A fractal is a shape that stays rough however closely you look. Zoom into a circle and it flattens into a line; zoom into a fern, a coastline or the Mandelbrot set and new detail keeps arriving, often in the form of smaller copies of the whole. This chapter is about what that means, how to tell, and why nature prefers it.
Common mix-up: self-similar does not mean fractal. A straight line is made of two half-size copies of itself, and so is a square, made of four; neither is a fractal. What makes a fractal is that the copies carry new detail at every scale, so the shape never smooths out under magnification. And many real fractals, from coastlines to the Mandelbrot set, contain no exact copies at all.
In 1975 the mathematician Benoit Mandelbrot published a book in French called Les objets fractals: forme, hasard et dimension. He needed a name for a family of shapes he had been collecting for years: shapes that are jagged, broken or fragmented, and that stay that way at every magnification. He built the word from the Latin adjective fractus, from the verb frangere, to break. The same root gives English "fracture" and "fraction." An English version, Fractals: Form, Chance and Dimension, followed in 1977, and in 1982 he expanded it into The Fractal Geometry of Nature, the book that made the word famous.
The shapes themselves were much older than the name. Through the late nineteenth and early twentieth centuries, mathematicians had built a series of strange objects, usually to show that some intuition was wrong. In 1872 Karl Weierstrass presented a curve that is continuous everywhere but has no tangent anywhere: it is all corners. Georg Cantor's set of 1883 is a line with infinitely many gaps that still has as many points as the line. Giuseppe Peano's curve of 1890 passes through every point of a square. Helge von Koch's curve of 1904, the one you saw grow into a snowflake above, was designed as a friendlier, drawable example of a curve without tangents. Wacław Sierpiński added his triangle in 1915 and his carpet in 1916.
At the time these were treated as exceptions, curiosities kept away from "real" geometry. They were often called monsters. Mandelbrot's insight was to turn that around. The monsters, he argued, are not exceptions at all. They are much better descriptions of coastlines, clouds, mountains and trees than the smooth lines, circles and cones of Euclid. Chapter 3 takes the monsters one at a time.
Not one that everyone uses. In The Fractal Geometry of Nature Mandelbrot offered a precise definition: a fractal is a set whose Hausdorff dimension is strictly greater than its topological dimension. The Koch curve qualifies: topologically it is a curve, dimension 1, but its Hausdorff dimension is about 1.26.
The definition turned out to be too narrow. Some shapes that everyone agrees are fractal have a whole-number dimension. The boundary of the Mandelbrot set, for example, has Hausdorff dimension exactly 2 (Mitsuhiro Shishikura proved this in 1998), and a plane-filling curve like Peano's also has dimension 2. Mandelbrot himself later preferred to leave the word loosely defined, much as biologists use "life" without a one-line definition.
The working list that most textbooks use, following Kenneth Falconer's Fractal Geometry, describes a typical fractal set F like this: it has detail at arbitrarily small scales; it is too irregular to describe with ordinary calculus or geometry, either locally or globally; it often has some form of self-similarity, perhaps approximate or statistical; its fractal dimension is usually larger than its topological dimension; and in many interesting cases it is defined very simply, often by repeating a rule. The rest of this chapter takes those properties one at a time. Chapter 2 is about the dimension.
The property people notice first is that part of a fractal looks like the whole. Break off a floret of broccoli and it looks like a small head of broccoli. Look at a branch of a tree and it looks like a small tree. That is self-similarity, and it comes in three strengths.
Exact self-similarity is the mathematician's version. The Sierpinski triangle is made of exactly three copies of itself, each half as wide. The Koch curve is exactly four copies of itself, each a third as long. You can write the whole shape down as a short list of shrink-and-move rules, and the shape is the one thing that those rules leave unchanged. John Hutchinson showed in 1981 that every such list of shrinking rules has exactly one shape like that, which is how the fern in the walkthrough can be stored in 24 numbers (Chapter 4).
Quasi-self-similarity is looser. The Mandelbrot set contains infinitely many small copies of itself, scattered along its filaments, but each copy is slightly distorted and surrounded by different decoration. Zoom in near one and you recognize it; lay it over the original and it doesn't quite match. Julia sets behave the same way: the dynamics that generate them copy small regions onto large ones, but not by a pure scaling.
Statistical self-similarity is the kind nature uses. A photograph of a rocky coastline from an airplane and one from a cliff top are not copies of each other, but you often can't tell which is which, because the wiggles have the same character at both scales. The same is true of a mountain range and a single rock, or a river network and one of its tributaries. No piece is a copy of the whole; the statistics of the roughness are what repeat.
| Kind | What repeats | Examples | How you'd check |
|---|---|---|---|
| Exact | The shape itself, precisely, at fixed zoom factors | Koch curve (×3), Sierpinski triangle (×2), Cantor set (×3), Barnsley fern | Overlay a scaled copy: it matches pixel for pixel |
| Quasi | Distorted copies, never quite identical | Mandelbrot set, Julia sets | Copies are recognizable but don't overlay exactly |
| Statistical | The character of the roughness | Coastlines, mountains, clouds, river networks, Brownian motion | Measurements like the box count follow the same power law at many scales |
| Self-affine | The shape, after stretching different directions by different amounts | The graph of Brownian motion, many terrain profiles | Rescale one axis by a different factor than the other and the statistics match |
Some shapes only look like themselves after you stretch them unevenly. The classic example is the graph of Brownian motion, the path of a particle being knocked about by molecules, recorded as position against time. Squeeze the time axis by a factor of 4 and you have to squeeze the position axis by a factor of 2, the square root, for the statistics to match. Mandelbrot called such shapes self-affine, after the affine maps that stretch x and y separately.
The distinction matters when you measure. A self-similar curve and a self-affine graph can look equally rough, but methods that assume equal scaling in every direction, such as walking the curve with a ruler, can give different answers for a self-affine one depending on the range of scales you test. That's a warning worth carrying into Chapter 2.
Pick a shape and zoom: scroll or pinch on the picture, drag to move, double-click (or double-tap) to zoom in where you point, or press Dive to fall toward a point where the picture repeats. The drawing is never a stored image. At every magnification the program re-applies the shape's rule as many times as it takes to reach single-pixel detail, and the readout tells you how many levels that was. That number climbs as you zoom, and it never stops climbing: the detail does not run out.
Every picture of a fractal is a lie in one specific way: it stops. A screen has pixels, a printer has dots, and a real fern has cells. The mathematical object doesn't stop. Whatever magnification you pick, there is structure smaller than that, and the structure is not just a blurrier version of what you saw before. It is new.
The Koch snowflake shows what that does to ordinary measurements. Start with an equilateral triangle whose sides are 1 unit long, so its perimeter is 3. Step one: cut each side into thirds and replace the middle third with two sides of a smaller triangle pointing outward. Each side is now made of four pieces a third as long, so it is 4/3 as long as before. Step two does the same thing to every one of those 12 pieces. Every step multiplies the perimeter by 4/3, so after n steps it is:
The area, though, settles down. Each step adds small triangles, but they get smaller faster than they get more numerous: the number of new triangles goes up by a factor of 4, and each one's area goes down by a factor of 9. The additions form a shrinking series that adds up to exactly 3/5 of the original triangle, so the finished snowflake has 8/5 of the original area. It fits comfortably inside a circle drawn through the first triangle's corners.
So you have a shape with a finite area and an infinitely long boundary. A circle can't do that, and neither can any shape whose edge is smooth. It is a first hint that "length" is the wrong question to ask about a fractal curve, which is where Chapter 2 picks up.
Call the starting triangle's area A. At step 1 you add 3 triangles, each with sides a third as long, so each has area A/9. Total added: 3A/9 = A/3.
At step 2 there are 12 sides, so 12 new triangles, each with area A/81. Total added: 12A/81 = (A/3)(4/9). In general step n adds (A/3)(4/9)ⁿ⁻¹: four times as many triangles as the step before, each a ninth of the area.
That's a geometric series with ratio 4/9. Its sum is (A/3) ÷ (1 − 4/9) = (A/3) × (9/5) = 3A/5. Add the original triangle and the total is A + 3A/5 = 8A/5. The perimeter's series has ratio 4/3, which is larger than 1, so it has no sum: that difference between 4/9 and 4/3 is the whole story.
If length is the wrong question, what is the right one? Here is the experiment that answers it. Walk along a curve with a pair of dividers, the drawing tool with two sharp points, set to some fixed opening r. Count the steps it takes to get around, multiply by r, and call that the length. Then close the dividers a little and do it again.
For a circle, the answers settle down fast. With a big opening you cut corners and get a polygon a little shorter than the circle; with a smaller opening the polygon hugs the curve, and the measured length closes in on π times the diameter. Halve the ruler again and the answer barely moves. That's what it means for a curve to have a length.
For the snowflake, every time you shrink the ruler it starts fitting into bumps it used to step over, and the measured length goes up. It never settles. But it goes up in a very orderly way: shrink the ruler by a factor of 3 and the length grows by a factor of 4/3, every time, at every scale. That fixed ratio is a number you can measure, and it describes how rough the curve is. Plot measured length against ruler size on logarithmic axes and you get a straight line; the steeper the line, the rougher the curve. The slope is how the fractal dimension is defined, and it's the subject of the next chapter.
This is exactly what Lewis Fry Richardson found with real maps. Measuring coastlines and land borders with smaller and smaller steps, he saw the lengths grow steadily instead of settling, and that the growth followed a power law. His results were published in 1961, after his death, and Mandelbrot made them famous in a 1967 paper in Science called "How Long Is the Coast of Britain?" Instrument 2 lets you repeat the experiment on a circle and a snowflake side by side.
Both shapes are the same size: the circle has a diameter of 1 unit, and the snowflake fits inside a circle of the same size. Drag the slider to shrink the ruler, or press Sweep. The program walks each outline with dividers of exactly that opening, counting steps the way a surveyor would, and draws the polygon it walked. The chart below records every measurement: the circle's line flattens out at π, the snowflake's keeps climbing.
Most familiar objects have a size. A coffee cup is about 10 centimeters tall, a door is about 2 meters, and a photograph of either tells you roughly how far away the camera was. A fractal doesn't have a size in that sense. A photograph of a cliff with no person, tree or car in it gives you no idea of scale: it could be a meter of rock or a kilometer. Geologists put a hammer or a coin in their photographs for exactly this reason.
The mathematical name for having no favorite size is scale invariance, and its signature is the power law. A quantity follows a power law when multiplying the scale by some factor always multiplies the quantity by the same fixed factor, wherever you start. The snowflake's measured length is like that: dividing the ruler by 3 multiplies the length by 4/3, whether the ruler was a meter or a micron. Compare an exponential process, such as radioactive decay, which has a built-in scale (the half-life). A power law has none.
Power laws turn up wherever fractals do. The number of craters on the Moon larger than a given size, the size of the lakes on a landscape, the energy released by earthquakes: each, over a wide range, has no characteristic scale. Finding a straight line on a log-log plot doesn't prove you have a fractal, but finding curvature is a good sign you don't.
Real objects are only scale invariant over a range. A coastline's roughness stops somewhere around the size of a grain of sand, and stops at the other end at the size of the continent. A tree branches perhaps ten or twenty times between trunk and twig, not forever. Physicists sometimes call these limits the inner and outer cutoffs. The mathematical fractal is an idealization that throws the cutoffs away, in the same way a mathematical circle throws away the atoms of a coin.
There is no agreed number, and it's an argument that comes up often in the scientific literature. A straight line on a log-log plot over a factor of 10 is weak evidence: plenty of curved relationships look straight over one decade. Over three or four decades, a factor of 1,000 to 10,000, a power law is much more convincing.
The instruments on this site have the same limits. The snowflake in Instrument 2 is built seven levels deep, so its smallest wiggle is 3⁷ = 2,187 times smaller than its sides. Below that the measured length would stop growing and the curve would behave like an ordinary polygon. The slider stops before you reach it.
Mandelbrot opened The Fractal Geometry of Nature with the sentence that became his motto: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." School geometry is built from lines, circles, planes and spheres. Those shapes are perfect for things people make, like wheels, roads and buildings. They are poor descriptions of almost everything that grew, eroded, cracked or flowed.
The reason is that natural shapes are usually made by a process that repeats across scales. A few patterns account for most of what you see.
There is often a practical reason to be fractal, too. A branching network is a good way to reach every point of a volume from one entrance, which is what blood vessels and airways have to do. A crinkled surface packs a lot of area into a small space, which is what lungs, gut linings and the folds of the brain need. Smoothness is the exception in nature because it is rarely useful.
| Thing | Euclidean cartoon | What it really does |
|---|---|---|
| Coastline | A smooth curve with a length | Length grows as the ruler shrinks (Richardson; Chapter 2) |
| Mountain | A cone | Ridges on ridges; a rock looks like a range |
| Tree | A stick with a ball on top | Branches split into smaller copies of the whole |
| Cloud | A sphere or blob | Edges stay ragged from kilometers down to meters |
| Lung | Two bags | About 23 generations of branching airways |
| Lightning | A straight line | A branching, jagged channel at every scale |
Because natural shapes are fractal, the fastest way to fake them on a computer is with a fractal rule. Film and game landscapes have been built from random fractal terrain since the early 1980s, and plants from rewriting rules like the L-systems of Chapter 5. A few lines of rule can stand in for millions of hand-placed polygons.
The same economy makes fractals good material for moving art. J's JellyDazzle kaleidoscope has about twenty fractal modes, from a Mandelbrot deep zoom to a growing fern; Part V of this site, on color and computer fractals, explains how they are drawn.
| Question | Smooth shape (circle, line) | Fractal (snowflake, coastline) |
|---|---|---|
| Zoom in | Flattens into a straight line | New detail keeps appearing |
| Length of the edge | A definite number | Grows as the ruler shrinks; infinite in the ideal |
| Tangent line | Exists at almost every point | Usually exists nowhere |
| Characteristic size | Its radius or length | None, over a range of scales |
| Dimension | A whole number: 1 for a curve | Often a fraction: about 1.26 for the Koch curve |
| How you describe it | A formula for the shape | A rule that is repeated |