02 September 2026

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.

01 September 2026

Atwood’s Machine — The Elegant Experiment That Slows Down Gravity

There are some pieces of physics apparatus that look almost too simple to be interesting.

Atwood’s machine is one of them.

Two masses hang on either side of a pulley. They are joined by a string. Make one mass slightly heavier than the other, release the system, and the heavier mass moves down while the lighter mass rises.

That is essentially the whole apparatus.

Yet from this simple arrangement we can investigate Newton’s laws, resultant force, acceleration, tension, conservation of energy, experimental uncertainty and even the rotational inertia of the pulley itself.

For me, that is what makes Atwood’s machine such an elegant classical physics experiment. It takes something that normally happens rather too quickly — acceleration under gravity — and slows it down enough for us to study it carefully.

And it raises a very useful question:

If gravity is pulling strongly on both masses, why can the resulting acceleration be surprisingly small?

A Machine Designed to Make Acceleration Measurable

The Atwood machine is named after the English mathematician and physicist George Atwood, who described the apparatus in the eighteenth century.

The problem Atwood faced was a practical one.

If you simply drop an object, it accelerates at approximately:

g = 9.81 m/s^2

That is quite a large acceleration.

Drop something through one metre and the whole event is over in less than half a second. That was particularly awkward in the days before electronic timers, light gates, motion sensors and high-speed video.

Atwood's clever idea was to allow gravity to act on two connected masses.

Most of the gravitational forces effectively oppose one another. Only the difference between them produces the acceleration of the system.

The result is a controllable acceleration that can be much smaller than g.

That makes Newton's second law much easier to investigate experimentally.

The Simplest Atwood Machine

Imagine two masses:

and

connected by a light string passing over a pulley.

Suppose:

m2 > m1

The second mass therefore travels downwards and the first mass travels upwards.

At first glance we might be tempted to say that the force accelerating the system is simply:

m2g

But that cannot be correct.

Gravity is also pulling downwards on m1.

Because the two masses are connected, the gravitational effects oppose each other as far as the motion of the complete system is concerned.

The driving force is therefore:

F = m2g - m1g

or:

F = (m2 - m1)g

But what mass is being accelerated?

Both masses are moving.

Therefore the total moving mass is:

m1 + m2

Newton's second law tells us:

F = ma

so:

(m2 - m1)g = (m1 + m2)a

and therefore:

a = (m2 - m1)g / (m1 + m2)

That is the central equation of the Atwood machine.

It is also a lovely example of why simply remembering F = ma is not enough. We have to decide which force and which mass belong in the equation.

A Small Imbalance Can Move a Large Mass

This is where the experiment becomes particularly interesting.

Suppose the two sides contain:

m1 = 0.245 kg

m2 = 0.255 kg

The total moving mass is:

0.245 + 0.255 = 0.500 kg

But the difference between the masses is only:

0.255 - 0.245 = 0.010 kg

The theoretical acceleration is:

a = 0.010 x 9.81 / 0.500

a = 0.1962 m/s^2

So although gravity itself produces an acceleration of about 9.81 m/s^2, our Atwood machine accelerates at only about:

0.20 m/s^2

That is around one-fiftieth of g.

Suddenly the motion is slow enough to watch.

That is the cleverness of the machine.

The Best Investigation: Keep the Total Mass Constant

One particularly satisfying experiment is to investigate how acceleration changes when the difference between the masses changes, while keeping the total mass constant.

This is important experimentally.

If we changed both the driving force and the total mass at the same time, interpreting the results would become more difficult.

Instead, imagine starting with:

250 g | 250 g

The machine is balanced.

Now transfer 5 g from one side to the other:

245 g | 255 g

The total mass is still:

500 g

but the mass difference is now:

10 g

Transfer another 5 g:

240 g | 260 g

The total remains 500 g, while the difference becomes 20 g.

We might continue with:

235 g | 265 g

230 g | 270 g

225 g | 275 g

The total mass remains unchanged throughout.

Only the imbalance changes.

What Are We Really Changing?

The driving force is:

F = (m2 - m1)g

So increasing the mass difference increases the resultant force.

At the same time:

m1 + m2

remains constant.

Newton tells us:

a = F / m

If mass remains constant, acceleration should therefore be directly proportional to force.

That gives us a prediction before we even perform the experiment:

Double the mass difference and the acceleration should approximately double.

That is excellent experimental physics.

We have a theory.

We make a prediction.

Then we collect measurements and see whether nature agrees.

A Possible Set of Results

Suppose the total moving mass is 0.500 kg.

We might obtain something resembling:

Mass difference	Driving force	Ideal acceleration
0.010 kg	0.0981 N	0.196 m/s^2
0.020 kg	0.1962 N	0.392 m/s^2
0.030 kg	0.2943 N	0.589 m/s^2
0.040 kg	0.3924 N	0.785 m/s^2
0.050 kg	0.4905 N	0.981 m/s^2

Plot:

acceleration against mass difference

and, for an ideal Atwood machine, we should obtain a straight line.

Alternatively plot:

acceleration against driving force

Again, Newton's second law predicts a straight line.

The gradient can even be connected to the total accelerated mass.

Now our very simple pulley has become an experimental test of:

F = ma

How Do We Measure the Acceleration?

There are several possibilities.

Method 1 — Measure Distance and Time

Release the masses from rest and measure the time taken to travel a known distance.

For motion starting from rest:

s = 1/2 at^2

Therefore:

a = 2s / t^2

If the mass travels 0.50 m in 1.80 seconds:

a = (2 x 0.50) / 1.80^2

a = 1.00 / 3.24

a = 0.309 m/s^2

The problem is timing.

Human reaction time becomes a significant source of uncertainty, particularly for short runs.

Method 2 — Video the Motion

A smartphone capable of slow-motion recording can make the experiment much better.

Place a scale beside one of the moving masses and record the motion.

Individual frames can then be used to determine:

position;
time;
velocity;
acceleration.

This is also an excellent opportunity to show students that physics experiments do not always require expensive specialist equipment.

Method 3 — Use Light Gates

Light gates make the experiment considerably more precise.

A card attached to one of the masses can interrupt one or more beams, allowing velocity and acceleration to be calculated electronically.

Method 4 — Use a Motion Sensor

With a suitable motion sensor or data-logging system, position can be recorded continuously.

This produces graphs of:

position against time

velocity against time

and possibly:

acceleration against time.

The velocity-time graph is particularly satisfying.

If acceleration is approximately constant, velocity increases linearly with time.

The gradient gives the acceleration directly.

Resolving the Forces on Each Mass

There is another way of analysing the machine, and this is particularly useful for A-level Physics.

Instead of treating the two masses as one system, consider each mass separately.

For the heavier mass m2:

Weight acts downward:

m2g

Tension acts upward:

Therefore:

m2g - T = m2a

For the lighter mass m1:

Tension acts upward.

Weight acts downward.

Therefore:

T - m1g = m1a

Now add the two equations:

m2g - T + T - m1g = m2a + m1a

The tensions cancel:

(m2 - m1)g = (m1 + m2)a

Giving us once again:

a = (m2 - m1)g / (m1 + m2)

This is a beautiful piece of mechanics because it demonstrates why choosing a complete system can simplify a problem.

The tension is very important to each individual mass.

But when we consider the complete two-mass system, tension becomes an internal force and disappears from the final equation.

That is a powerful idea that appears repeatedly in mechanics.

Tension Is Not Simply Equal to the Weight

Another useful misconception can be explored using Atwood's machine.

Students often become accustomed to situations where:

T = mg

But that is only true when the mass has zero acceleration.

In an Atwood machine the masses accelerate.

For the lighter mass:

T - m1g = m1a

Therefore:

T = m1(g + a)

So the tension is greater than the weight of the lighter mass.

For the heavier mass:

m2g - T = m2a

Therefore:

T = m2(g - a)

The tension is less than the weight of the heavier mass.

It must, of course, be the same tension throughout an ideal light string.

Again, a very simple experiment exposes some quite subtle mechanics.

The Real Machine Is More Interesting Than the Ideal One

Classroom calculations usually make several assumptions:

the pulley is frictionless;
the pulley has negligible mass;
the string has negligible mass;
the string does not stretch;
the string does not slip over the pulley;
air resistance is negligible.

Real apparatus politely refuses to obey all of these assumptions.

And that is where the experiment becomes even more interesting.

Friction in the Pulley

A real pulley has bearings.

Those bearings have friction.

When the mass difference is very small, you may find that the machine does not move at all.

Perhaps:

249 g | 251 g

remains stationary.

Increase the imbalance slightly and suddenly it begins to move.

That immediately tells us something important.

The driving force must first overcome friction.

If we plot acceleration against driving force, our experimental line may therefore fail to pass exactly through the origin.

Far from being a failed experiment, this is useful evidence about the real apparatus.

The Pulley Has to Accelerate Too

There is another wonderfully subtle effect.

The pulley rotates.

That means some of the driving force is being used not merely to accelerate the hanging masses but also to accelerate the rotation of the pulley.

In the ideal equation we wrote:

a = (m2 - m1)g / (m1 + m2)

But for a real pulley with rotational inertia, the acceleration will usually be slightly smaller.

At a more advanced level the pulley can effectively contribute an additional inertial term.

For a pulley of moment of inertia I and radius r:

a = (m2 - m1)g / (m1 + m2 + I/r^2)

We have not lost Newton's laws.

We have simply discovered that there was another part of the system whose acceleration we originally ignored.

This is one of the reasons I particularly like classical experiments: increasing the precision does not merely improve the same answer. It often reveals another layer of physics.

An Interesting Extension: Keep the Difference Constant Instead

We can reverse the experiment.

Suppose we maintain the same difference between the masses but increase the total mass.

For example:

95 g | 105 g

195 g | 205 g

295 g | 305 g

In each case the mass difference is:

10 g

Therefore the driving force is approximately constant.

But the total mass being accelerated becomes progressively larger.

Since:

a = F / m

we expect the acceleration to decrease.

This allows students to investigate the other half of Newton's second law.

The first investigation asks:

What happens when force changes but mass remains constant?

The second asks:

What happens when mass changes but force remains approximately constant?

Together, they provide a remarkably complete investigation of F = ma.

Could We Measure g Using an Atwood Machine?

Yes.

Rearrange:

a = (m2 - m1)g / (m1 + m2)

to give:

g = a(m1 + m2) / (m2 - m1)

Measure the masses accurately and determine the acceleration experimentally.

You can then calculate an experimental value of gravitational field strength.

It probably will not produce 9.81000 m/s^2.

That is not the point.

The interesting question becomes:

Why doesn't it?

Possible explanations include:

pulley friction;
pulley rotational inertia;
inaccurate masses;
timing uncertainty;
string mass;
air resistance;
the release method;
the pulley not being perfectly aligned.

That turns a demonstration into a genuine experimental investigation.

Don't Push the Masses

One seemingly trivial experimental detail is actually extremely important.

The masses must be released, not pushed.

Giving the system even a small initial velocity affects measurements based on:

s = 1/2 at^2

because that equation assumes:

u = 0

A good release mechanism makes the experiment significantly more repeatable.

This is exactly the sort of small practical detail that students begin to appreciate when they actually perform experiments rather than merely reading descriptions of them.

Repeat the Measurements

One timing result tells us surprisingly little.

Repeat each mass configuration several times.

For example:

5 trials for each mass difference.

Calculate the mean acceleration and investigate the spread of the measurements.

If one measurement is very different, do not simply delete it because it is inconvenient.

Ask why.

Did the string catch?

Did the mass swing?

Was the release poor?

Did the pulley hesitate before moving?

Experimental physics is partly about collecting numbers.

It is much more about deciding which numbers we should trust.

Why This Experiment Is Such Good Teaching Physics

What I particularly like about Atwood's machine is that almost everyone can understand the apparatus immediately.

There is no mysterious black box.

You can see the masses.

You can see the string.

You can see the pulley.

You can see which side is heavier.

Yet from that simple arrangement we can explore:

Newton's first and second laws;
resultant force;
acceleration;
tension;
free-body diagrams;
simultaneous equations;
experimental uncertainty;
friction;
rotational motion;
energy;
modelling assumptions.

It is an excellent example of how sophisticated physics does not necessarily require sophisticated-looking equipment.

The Physics of a Nearly Balanced System

There is also something slightly counter-intuitive about watching an Atwood machine.

Imagine 500 g of mass being accelerated by an imbalance of only 10 g.

Gravity is pulling on both sides with forces of several newtons.

Yet the resultant force may be less than one tenth of a newton.

Large forces can therefore exist within a system while producing only a small overall acceleration.

That idea matters far beyond pulley experiments.

It appears whenever opposing forces nearly balance:

vehicles moving at constant speed;
aircraft in level flight;
terminal velocity;
objects floating;
structures under load;
orbital mechanics;
mechanical control systems.

The motion depends not on how large the individual forces are, but on their resultant.

From an Eighteenth-Century Pulley to Modern Physics

Atwood designed his machine at a time when accurately measuring rapidly changing motion was extremely difficult.

Today we can attach sensors, use computer data logging, film the experiment at hundreds of frames per second and fit mathematical models to thousands of measurements.

Yet the underlying apparatus hardly needs to change.

Two masses.

One string.

One pulley.

That longevity tells us something.

A great experiment does not merely demonstrate an effect.

It isolates an idea.

Atwood's machine isolates Newton's second law beautifully.

A Simple Apparatus With a Lot to Say

Atwood's machine may never compete visually with exploding chemicals, giant electrical discharges or dramatic vacuum demonstrations.

But that is part of its charm.

Transfer a tiny mass from one side to the other and the entire system begins to accelerate.

Increase that imbalance and the acceleration increases predictably.

Keep the force constant but increase the mass and the acceleration falls.

Measure carefully enough and even the imperfections — friction, pulley inertia and experimental uncertainty — become physics worth investigating.

The experiment begins with one of the most familiar equations in science:

F = ma

But actually performing it reveals what that equation really means.

For me, that is classical experimental physics at its best: simple apparatus, a clear question, a prediction that can be tested, and enough hidden complexity to reward anyone who decides to look a little more closely.

31 August 2026

Darwin’s Phototropism Experiment — How Does a Plant Know Where the Light Is?

Every plant on a windowsill seems to know something that it has never been taught: where the light is.

Leave a young plant beside a window and, given enough time, its shoots begin to lean towards the glass. Turn the pot around and, remarkably, the plant gradually changes direction again.

We usually accept this without thinking very much about it.

But there is a wonderful biological question hidden inside that simple observation:

How does a plant know where the light is?

Plants have no eyes, no brain and no nervous system remotely like ours. Yet they can detect the direction of light and alter their growth accordingly.

More than 140 years ago, Charles Darwin and his son Francis investigated this problem using an experiment so simple that a version of it can still be carried out on a kitchen table, windowsill or in a small home laboratory.

And the really fascinating discovery was not merely that plants grow towards light.

It was that the part of the plant that detects the light is not necessarily the part that does the bending.

That observation eventually helped open the door to the discovery of plant hormones.

A Classic Experiment With a Very Modern Question

Charles and Francis Darwin described their investigations of plant movement in their 1880 book The Power of Movement in Plants. They studied young grass seedlings and discovered that directional light was detected principally at the tip of the emerging shoot. Covering that tip with an opaque cap could prevent the normal bending response, even though the region lower down the shoot remained illuminated.

Today we call growth towards light positive phototropism.

The experiment is beautifully suited to teaching because the apparatus is almost embarrassingly simple.

You need seedlings, some small caps and a light source.

But from those simple materials comes a surprisingly sophisticated idea about communication within a living organism.

Recreating Darwin’s Experiment at Home

For a practical version, I would use a fast-germinating grass such as oats or wheat.

You want the seedlings to be young enough that the emerging grass shoot, or coleoptile, is still clearly visible.

A coleoptile is the protective sheath that surrounds the first emerging shoot of grasses. It is particularly useful for experiments such as this because its growth response is relatively easy to observe.

What you need

You could use:

oat, wheat or another suitable grass seed;
several small pots, trays or sections of damp paper;
compost, vermiculite or moist cotton wool;
a cardboard box;
a desk lamp or other directional light source;
aluminium foil or black paper;
transparent plastic film;
small pieces of paper or drinking straw from which to make caps;
ruler;
protractor or photographs for measuring curvature;
labels.

I would recommend growing several seedlings for each treatment rather than relying upon one plant.

Living organisms vary.

One seedling may grow rapidly while another hardly grows at all. Replication turns a demonstration into a much better scientific investigation.

Stage One — Grow the Seedlings in Darkness

Germinate the seeds and keep them in darkness until shoots have emerged.

Growing them initially in darkness has an advantage.

The shoots tend to become relatively long and pale as they search for light, making subsequent curvature easier to see.

Once you have reasonably similar seedlings, divide them into experimental groups.

The Four Key Treatments

1. The uncovered seedling

Leave the entire shoot exposed.

This is your basic control.

When illuminated from one side, it should bend towards the light.

2. Cover the tip with an opaque cap

Make a tiny cap from foil or another completely opaque material and place it over the very top of the shoot.

Do not crush the seedling.

The lower part of the shoot remains exposed to exactly the same directional light as before.

The intriguing result is that the seedling should now show greatly reduced phototropic bending.

And immediately we have a puzzle.

The lower part of the shoot can still see the light — so why doesn't it respond normally?

3. Cover the tip with a transparent cap

This is one of my favourite controls in the experiment because it answers an obvious criticism.

Perhaps putting a cap on the plant simply interferes mechanically with its growth.

So make another cap from transparent plastic.

The tip is enclosed just as before, but light can pass through it.

The plant should still show phototropic curvature.

That tells us that it is not simply the presence of the cap that matters.

Blocking the light reaching the tip is important.

4. Cover the lower shoot but leave the tip exposed

Now reverse the arrangement.

Wrap an opaque shield around part of the lower coleoptile while leaving the tip exposed.

We might expect this plant not to bend because the region where much of the bending actually occurs has been hidden from the directional light.

Yet it can still respond.

The tip detects the direction of illumination and somehow sends information downwards to the growing region.

This is the crucial observation.

The Darwins' work showed that the perception of light and the resulting growth response could occur in different regions of the plant.

The Results You Might Expect

Treatment	Tip receives directional light?	Expected response
Uncovered shoot	Yes	Strong bending towards light
Opaque cap on tip	No	Little or greatly reduced bending
Transparent cap on tip	Yes	Bends towards light
Lower shoot covered, tip exposed	Yes	Bends towards light

An excellent optional fifth treatment is simply to remove the tip.

Again, phototropic bending is greatly reduced.

But be careful about interpretation: removing the tip also removes actively growing tissue and potentially damages the seedling, so the opaque-cap experiment is in many ways the more elegant demonstration.

Why the Experiment Is So Clever

At first glance this looks like an experiment about light.

It is actually an experiment about information.

Imagine that we separate the plant into two regions.

At the top is the detector.

Further down is the machinery producing the response.

Something therefore has to connect the two.

The Darwins did not discover auxin in this experiment. Their work suggested that some influence was transmitted from the light-sensitive region towards the region where bending occurred. Later experiments by researchers including Peter Boysen-Jensen and Frits Went helped establish the idea of a mobile chemical growth signal and contributed to the development of our understanding of auxin.

That historical distinction is worth making.

Science often advances this way.

One experiment rarely produces the whole explanation.

Instead:

Darwin asks where the stimulus is detected.

Then someone else asks:

What carries the message?

Then another scientist asks:

What molecule is responsible?

And modern biology asks:

What receptors, proteins and genes control the whole system?

One simple plant bending beside a lamp becomes decades of biology.

So What Is Auxin Doing?

For an A-level student, this is where the experiment becomes especially useful.

One of the major auxins in plants is indole-3-acetic acid, usually abbreviated to IAA.

When a young shoot receives light predominantly from one direction, blue-light receptors called phototropins help detect this uneven illumination. This ultimately results in an unequal distribution of auxin across the shoot, with more growth-promoting auxin activity on the shaded side.

In a shoot, auxin promotes cell elongation.

So imagine the stem viewed from above:

Light side: cells elongate less.

Shaded side: cells elongate more.

If one side becomes longer than the other, the shoot cannot remain straight.

It curves.

And because the shaded side has grown more, the shoot curves towards the light.

That distinction is important.

Auxin does not somehow "pull" the plant towards the lamp.

The plant bends because of differential growth.

One side grows faster than the other.

The Plant Is Not Turning — It Is Growing Unequally

This is something students can easily overlook.

An animal can see food, turn its body and walk towards it.

A young plant cannot do that.

Once rooted into the soil, it has another solution.

It alters where growth occurs.

That makes tropisms fundamentally different from most animal movements.

When a sunflower appears to have "turned" or a seedling leans towards a window, at least some of what we are observing is the consequence of controlled differences in cell growth.

This is a wonderfully economical biological system.

No muscles.

No brain.

No conscious decision.

Just receptors, chemical signalling and changes in cell growth.

What Happens Inside the Cells?

For younger students, saying that auxin causes the shaded side to grow faster is sufficient.

For A-level students, we can go another step.

Auxin can stimulate proton pumps in cell membranes. These move H+ ions into the cell wall region, reducing the pH.

That acidic environment activates proteins including expansins, which loosen interactions within the cell wall.

The cell wall becomes more capable of expanding.

Water entering the cell maintains turgor pressure, and the cell can elongate.

So we can follow the mechanism from something enormous — a plant bending visibly towards a window — all the way down to processes occurring across microscopic cell membranes.

Directional light → photoreceptor signalling → altered auxin distribution → increased cell elongation on the shaded side → bending towards light.

That is a lovely chain of cause and effect for an A-level Biology student to practise explaining.

Turn the Demonstration Into a Proper Investigation

Rather than simply deciding whether each plant "bent" or "didn't bend", we can collect numerical data.

Photograph the seedlings from exactly the same position at regular intervals.

For example:

immediately before illumination;
after 2 hours;
after 4 hours;
after 8 hours;
after 24 hours.

Measure the angle between the original vertical direction and the direction of the shoot tip.

You could then calculate a mean angle for perhaps five seedlings in each treatment.

That produces genuine quantitative data.

A student could graph:

Time on the horizontal axis

against

mean angle of curvature on the vertical axis.

Suddenly Darwin's Victorian botanical experiment has become an excellent modern exercise involving biology, experimental design, statistics, photography and data analysis.

One Seedling Is Not an Experiment

This is also an ideal opportunity to explain biological variation.

Suppose our uncovered plant bends 35 degrees towards the light while our capped plant bends 4 degrees.

That looks convincing.

But perhaps the second seedling was simply unhealthy.

Instead, try five or ten plants in each condition.

If most uncovered plants bend strongly and most opaque-capped plants do not, our evidence becomes much more convincing.

If doing this with an A-level student, I would consider calculating:

mean curvature;
range;
standard deviation if appropriate;
perhaps statistical significance in a larger investigation.

For a family experiment, however, photographs and a simple comparison are more than adequate.

Keep the Light Directional

One practical difficulty is that modern houses are full of stray light.

A seedling beside a lamp may also receive daylight from a window and artificial light from several other directions.

The easiest solution is a cardboard box.

Cut a window in one side and place the light beyond it.

Now most of the useful illumination arrives from one direction.

You have effectively constructed a miniature controlled environment chamber for almost nothing.

A Few Things That Can Go Wrong

This experiment is straightforward, but biology rarely behaves as neatly as a textbook diagram.

The seedlings don't bend very much

They may not have been exposed for long enough, or the directional difference in light intensity may not be strong enough.

Move the light closer, while avoiding excessive heating.

Everything bends before the experiment starts

The seedlings may already have received directional light.

Grow them in darkness or uniform lighting first.

The opaque caps fall off

They need to be extremely light.

A heavy cap can bend or damage the shoot and introduce an entirely different variable.

The capped seedlings stop growing completely

Check that the cap is not physically constricting the shoot.

This is exactly why the transparent-cap control matters.

Different seedlings give different results

Good!

That is biology.

Use more seedlings, calculate averages and discuss variation rather than hiding it.

An Excellent Investigation of Controls

From a teaching point of view, the real strength of this experiment may be its experimental design.

Ask a student:

"Why do we need the transparent cap?"

They should eventually realise that it separates two possible explanations.

If both opaque and transparent caps stopped bending, then perhaps the physical cap itself was responsible.

But if the transparent cap allows bending while the opaque cap prevents it, the evidence points towards light reaching the tip as the important variable.

Then ask:

"Why cover the bottom of the shoot?"

Because it distinguishes the region detecting light from the region producing the response.

This is experimental science at its best: changing one thing at a time to distinguish between competing explanations.

Darwin Didn't Have Our Modern Equipment

This is another aspect of these classic experiments that I find particularly appealing.

We sometimes give students the impression that important science requires enormously expensive equipment.

Modern science certainly can.

But some profound scientific discoveries began with extraordinarily simple apparatus.

Darwin did not have digital light sensors, data loggers, time-lapse cameras or image-analysis software.

Today, however, we can repeat the underlying investigation and add all of those things if we wish.

A smartphone placed on a tripod could produce a time-lapse sequence.

The resulting video would make the movement dramatically more obvious than watching the seedling directly.

Image-analysis software could measure the curvature.

A light sensor could measure illumination.

A temperature sensor could check whether the lamp is heating one side of the apparatus.

It becomes an excellent example of combining a classic scientific question with modern measurement technology.

A Family Version: Make It a Prediction Game

For children, I would not begin by explaining auxin.

I would show them four seedlings and ask them to predict what will happen.

Which plant will bend?

Perhaps draw the expected shape of each shoot before switching on the lamp.

Then return the following day.

The opaque-tip experiment is particularly satisfying because the answer is not necessarily obvious.

Most people initially assume that the part of the plant doing the bending must also be the part detecting the light.

When the experiment contradicts that prediction, we have one of the most important experiences in science:

"That isn't what I expected."

That sentence is often where interesting science starts.

A-Level Extension — Follow the History of the Signal

The experiment can also become the beginning of a much larger sequence.

Once we know the tip is important, the obvious next question is:

What travels from the tip to the rest of the plant?

That led to further classic experiments.

Peter Boysen-Jensen placed different materials between the tip and the lower part of the coleoptile. A permeable gelatin barrier allowed the response to continue, whereas an impermeable barrier could prevent it. This supported the idea of a mobile chemical signal rather than something requiring direct cellular continuity.

Frits Went's subsequent work with coleoptile tips and agar blocks helped establish the existence of a diffusible growth-promoting substance associated with auxin research.

So Darwin's experiment can become the first chapter in a scientific detective story:

Where is light detected?

How does the message move?

What is the message?

How does that message alter cell growth?

That progression is far more memorable than simply learning:

"Auxin causes phototropism."

Could We Test Colour as Well as Direction?

A particularly interesting extension would be to illuminate seedlings using different wavelengths of light.

Modern plant phototropism is strongly associated with blue-light photoreceptors called phototropins.

So we could investigate whether seedlings respond equally strongly to:

blue light;
red light;
green light;
white light.

Care would be needed because different lamps may have very different intensities. Merely placing coloured plastic over a lamp would not necessarily produce a fair quantitative experiment.

But it creates another excellent question:

Does a plant simply measure how much light there is, or can it distinguish different wavelengths?

Once again the apparently simple plant on the windowsill turns out to possess remarkably sophisticated sensory biology.

Plants Know Far More About Their Environment Than They Appear To

This experiment also challenges a very human way of thinking.

Because plants do not move around rapidly, we often treat them as passive.

They are anything but passive.

Plants continually detect and respond to:

light;
gravity;
water;
touch;
temperature;
chemicals;
day length;
damage;
neighbouring organisms.

They do not experience these things as we do, but they possess molecular sensing and signalling systems capable of changing growth and development.

Phototropism is simply one particularly visible example.

The Real Lesson From Darwin's Seedlings

We could describe this as an experiment demonstrating phototropism and auxin.

But I think that undersells it.

The more interesting lesson is about how biology discovers hidden processes.

We cannot see a plant detecting blue light.

We cannot watch auxin molecules redistributing with the naked eye.

We cannot directly see individual cells on one side of the stem elongating faster than those on the other.

Yet by performing a carefully designed set of experiments, we can infer that something must be happening.

Cover the top.

The response disappears.

Use a transparent cover.

It returns.

Cover another region instead.

The response remains.

Each experiment eliminates an explanation and narrows the possibilities.

That is scientific reasoning.

And it can be demonstrated with little more than some grass seedlings, cardboard, foil and a lamp.

Conclusion — How Does a Plant Know Where the Light Is?

A plant does not "know" where the light is in the conscious sense.

Instead, cells near the growing tip contain molecular systems capable of detecting differences in illumination.

That information alters growth signalling, including the distribution and action of auxin.

Cells on the shaded side elongate more strongly than those on the illuminated side.

The shoot therefore curves towards the light.

What makes the Darwin experiment so memorable is the discovery that sensing and responding can happen in different places.

The tip detects.

A message travels.

Cells further down respond.

The plant bends.

And from that remarkably simple observation grew an entire field of plant physiology.

The next time a houseplant starts leaning towards a window, it is worth remembering that you are watching the result of an extraordinarily sophisticated biological signalling system.

The plant has no eyes — but it can certainly detect where the light is.