Fractal Mathematics
Types of Fractals: 7 Classes Explained
Mathematicians sort fractals two ways — by how their parts repeat (exact, quasi, and statistical self-similarity) and by how they are built (iterated function systems, escape-time formulas, strange attractors, and L-systems). Here are the seven classes, with real examples and their fractal dimensions.
Ask three mathematicians to name the “types” of fractals and you may get three different lists — because there are two equally valid ways to sort them. You can classify a fractal by how its parts relate to the whole (the kind of self-similarity it shows) or by how it is generated (the algorithm that produces it). The standard reference taxonomy, used by Wikipedia and most textbooks, combines both. This guide walks through seven well-defined classes that together cover almost every fractal you will encounter, from the Koch snowflake to the cosmic web.
Before the categories, one anchor. Benoit Mandelbrot's 1982 working definition is that a fractal is a set whose Hausdorff–Besicovitch dimension strictly exceeds its topological dimension — in plainer terms, a shape rougher and more space-filling than its naive dimension suggests. (He later loosened this to “a rough or fragmented geometric shape that can be split into parts, each a reduced-size copy of the whole.”) If you want the underlying number that makes a shape qualify, read our explainer on fractal dimension. Here we focus on the families.
Key takeaway: Fractals are classified two ways. By self-similarity there are three grades — exact (Koch, Sierpiński), quasi (Mandelbrot set), and statistical (coastlines, clouds). By generation method the major families are iterated function systems, escape-time fractals, strange attractors, and L-systems. Those seven classes, taken together, describe nearly every fractal in math and nature.
What are the three types of self-similarity?
The first axis of classification asks a single question: when you zoom in, does the small piece look exactly, approximately, or merely statistically like the whole? This produces the three grades of self-similarity that nearly every fractal text repeats.
1. Exact self-similar fractals
The strongest grade. Magnify any sub-region and you find a perfect, undistorted copy of the entire figure. These are the textbook fractals, almost always built by a deterministic rule applied infinitely. The von Koch snowflake — an equilateral triangle with a smaller triangle grafted onto the middle third of every edge, forever — is the canonical example, and so is the Sierpiński triangle. The Cantor set (repeatedly delete the middle third of a line) belongs here too. Because the generating rule never changes, the structure is genuinely identical at every scale.
2. Quasi-self-similar fractals
A looser grade. Zoom in and you find near-copies of the whole — recognizable, but distorted or degenerate. Fractals defined by a recurrence relation typically fall here, and the most famous of all, the Mandelbrot set, is the textbook case: its boundary is studded with tiny “baby Mandelbrots” that resemble but never perfectly duplicate the parent. As the Mandelbrot set entry notes, the satellites are similar, not congruent.
3. Statistically self-similar fractals
The weakest — and most common in the real world. Here no piece reproduces the whole, but a numerical measure such as the fractal dimension stays constant across scales. This is the self-similarity of nature: coastlines, mountain ranges, clouds, river networks, and the branching of your own lungs. They are sometimes called random or approximate fractals, and they hold their fractal character only across a finite band of scales — usually two to four orders of magnitude — rather than to infinity. For dozens of worked examples, see our guide to fractals in nature.
A fourth grade, multifractals, deserves a mention: these objects need more than one scaling exponent to describe them — a whole spectrum of fractal dimensions rather than a single number. Turbulence, financial price series, and rainfall fields are typically multifractal. Most introductory courses fold them into the “statistical” bucket, which is why the headline count is three grades of self-similarity, not four.
How are fractals classified by how they are generated?
The second axis is constructive: what algorithm draws the shape? This is where the remaining four of our seven classes live. The same fractal can sit in two boxes at once — the Koch snowflake is both exactly self-similar and an L-system fractal — because the two axes answer different questions.
4. Iterated function systems (IFS)
An IFS builds a fractal from a small set of contraction maps — affine transformations that shrink, rotate, and shift the plane — applied over and over. Because every map contracts (its scaling factor is below 1), the process converges on a unique limiting shape called the attractor. Michael Barnsley's Barnsley fern, from his 1988 book Fractals Everywhere, is the showcase: four affine maps, each chosen at random with a fixed probability and applied to a moving point — the so-called chaos game — trace out a strikingly realistic Black Spleenwort fern. The Sierpiński triangle is also an IFS (three half-scale maps to the corners of a triangle), which is why a single shape can be both exactly self-similar and IFS-generated.
5. Escape-time fractals
These are defined by iterating a formula at every point of a space and asking how fast the orbit runs to infinity. Points that never escape form the set; points that escape are colored by how quickly they did, producing the iconic glowing halos. The Mandelbrot set — iterate zn+1 = zn2 + c on the complex plane — and its cousins the Julia sets are the defining members. Escape-time fractals are deterministic (no randomness) yet only quasi-self-similar, neatly illustrating how the two classification axes cut across each other.
6. Strange attractors
When you plot the long-term trajectory of certain chaotic dynamical systems, the path never repeats yet stays bounded, settling onto a set with fractal structure: a strange attractor. The Lorenz attractor — Edward Lorenz's 1963 simplified model of atmospheric convection — is the iconic example, tracing its famous butterfly shape in three-dimensional space. Strange attractors are the geometric face of chaos theory: they show that deterministic equations can produce behavior that is unpredictable in detail yet fractal in overall form.
7. L-system (Lindenmayer) fractals
An L-system is a string-rewriting grammar invented by biologist Aristid Lindenmayer in 1968 to model plant growth. Starting from an axiom, symbols are recursively replaced by production rules; the resulting string is then read as turtle-graphics instructions that draw the figure. L-systems generate branching plant forms, the Koch and dragon curves, and space-filling curves such as the Hilbert curve (David Hilbert, 1891) and the Peano curve (Giuseppe Peano, 1890) — continuous curves that pass through every point of a square, achieving fractal dimension 2.
A seventh constructive family, finite subdivision rules, rounds out the standard list: recursive algorithms that refine a tiling into ever-finer cells. They underlie the formal study of self-similar tilings, but for an introductory map the four families above — IFS, escape-time, strange attractors, and L-systems — cover the fractals most people meet.
What are the fractal dimensions of the most famous types?
The clearest way to compare classes is by their fractal (Hausdorff) dimension — the non-integer number that quantifies roughness. Note that the dimension is independent of the generation method: the same value can describe shapes from different families. The figures below come from the list of fractals by Hausdorff dimension.
| Fractal | Self-similarity | Generated by | Hausdorff dimension |
|---|---|---|---|
| Cantor set | Exact | Deletion rule / IFS | log₃2 ≈ 0.6309 |
| Koch curve / snowflake | Exact | L-system / replacement | log₃4 ≈ 1.2619 |
| Sierpiński triangle | Exact | IFS | log₂3 ≈ 1.5850 |
| Sierpiński carpet | Exact | IFS | log₃8 ≈ 1.8928 |
| Menger sponge | Exact | IFS (3D) | log₃20 ≈ 2.7268 |
| Mandelbrot set boundary | Quasi | Escape-time | 2 (exactly) |
| British coastline | Statistical | Natural process | ≈ 1.25 |
The Cantor set's dimension below 1 captures something profound: it is “more than nothing” (infinitely many points) yet “less than a line” (zero total length). At the other extreme, the Menger sponge — Karl Menger's 1926 three-dimensional generalization of the Sierpiński carpet — has a dimension of about 2.73, sitting between a surface and a solid: its volume shrinks to zero while its surface area grows without bound.
Why does the same fractal fall into more than one type?
This is the single most common point of confusion, so it is worth stating plainly. The two classification axes are orthogonal — they measure unrelated things. Self-similarity describes the finished object's geometry; generation method describes the recipe that produced it. A fractal therefore carries a label on each axis at once.
- The Koch snowflake is exactly self-similar (geometry) and an L-system / replacement fractal (recipe).
- The Sierpiński triangle is exactly self-similar and an IFS fractal — and can also be drawn by the chaos game or by Pascal's triangle mod 2.
- The Mandelbrot set is quasi-self-similar and an escape-time fractal.
- A coastline is statistically self-similar and produced by a natural (random) process.
Once you see the two axes as a grid rather than a single list, the apparent contradictions dissolve. For the deeper mechanics behind the geometry axis, see self-similarity; for the founding history of the whole field, start at our pillar guide, what is a fractal.
Frequently asked
What are the main types of fractals?
Fractals are classified along two independent axes. By self-similarity there are three grades: exact self-similar fractals (perfect copies at every scale, like the Koch snowflake and Sierpiński triangle), quasi-self-similar fractals (near-copies that are slightly distorted, like the Mandelbrot set), and statistically self-similar fractals (only a numerical measure is preserved, like coastlines and clouds). By generation method the major families are iterated function systems (IFS), escape-time fractals, strange attractors, and L-systems. Together these seven classes cover almost every fractal in mathematics and nature. The same shape can belong to one class on each axis at once.
What is the difference between exact and statistical self-similarity?
Exact self-similarity means that when you magnify any part of the fractal you find an identical, undistorted copy of the whole figure — this happens with deterministic mathematical fractals such as the Koch snowflake, the Sierpiński triangle, and the Cantor set. Statistical self-similarity is far weaker: no piece reproduces the whole, but a numerical property — usually the fractal dimension — stays roughly constant across scales. This is the kind found throughout nature, in coastlines, mountains, clouds, and river networks. Natural fractals are also bounded: they hold their fractal character across only two to four orders of magnitude, not to infinity like their mathematical counterparts.
Is the Mandelbrot set self-similar?
The Mandelbrot set is quasi-self-similar rather than exactly self-similar. When you zoom into its boundary you encounter countless tiny “baby Mandelbrots” that closely resemble the full set, but they are never perfect, congruent copies — each is distorted and embedded in different surrounding detail. This near-but-not-exact repetition is the hallmark of fractals defined by a recurrence relation, which is how the Mandelbrot set is built: it iterates the formula z² + c at each point of the complex plane. So while it shows endless structure under magnification, it does not satisfy the strict exact-self-similarity that the Koch snowflake or Sierpiński triangle do.
What is an iterated function system (IFS)?
An iterated function system is a method of building a fractal from a small collection of contraction maps — affine transformations that shrink, rotate, and shift space, each with a scaling factor below 1. Applying these maps repeatedly converges on a single limiting shape called the attractor, which is the fractal. Michael Barnsley popularized the technique in his 1988 book Fractals Everywhere; his Barnsley fern uses just four affine maps chosen at random by probability and applied to a moving point, an approach called the “chaos game.” The Sierpiński triangle is also an IFS, generated by three half-scale maps toward the corners of a triangle. IFS fractals are typically exactly or quasi self-similar.
What are escape-time fractals?
Escape-time fractals are defined by iterating a formula at every point in a space and recording how quickly the resulting sequence “escapes” toward infinity. Points whose orbits stay bounded belong to the set and are usually colored black; points that escape are colored according to how fast they did, which creates the glowing gradient halos seen in classic renderings. The Mandelbrot set — built by iterating z¹₊¹ = z² + c on the complex plane — and the related Julia sets are the defining examples. Escape-time fractals are fully deterministic, with no randomness involved, yet they are only quasi-self-similar, which shows that a fractal's generation method and its grade of self-similarity are separate properties.
What is a strange attractor?
A strange attractor is the fractal-shaped set that the long-term trajectory of a chaotic dynamical system settles onto. As the system evolves, its path never exactly repeats yet remains bounded, weaving forever through the same fractal region of state space. The most famous example is the Lorenz attractor, which Edward Lorenz derived in 1963 from a simplified model of atmospheric convection; plotted in three dimensions it forms the celebrated butterfly shape. Strange attractors are the geometric signature of chaos theory: they demonstrate that perfectly deterministic equations can generate motion that is impossible to predict in detail over the long run while still possessing an elegant, self-similar fractal structure.
Are L-systems considered fractals?
Yes — L-systems are one of the standard fractal-generation families. An L-system, or Lindenmayer system, is a string-rewriting grammar that biologist Aristid Lindenmayer introduced in 1968 to model plant growth. Beginning from a starting string called the axiom, symbols are repeatedly replaced according to production rules, and the final string is interpreted as turtle-graphics drawing commands. This recursive rewriting naturally produces self-similar output, which is why L-systems can render realistic branching plants as well as classic fractals like the Koch curve, the dragon curve, and space-filling curves such as the Hilbert and Peano curves. Those space-filling curves are remarkable because they pass through every point of a square, reaching fractal dimension 2.
How many types of fractals are there really?
There is no single official count, because fractals are sorted along two different axes at once. By self-similarity the standard reference taxonomy lists three grades — exact, quasi, and statistical — sometimes adding multifractals as a fourth. By generation method the commonly cited families are iterated function systems, escape-time fractals, strange attractors, L-systems, random fractals, and finite subdivision rules. Combining the three most useful self-similarity grades with the four most common construction families gives the practical “seven classes” framework most introductions use. The key insight is that any given fractal carries one label from each axis simultaneously — the Koch snowflake, for instance, is both exactly self-similar and L-system-generated.