Unit 4 of 4
Fractals, Patterns Without End
- 22 chapters
- Video 4, chapters 1 to 22
This video builds fractals from simple repeated rules, measures them with fractal dimension and follows them into chaos and the complex plane, with no complex numbers needed in advance.
The Koch snowflake, the Cantor set and Sierpinski's triangle each follow a single rule, and the same triangle returns in Pascal's triangle, the chaos game and iterated function systems, the recipe behind Barnsley's fern. L-systems and the dragon curve grow from short rules. Similarity dimension and box counting measure them, and Hilbert's curve, the Menger sponge and coastlines test what length, area and volume mean. The logistic map, period doubling and the Lorenz attractor bring iteration to changing systems, and complex numbers lead to Julia sets, the Mandelbrot set and Newton fractals.
Every chapter will be part of one of 4 long videos. The videos are coming soon, and each chapter page already has the definitions and an example for every key term.
One rule, repeated
A Shape Made from Smaller Versions of Itself
What a fractal is, explained with a triangle made of three half-size copies of itself, covering self-similarity, the iteration of one rule, and the difference between a finite approximation and its limit.
Video 4, chapter 1
The Koch Snowflake, a Boundary That Keeps Growing
How the Koch snowflake, made of three Koch curves, gets an infinite perimeter because each step multiplies its perimeter by 4/3, while a geometric series keeps its area finite at 8/5 of the starting triangle.
Video 4, chapter 2
The Cantor Set and What Remains After Removal
How the middle-thirds Cantor set is built by removing open intervals, why it has measure zero, and how ternary expansions with only the digits 0 and 2 name the points that remain.
Video 4, chapter 3
Sierpinski's Triangle, Holes Inside Holes
How recursive removal of middle triangles builds the Sierpinski triangle, also called the Sierpinski gasket, why its area is zero, and why the set that remains is still connected.
Video 4, chapter 4
Different recipes, the same structure
Pascal's Triangle Turns into a Fractal
Why shading the odd entries of Pascal's triangle modulo 2 draws Sierpinski's triangle, explained by the parity of binomial coefficients and by modular arithmetic, which makes the pattern copy itself at each power of two.
Video 4, chapter 5
The Chaos Game
How the chaos game, which moves halfway toward randomly chosen corners of a triangle, draws Sierpinski's triangle, why random iteration of halving maps forgets its start, and what its attractor is.
Video 4, chapter 6
Iterated Function Systems, a Tiny Recipe for a Shape
How an iterated function system of contraction mappings describes a shape as its invariant set, and why Hutchinson's theorem gives the uniqueness of the attractor for every closed and bounded starting shape.
Video 4, chapter 7
Barnsley's Fern
How Barnsley's fern arises from four affine transformations, why it is a self-affine set, and how the map probabilities of the chaos game change where the points pile up but not the shape itself.
Video 4, chapter 8
L-Systems, a Grammar That Grows Branches
How an L-system, or Lindenmayer system, grows a branching picture by parallel string rewriting from a starting string, how turtle graphics turns the symbols into a drawing, and how it differs from a formal grammar.
Video 4, chapter 9
Measuring the unfamiliar
The Dragon Curve, Folding a Path
How folding a strip in half again and again gives the Heighway dragon, why its turn sequence, the paperfolding sequence, grows by a reversed and exchanged copy, and how the paths approach the dragon curve.
Video 4, chapter 10
Fractal Dimension, Between a Line and an Area
How the similarity dimension log N over log s comes from counting scaled copies, why Sierpinski's triangle has fractal dimension about 1.585, and how logarithms, Hausdorff dimension and topological dimension fit in.
Video 4, chapter 11
Box Counting, Measuring a Pattern Across Scales
How the box-counting dimension measures a shape with grids of shrinking boxes, why it appears as the slope of a log-log plot, and how grid resolution, grid position and fit range move a finite dimension estimate.
Video 4, chapter 12
Hilbert's Space-Filling Curve
How the Hilbert curve, the limit of paths through finer and finer grids of cells, reaches every point of a square through nested squares, and why this space-filling curve is a continuous surjection but not one-to-one.
Video 4, chapter 13
The Menger Sponge
How the Menger sponge, a three-dimensional fractal built by removing 7 of 27 smaller cubes at every stage, keeps a connected framework while its volume tends to zero, and why its similarity dimension is about 2.727.
Video 4, chapter 14
From shapes to changing systems
Coastlines and the Scale of a Measurement
Why the coastline paradox happens, how the measured length of a rough boundary depends on the ruler length and the measurement scale, and how exact and statistical self-similarity differ from a real shoreline.
Video 4, chapter 15
The Logistic Map, Repeating One Equation
How the orbits of the logistic map, the rule that multiplies x by 1 minus x and by a parameter r, can settle at a stable fixed point or on a two-cycle, and how a cobweb diagram draws each step of this discrete dynamical system.
Video 4, chapter 16
Period Doubling and the Route to Chaos
How the logistic map follows the period-doubling route to chaos through a cascade of period-doubling bifurcations, how to read its bifurcation diagram, and why periodic windows interrupt the deterministic chaos.
Video 4, chapter 17
The Lorenz Attractor and Order Inside Chaos
How the Lorenz system of three equations produces the Lorenz attractor, a bounded two-lobed shape in phase space, and how sensitive dependence on initial conditions, the butterfly effect, limits forecasts on it.
Video 4, chapter 18
A map of mathematical worlds
Complex Numbers and the Plane Behind the Pictures
How a complex number a + bi becomes a point of the complex plane, why adding slides it, why multiplying by the imaginary unit i turns it, and why squaring squares its modulus and doubles its argument.
Video 4, chapter 19
Julia Sets, One Rule and Many Starting Points
Which starting points stay bounded when the rule z squared plus c is repeated with c fixed, how the filled Julia set and the Julia set answer it, and how an escape radius proves that an orbit escapes.
Video 4, chapter 20
The Mandelbrot Set, a Map of Julia Worlds
How the Mandelbrot set fixes the start 0 and varies c in the rule z squared plus c, why radius 2 proves escape, what escape-time pictures can and cannot show, and how the set decides which Julia sets are connected.
Video 4, chapter 21
Newton Fractals and Where an Approximation Will Land
How Newton's method sorts the starting points of the complex plane by the cube root of unity they approach, why these basins of attraction share one boundary, the Newton fractal, and where the method fails.
Video 4, chapter 22