# 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.

*Published 2026-06-25 · Updated 2026-06-25 · By Dr. Elena Marchetti*

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](https://en.wikipedia.org/wiki/Fractal) 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](https://fractal.us/famous-fractals/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](https://fractal.us/mathematics/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](https://fractal.us/famous-fractals/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](https://fractal.us/famous-fractals/sierpinski-triangle). The [Cantor set](https://en.wikipedia.org/wiki/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](https://fractal.us/famous-fractals/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](https://en.wikipedia.org/wiki/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](https://fractal.us/mathematics/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](https://fractal.us/nature/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](https://en.wikipedia.org/wiki/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](https://fractal.us/famous-fractals/mandelbrot-set) — iterate **zn+1 = zn2 + c** on the complex plane — and its cousins the [Julia sets](https://en.wikipedia.org/wiki/Julia_set) 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](https://en.wikipedia.org/wiki/Lorenz_system) — 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](https://en.wikipedia.org/wiki/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](https://mathworld.wolfram.com/LindenmayerSystem.html) 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](https://en.wikipedia.org/wiki/Hilbert_curve) (David Hilbert, 1891) and the [Peano curve](https://en.wikipedia.org/wiki/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](https://fractal.us/mathematics/fractal-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](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension).

FractalSelf-similarityGenerated byHausdorff dimension

Cantor setExactDeletion rule / IFSlog₃2 ≈ 0.6309
Koch curve / snowflakeExactL-system / replacementlog₃4 ≈ 1.2619
Sierpiński triangleExactIFSlog₂3 ≈ 1.5850
Sierpiński carpetExactIFSlog₃8 ≈ 1.8928
Menger spongeExactIFS (3D)log₃20 ≈ 2.7268
Mandelbrot set boundaryQuasiEscape-time2 (exactly)
British coastlineStatisticalNatural 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](https://fractal.us/famous-fractals/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](https://fractal.us/mathematics/self-similarity); for the founding history of the whole field, start at our pillar guide, [what is a fractal](https://fractal.us/what-is-a-fractal).

## Sources

1. [https://en.wikipedia.org/wiki/Fractal](https://en.wikipedia.org/wiki/Fractal)
2. [https://mathworld.wolfram.com/Fractal.html](https://mathworld.wolfram.com/Fractal.html)
3. [https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension)
4. [https://en.wikipedia.org/wiki/Mandelbrot_set](https://en.wikipedia.org/wiki/Mandelbrot_set)
5. [https://en.wikipedia.org/wiki/Julia_set](https://en.wikipedia.org/wiki/Julia_set)
6. [https://en.wikipedia.org/wiki/Barnsley_fern](https://en.wikipedia.org/wiki/Barnsley_fern)
7. [https://en.wikipedia.org/wiki/Cantor_set](https://en.wikipedia.org/wiki/Cantor_set)
8. [https://en.wikipedia.org/wiki/Menger_sponge](https://en.wikipedia.org/wiki/Menger_sponge)
9. [https://en.wikipedia.org/wiki/Lorenz_system](https://en.wikipedia.org/wiki/Lorenz_system)
10. [https://en.wikipedia.org/wiki/Chaos_theory](https://en.wikipedia.org/wiki/Chaos_theory)
11. [https://en.wikipedia.org/wiki/Hilbert_curve](https://en.wikipedia.org/wiki/Hilbert_curve)
12. [https://en.wikipedia.org/wiki/Peano_curve](https://en.wikipedia.org/wiki/Peano_curve)
13. [https://mathworld.wolfram.com/LindenmayerSystem.html](https://mathworld.wolfram.com/LindenmayerSystem.html)

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Source: https://fractal.us/mathematics/types-of-fractals
Index: https://fractal.us/llms.txt · Full text: https://fractal.us/llms-full.txt