Fractal Tree

About this simulation

This simulation builds a tree from a simple rule: each branch splits into two shorter branches, and the rule repeats. Students set the depth from 1 to 10, the split angle, and the length ratio r, and can turn on wind sway. Readouts show the number of tips at depth n, the total number of branches and the total branch length. The tree twists by the golden angle at each level so it fills 3D space. Teachers can use it to teach self-similarity, powers of two and geometric series.

What students can explore

Raise Depth from 1 to 10 and watch the Tips readout double each time.
Check that All branches equals two to the n plus one, minus one.
Change Split angle and describe how the tree shape changes.
Set Length ratio r above 0.5 and watch the Total branch length readout.
Turn Wind sway on and off to see the angles move level by level.

Learning objectives

1Describe self-similarity in a fractal.
2Predict the number of tips and branches for a given depth using powers of two.
3Explain how a length ratio affects total branch length as a geometric series.
4Investigate how angle and ratio change the shape of a tree.

Questions teachers ask

Is the fractal tree simulation free to use?

Yes. It is completely free and runs in any modern web browser, including on interactive whiteboards, tablets, Chromebooks and phones. There is nothing to install and no sign-up or account is needed.

What makes a tree a fractal?

Each branch is a smaller copy of the whole tree. Zooming in on any branch shows the same splitting pattern, which is called self-similarity.

How many branches are there at depth n?

There are two to the power n tips and two to the power n plus one, minus one branches in total. At depth 10 that is 1,024 tips and 2,047 branches.

How do I use the fractal tree simulation?

Start at depth 1 and increase it one step at a time, recording tips and branches. Then change the angle and ratio and describe the new shape.

Embed this simulation

Drop it into your LMS, blog, or class site — free.

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Continue exploring

Type any function and see its shape straight away. Plot up to three at once in three colours to compare them, drag the trace point to read a value off a curve, and see where each one crosses zero.

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Spin a spinner, flip a coin or roll a die and watch two sets of bars: the chances the shapes predict, and the results you actually got. Ten spins prove nothing; a few hundred and the two are hard to tell apart.

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Move m and watch the line tilt; move b and watch it slide. Or drag the two points on the line and read the equation it writes back. A rise-over-run staircase climbs the line so slope becomes something you can count.

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Preview of Two-Body, Three-Body & N-Body Gravity

Compare a two-body orbit with three-body figure-eight motion, then add up to twelve bodies. Change initial masses, velocities and positions; watch orbit trails and numerical energy error.

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Preview of Riemann Sums and the Integral Simulation

Approximate the area under a curve with strips and choose how each takes its height: left edge, right edge, middle, or a sloping top. Push the strip count up and watch the error fall, with the running total plotted as its own curve.

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From model to evidence

Use a worksheet to capture predictions, observations, and explanations after students explore.