Science labs › Maths › Fractals
One simple rule, repeated, grows a whole tree. How many twigs does it make, and how much space does it fill?
Fractals lab · Last updated: 16 October 2026
One simple rule, repeated, can grow a whole tree. In this 3D lab, you apply a branching rule again and again, count the twigs and see how the shape fills space. It shows how nature builds ferns, rivers, lungs and lightning from patterns that repeat at every scale.
What you'll discover
- A fractal is a shape whose small parts look like smaller copies of the whole.
- If every branch splits in two, the number of twigs doubles with each round, growing very quickly.
- A fractal tree fills space more than a line does but less than a solid area, which a fractal dimension can describe.
The science in plain words
A fractal is a pattern that repeats itself at different sizes. Zoom in on a branch of a fractal tree and it looks like a smaller tree; zoom in on a twig and it looks like a smaller branch. This property is called self-similarity. Fractals can be made by taking a simple rule and repeating it over and over, a process called iteration.
Take a trunk and add two smaller branches at its tip, angled apart. Then apply the same rule to each new branch, and again to each of theirs. Each round doubles the number of tips: after 1 round there are 2, after 5 rounds 32 and after 10 rounds 1,024. Meanwhile the branches shrink, so the tree stays a manageable size while becoming ever more detailed.
Fractals fill space in an unusual way. A straight line is one-dimensional and a filled square is two-dimensional, but a fractal tree or a crinkly coastline lies somewhere in between. Mathematicians measure this with a fractal dimension, which can be a number such as 1.5. Nature uses fractal branching because it is an efficient way to reach a large area or volume: trees spread leaves to catch light, lungs pack an enormous surface into the chest to absorb oxygen, and blood vessels reach every cell. Ferns, river networks, lightning and romanesco broccoli show fractal patterns too.
Where it fits
- India, NCERT: A maths topic beyond the NCERT science syllabus for classes 6 to 10.
- United States: Grades 6 to 12
Common questions
What is a fractal?
A fractal is a shape made of parts that look like smaller copies of the whole, at many different sizes. Fractals are often created by repeating one simple rule again and again. Ferns, snowflakes, coastlines, trees and broccoli all show fractal patterns.
Where can you find fractals in nature?
Fractals appear in tree branches, fern leaves, river networks, lightning bolts, coastlines, snowflakes and romanesco broccoli. Inside the body, the airways of the lungs and the network of blood vessels branch in fractal patterns, which helps them reach every part efficiently.
What is a fractal dimension?
A fractal dimension measures how completely a fractal fills space. A line has dimension 1 and a flat filled shape has dimension 2. A wiggly fractal curve or branching tree can have a dimension in between, such as 1.3 or 1.6, showing it is more than a line but less than a full area.
Related labs
- Fibonacci Spiral: Count the spirals in a sunflower and you find 21, 34, 55… Why does a flower know these numbers?
- Chaos Theory: Two waterwheels start almost exactly the same. How long until they do something completely different?
- Storm over the Town: Where will the lightning strike? Watch it feel its way down, then add a lightning rod.
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Try it yourself
The Fractals lab is one of the interactive 3D science labs inside Taz. Change one thing, predict what will happen, then run it and watch. Labs open after you sign in.