Fractals — Measuring Shapes That Live Between Dimensions

What dimension is a coastline? The answer may not be 1 or 2.

At school, dimensions initially seem wonderfully straightforward.

A line is one-dimensional.

A square is two-dimensional.

A cube is three-dimensional.

That feels like the end of the story.

But mathematics has an irritating and rather wonderful habit of taking ideas that appear completely settled and asking one more question.

Must a dimension actually be a whole number?

Could something have dimension 1.26?

Or 1.58?

At first that sounds impossible. What would it even mean to be more than a line but less than a surface?

This question leads us into fractal geometry, an area of mathematics that provides a very different way of thinking about shape, scale and complexity.

And unlike some branches of advanced mathematics, fractals are remarkably easy to begin investigating. You can draw them with pencil and paper, construct them with a spreadsheet, generate them with a short computer program — and then walk outside and start seeing similar structures everywhere.

The Comfortable World of 1D, 2D and 3D

We normally associate dimension with the number of directions in which something extends.

A line has length but no width, so we call it one-dimensional.

A square has length and width, so it is two-dimensional.

A cube has length, width and height, making it three-dimensional.

For ordinary geometry this works perfectly well.

But nature does not always produce ordinary geometric shapes.

A coastline is not a perfectly smooth curve.

A tree is not a cylinder.

A lung is not simply a hollow sphere.

A lightning bolt is certainly not a straight line.

These structures contain detail at many different scales.

Zoom in and you often find more complexity.

Zoom in again and there may be still more.

This was one of the ideas that led mathematicians to investigate fractals.

Start with the Koch Curve

One of the easiest fractals to understand is the Koch curve, developed by Swedish mathematician Helge von Koch in the early twentieth century.

Begin with a straight line:

────────────

Now divide it into three equal sections.

Remove the middle third and replace it with two lines forming the other sides of an equilateral triangle.

Instead of one straight section you now have four shorter sections.

Then repeat the same process on every one of those four sections.

Then repeat it again.

And again.

And theoretically, forever.

The result becomes increasingly intricate.

Something Very Strange Happens to the Length

Suppose our original line has length 1.

After the first stage, each section has length 1/3 and there are four of them.

So the total length becomes:

4 x 1/3 = 4/3

At the next stage there are 16 sections, each of length 1/9.

Total length:

16 x 1/9 = 16/9

After another stage:

64 x 1/27 = 64/27

Every time the construction is repeated, the total length is multiplied by:

4/3

So after n stages:

Length = (4/3)^n

As n becomes larger, the length increases without limit.

That is already peculiar.

We have created a curve contained within a limited region of space, but whose mathematical length eventually becomes arbitrarily large.

And that is only the beginning.

Turn It into a Snowflake

Instead of beginning with one straight line, begin with an equilateral triangle.

Apply the Koch process to all three sides.

You obtain the Koch snowflake.

Repeat the construction and the edge becomes increasingly elaborate.

Here is the remarkable result:

The perimeter tends towards infinity, but the enclosed area remains finite.

That is one of those statements that initially feels as though mathematics has gone wrong.

How can something have an infinitely long boundary while enclosing only a finite amount of space?

Yet mathematically, that is exactly what happens.

It is a wonderful example for students because it challenges an assumption we rarely realise we are making:

A shape with a bigger and bigger perimeter does not necessarily need a bigger and bigger area.

So What Dimension Is the Koch Curve?

This is where things become particularly interesting.

A straight line has dimension 1.

A filled area has dimension 2.

The Koch curve is clearly more complicated than an ordinary line.

It folds around space so densely that simply calling it one-dimensional does not entirely describe its behaviour.

But it does not completely fill an area either.

Its fractal dimension is approximately:

1.2619

So mathematically it behaves as though it exists somewhere between a line and a surface.

That sounds bizarre until we think about what dimension is really trying to measure.

Dimension as a Measure of How Space Is Filled

One way to think about fractal dimension is to ask:

When I look at the object at a smaller scale, how much additional detail appears?

For some self-similar fractals we can calculate this quite neatly.

If an object divides into N smaller copies, each reduced by a scale factor s, then:

D = log(N) / log(s)

where D is the fractal dimension.

For the Koch curve:

N = 4
s = 3

Therefore:

D = log(4) / log(3)

which gives approximately:

D = 1.262

That number now has an interpretation.

The Koch curve fills space more effectively than a simple one-dimensional line, but not enough to become a two-dimensional region.

Compare It with the Sierpinski Triangle

Another beautiful fractal that students can easily construct is the Sierpinski triangle.

Begin with a large equilateral triangle.

Divide it into four smaller equilateral triangles.

Remove the middle one.

Now repeat the same operation on each of the three remaining triangles.

Continue repeating.

Each stage produces more and more holes.

At each scale we have:

N = 3 copies

and each copy has been scaled by:

s = 2

Therefore:

D = log(3) / log(2)

which is approximately:

1.585

Again, it lies between dimensions 1 and 2.

The Sierpinski triangle fills more of the plane than the Koch curve, so its fractal dimension is larger.

That provides a surprisingly intuitive way of thinking about these apparently strange decimal dimensions.

And Then There Is the Cantor Set

The Cantor set initially looks even stranger.

Draw a line segment.

Divide it into three equal parts.

Remove the middle third.

You now have two line segments.

Take each remaining segment and remove its middle third.

Repeat indefinitely.

Eventually the structure becomes an extraordinary collection of points.

At each stage:

N = 2
s = 3

So:

D = log(2) / log(3)

approximately:

0.631

A fractal whose dimension is less than 1.

It is more substantial than a collection of isolated points, but it does not behave like a continuous line.

For an able GCSE or A-level student, this is a wonderful reminder that mathematics becomes much more interesting once we stop assuming that familiar categories are the only categories possible.

The Coastline Paradox

Fractals become even more intriguing when we move away from deliberately constructed mathematical objects and look at the real world.

Imagine that I ask:

How long is the coastline of Britain?

It sounds as though there should be a straightforward numerical answer.

There isn't.

Or, more accurately, the answer depends on how you measure it.

Suppose we use a measuring stick 100 km long.

We work our way around Britain and calculate the total.

Now repeat the measurement using a 10 km ruler.

The new ruler fits into smaller bays and around more headlands.

The measured coastline gets longer.

Use a 1 km ruler and we capture still more detail.

Use a 100 m ruler.

Then 1 m.

Then 1 cm.

At increasingly small scales we discover additional bumps, rocks, cracks and irregularities.

The measured length keeps changing.

This is known as the coastline paradox.

How Long Is Britain?

The question therefore needs another piece of information.

Not simply:

"How long is the British coastline?"

but:

"At what scale are you measuring it?"

This idea was famously explored by mathematician Benoit Mandelbrot, whose work helped develop modern fractal geometry.

The important lesson is not that a coastline literally continues displaying identical patterns forever. Natural objects have physical limits.

Eventually we reach grains of sand, crystals, molecules and atoms.

But across a useful range of scales, many natural structures display behaviour resembling fractals.

Fractals Are Everywhere in Nature

Once students understand the idea, it becomes remarkably easy to find examples.

Trees

A tree has a trunk.

The trunk divides into branches.

Branches divide into smaller branches.

Those divide into twigs.

The same general branching pattern appears repeatedly at different scales.

Look at a photograph of a bare tree and then zoom in on one branch.

The branch often resembles a smaller version of the entire tree.

River Networks

A large river is fed by tributaries.

Those tributaries are fed by smaller streams.

Those streams may be fed by still smaller channels.

The resulting drainage network has a branching structure that resembles other fractal systems.

Interestingly, the pattern is rather like a tree turned upside down.

Blood Vessels

Our circulatory system faces a fascinating engineering problem.

A relatively small number of major blood vessels must ultimately supply enormous numbers of cells.

So large arteries branch into smaller arteries.

They branch into arterioles.

Then into tiny capillaries.

A branching network allows material to be distributed throughout a three-dimensional body efficiently.

Your Lungs Are an Extraordinary Fractal-Like Structure

The lungs provide an even more impressive example.

Air enters through the trachea.

The airway divides into bronchi.

These divide into smaller bronchioles.

Those divide repeatedly before eventually leading towards the tiny structures where gas exchange occurs.

Why?

Because exchanging oxygen and carbon dioxide requires a very large surface area.

If our lungs were simply two hollow bags, their internal surface area would be far too small.

Repeated branching and subdivision allow an enormous exchange surface to fit inside the relatively limited volume of the chest.

Fractal-like geometry is therefore not merely mathematically attractive.

It can be biologically useful.

Lightning

Lightning provides another visually dramatic example.

A lightning channel does not usually travel from cloud to ground as one perfectly straight line.

It branches.

Those branches may branch again.

The result can look remarkably similar to:

tree branches;
river systems;
blood vessels;
cracks;
electrical discharge patterns.

This raises a fascinating question.

Why do such similar patterns appear in completely different physical systems?

Sometimes the underlying processes are very different.

But branching is often an effective way for something to spread through space, collect material, distribute material or find a pathway through a complicated environment.

Practical Investigation 1: Make a Koch Snowflake

This is easily done with paper, ruler and pencil.

Stage 0

Draw an equilateral triangle.

Stage 1

Divide each side into three equal sections.

Replace the middle section with the two sides of a smaller equilateral triangle pointing outwards.

Stage 2

Repeat the operation for every new line segment.

Stage 3

Repeat again — assuming your patience and pencil remain intact.

Students can record:

number of sides;
length of each side;
total perimeter;
area.

A spreadsheet is excellent for extending the pattern without having to continue drawing it.

Students might discover:

Stage	Number of edges	Relative edge length	Relative perimeter
0	3	1	3
1	12	1/3	4
2	48	1/9	16/3
3	192	1/27	64/9

This opens the door naturally to sequences, powers and geometric progression.

Practical Investigation 2: The Sierpinski Triangle

This one is particularly suitable for younger students.

Draw a triangle.

Find the midpoints of all three sides.

Join them.

Shade or remove the central triangle.

Then repeat the process within every remaining triangle.

A fascinating extension is to count the number of remaining triangles.

The sequence is:

1, 3, 9, 27, 81, ...

So at stage n:

Number of triangles = 3^n

At the same time, the scale of each triangle becomes:

1/(2^n)

The picture therefore provides an excellent visual route into powers and sequences long before students need to encounter the formal idea of fractal dimension.

Practical Investigation 3: Grow a Fractal Tree

Draw a trunk.

At its end create two branches.

At the end of each branch create another two.

Continue.

Experiment with:

branch angle;
branch length;
number of branches;
rate at which branch length decreases.

A simple mathematical rule can produce surprisingly organic-looking trees.

For example:

New branch length = old branch length x 0.7

with each branch rotated by perhaps 25 degrees from the previous direction.

Change 0.7 to 0.8 and the tree spreads differently.

Change 25 degrees to 40 degrees and its whole appearance changes.

This is a beautiful example of complexity emerging from very simple rules.

Practical Investigation 4: Generate Fractals with a Spreadsheet

A spreadsheet is an excellent bridge between school mathematics and computational mathematics.

Students could create columns containing:

iteration number;
number of pieces;
length of each piece;
total length;
scaling factor.

For the Koch curve, for example:

Number of pieces = 4^n

Length of each piece = (1/3)^n

Total length = (4/3)^n

Students can graph the perimeter against iteration number.

They will immediately see that it continues increasing.

This gives fractals connections with:

indices;
logarithms;
sequences;
graphs;
exponential growth;
limits.

What initially looks like an exotic branch of geometry suddenly connects to large parts of GCSE and A-level mathematics.

Practical Investigation 5: Write a Small Computer Program

For students studying Computer Science as well as Mathematics, fractals provide an excellent programming exercise.

The underlying logic of many fractals is recursive.

In simplified form:

Draw something.
Replace each part according to a rule.
Apply the same rule to the new parts.
Repeat.

That is almost the definition of a recursive algorithm.

A fractal tree, for example, can be thought of as:

Draw a branch, then draw two smaller versions of the whole tree from its end.

It is a lovely demonstration of how mathematics and programming can describe the same idea in different languages.

Can We Measure the Fractal Dimension of a Real Object?

Natural objects are not exact mathematical fractals, but we can still estimate something resembling a fractal dimension.

One popular technique is box counting.

Imagine placing a grid over a map of a coastline.

Count how many squares contain some coastline.

Then use smaller squares.

Count again.

Then use still smaller squares.

If the number of occupied boxes rises predictably as the box size decreases, we can estimate a fractal dimension.

This is an excellent investigation for students because it turns what sounds like rather abstract mathematics into something measurable.

You could potentially do it using:

printed maps;
aerial photographs;
photographs of trees;
leaf edges;
river networks;
cracks in dried mud.

Even if students never calculate the dimension precisely, the investigation teaches an important scientific lesson:

measurement itself depends upon scale.

A Surprisingly Important Idea: Scale Matters

This may be the most valuable concept in the whole subject.

At school we are sometimes encouraged to think that objects simply possess fixed measurements.

A table has a length.

A circle has a circumference.

A field has an area.

But real-world measurement is more subtle.

Ask how long a table is and millimetre precision may be perfectly adequate.

Ask about a coastline and suddenly the definition of "length" becomes much less comfortable.

At what scale are you measuring?

Which bays count?

Which rocks?

Which cracks between rocks?

Fractals remind us that mathematics is not merely about performing calculations.

It is also about deciding what the calculation actually means.

Why I Think Fractals Are Worth Showing Students

One reason I enjoy topics such as this is that they reveal a side of mathematics students do not always see in examination courses.

School mathematics can sometimes give the impression that every problem has already been neatly defined.

Here is the triangle.

Here are its measurements.

Calculate x.

There is nothing wrong with that — those skills matter enormously.

But mathematics did not develop because mathematicians spent centuries solving examination questions.

It developed because people kept asking awkward questions.

What is infinity?

What happens if parallel lines behave differently?

Can a function be continuous everywhere but differentiable nowhere?

And:

Does dimension have to be a whole number?

Fractals give GCSE and A-level students a glimpse of that wider mathematical world without requiring years of university mathematics first.

From Geometry to Biology, Physics and Computing

Fractals are particularly useful because they refuse to remain inside one school subject.

A mathematician sees scaling and dimension.

A biologist sees lungs and blood vessels.

A geographer sees river networks and coastlines.

A physicist sees lightning, turbulence and growth patterns.

A computer scientist sees recursion and algorithms.

An artist sees extraordinary patterns.

That is exactly the sort of mathematics I like students to encounter.

Not mathematics isolated on a worksheet, but mathematics acting as a language connecting apparently unrelated parts of the world.

A Challenge for Students

Try investigating four fractals:

1. Koch snowflake
Can you calculate how the perimeter changes after each iteration?

2. Sierpinski triangle
Can you predict how many triangles remain after ten stages?

3. Cantor set
How much total line length remains after each stage?

4. Fractal tree
How does changing the branching angle or scaling factor alter the final structure?

Then go outside.

Photograph:

a tree;
a leaf;
clouds;
branching cracks;
a river system if you can find one on a map.

Ask yourself:

Is this genuinely fractal, approximately fractal, or does it merely look fractal?

That final question is arguably more interesting than simply generating another pretty pattern.

Mathematics Between the Dimensions

A fractal dimension such as 1.26 initially sounds nonsensical because our everyday experience encourages us to think only in whole-number dimensions.

But fractal dimension is not claiming that someone has discovered a mysterious direction that is 26% of another direction.

It is describing how an object behaves as its scale changes and how effectively it fills space.

The Koch curve is more complicated than an ordinary line but does not fill a plane.

The Sierpinski triangle fills still more of the plane.

Natural objects such as coastlines, river networks, lungs and trees show similar scale-dependent complexity.

And that is perhaps the most important lesson.

Sometimes mathematics advances not by calculating a more accurate answer to an old question, but by realising that we have been asking the question in the wrong way.

So perhaps the question is not simply:

How long is the coastline?

Perhaps it should be:

At what scale?

And perhaps the question is not:

Is this object one-dimensional or two-dimensional?

Perhaps occasionally the answer really can be:

Somewhere in between.

02 September 2026