unity in quadratic order to draw a complex exponential sum.
04Explore ↗Unexpected geometries
Integral Apollonian gasket
Exact integer bends build a fractal lace of tangent circles.
05Explore ↗Hidden sequences
Tribonacci Rauzy tile
Project the letter counts of an endless substitution word into three interlocking regions.
06Explore ↗Unexpected geometries
Conway’s quadratic river
A quadratic sign change threads a river through rational hyperbolic triangles.
07Explore ↗Hidden sequences
Fibonacci word fractal
An exact odd-even drawing rule builds repeating bays and corridors from a binary word.
08Explore ↗Quadratic worlds
The quadratic road to chaos
A real parabola grows a branching atlas of long-term states.
09Explore ↗
10Explore ↗Unexpected geometries
Theodorus’s square-root sail
Unit steps build a spiral whose spoke lengths are square roots.
11Explore ↗Unexpected geometries
Integer gear filigree
Two integer radii close a rolling-circle curve into intricate filigree.
12Explore ↗
13Explore ↗
14Explore ↗
15Explore ↗Polynomials & arithmetic
Euler’s luminous parabola
A polynomial draws its own diagonal through the integers.
16Explore ↗
17Explore ↗
18Explore ↗
19Explore ↗
20Explore ↗
21Explore ↗
22Explore ↗
23Explore ↗
24Explore ↗Unexpected geometries
The visible lattice orchard
A point is visible from the origin precisely when its coordinates are coprime.
25Explore ↗Unexpected geometries
Rational billiard embroidery
A rational slope folds a straight line into a closed square orbit.
26Explore ↗Unexpected geometries
Hilbert’s binary labyrinth
Binary digits guide a continuous traversal of a square lattice.
27Explore ↗
28Explore ↗Unexpected geometries
Paley residue rosette
Quadratic-residue differences decide which vertices are neighbors.
29Explore ↗Unexpected geometries
Hamming cube constellation
Binary words become vertices; flipping one bit creates an edge.
30Explore ↗Unexpected geometries
Finite projective incidence
A prime field builds a geometry where every pair of lines meets.
31Explore ↗Unexpected geometries
Pisano’s Fibonacci return
Consecutive Fibonacci remainders close into an exact finite orbit.
32Explore ↗Hidden sequences
Rudin–Shapiro turning tapestry
Read the sequence as a turtle path: turn by each term times 90°, then move one unit.
33Explore ↗Hidden sequences
Paperfolding dragon trace
Read the sequence as a turtle path: turn by each term times 90°, then move one unit.
34Explore ↗
35Explore ↗
36Explore ↗
37Explore ↗
38Explore ↗
39Explore ↗
40Explore ↗No shapes match that search. Try “prime”, “quadratic”, or “fractal”.