03 September 2026

How Do You Discover a Planet You Cannot See?

Detecting Exoplanets by Watching a Star Blink

Astronomy has a rather wonderful problem.

The objects we most want to investigate are often unimaginably far away, extremely faint and sitting beside something enormously brighter.

An exoplanet may be hundreds of light-years from Earth. It does not conveniently appear in a telescope photograph as a neat little sphere next to its star. In many cases, we discover that the planet is there without ever seeing the planet itself.

Instead, we watch the star.

And occasionally, almost imperceptibly, the star becomes slightly dimmer.

That tiny dip in brightness can be enough to reveal an entire world.

This makes exoplanet detection an excellent example of science beyond the normal school syllabus because it combines astronomy, physics, data analysis, graph interpretation and experimental design with one of the most important ideas in science:

You do not always have to see something directly to discover that it exists. You can measure the effect it has on something else.

NASA describes the transit method in essentially these terms: when a planet passes between its star and us, it blocks a small fraction of the star's light. Plotting the measured brightness against time produces a light curve, and a planetary transit appears as a dip in that curve.

And we can reproduce the basic idea on a laboratory bench.

A Star That Apparently Blinks

Imagine watching a distant star continuously.

For most of the time its measured brightness remains approximately constant.

Then this happens:

Normal brightness -> slight fall -> minimum brightness -> rise -> normal brightness

Nothing necessarily happened to the star itself.

Instead, a planet may have crossed the face of it.

From Earth, we see something rather like a very small eclipse.

The crucial word is small.

A planet is normally considerably smaller than its parent star, so only a fraction of the star's light is removed. Detecting exoplanets therefore depends on making extremely precise measurements and deciding whether a tiny change is genuine or merely noise.

That immediately makes this much more interesting than simply moving a ball in front of a lamp.

The real experiment is about measurement.

Can we detect the change?

Can we distinguish it from random fluctuations?

Can we extract information about our "planet" from the graph?

Building a Model Exoplanet System

The simplest version needs surprisingly little equipment.

You need:

a bright lamp or LED source;
preferably a translucent diffuser to create a circular illuminated "star";
several opaque balls or discs of different diameters;
a light sensor, lux sensor or data logger;
some way of moving the model planet steadily across the star;
software capable of recording light intensity against time.

A data logger is particularly useful because it turns the demonstration into something very close to the way astronomical observations are actually treated: a sequence of brightness measurements taken over time.

Why I Would Not Use a Bare LED

There is a useful experimental-design point here.

A bare LED is very nearly a small point source. Put an opaque object directly in front of it and you may simply block most or all of its light.

That is not a very good model of a planet crossing a star.

A better arrangement is to illuminate a circular translucent screen from behind. The whole circle then becomes the visible surface of our model star.

Now a small disc passing across it blocks only part of the illuminated area.

That gives us something much closer to a genuine transit.

First Experiment: Find the Planet

Begin with the detector recording a steady brightness.

Do nothing for perhaps five or ten seconds.

Then move the model planet steadily across the illuminated disc.

Continue recording for another five or ten seconds after it has left.

When the data are plotted, students should see something resembling:


Light
intensity

100 |____________             ____________
 98 |            \           /
 96 |             \_________/
 94 |
    +------------------------------------> time

We have created our first transit light curve.

The flat section before the transit represents the normal brightness of the star.

The falling section represents the planet beginning to move across the stellar disc.

The lower section occurs while much of the planet is in front of the star.

The brightness then rises again as the planet moves away.

The planet itself has never been detected by the light sensor.

We detected its shadow.

That is a deceptively profound scientific idea.

Can the Light Curve Tell Us How Big the Planet Is?

Now we can start doing some mathematics.

Suppose the star has radius Rs and the planet has radius Rp.

Ignoring complications such as the star being brighter in its centre than around its edge, the approximate fraction of light blocked is:

Transit depth = (Rp / Rs)^2

This occurs because the amount of light blocked depends approximately on the ratio of the areas, not simply the diameters.

Area is proportional to radius squared.

For example, suppose our model star has a diameter of 15 cm and our model planet has a diameter of 3 cm.

The radius ratio is:

3 / 15 = 0.20

So:

Transit depth = 0.20^2

Transit depth = 0.04

The expected brightness decrease is therefore about:

If the normal sensor reading were 1,000 arbitrary units, we might expect it to fall to roughly 960 during the central part of the transit.

Suddenly a small dip on a graph contains physical information about an object we cannot see.

NASA uses exactly this principle with real transit observations: if astronomers know the size of the star, the depth of the transit helps them determine the radius of the planet.

Investigation 1: Bigger Planet, Bigger Dip

Now repeat the experiment with different-sized balls or discs.

Perhaps use:

1 cm;
2 cm;
3 cm;
4 cm;
5 cm.

Keep everything else approximately constant.

Students can record:

Planet diameter	Minimum brightness	Percentage brightness decrease
1 cm
2 cm
3 cm
4 cm
5 cm

They should discover that increasing the planet's diameter does not produce a simply proportional increase in the light lost.

Doubling the radius means approximately four times the area.

That gives a lovely connection between familiar school mathematics and modern observational astronomy.

Investigation 2: What Does Orbital Speed Do?

Use the same planet but move it across the star at different speeds.

Importantly, if the planet follows the same path, the depth of the transit should remain broadly similar.

What changes is its duration.

A slowly moving planet produces a wider dip.

A fast-moving planet produces a narrower one.

This introduces another important feature of astronomical light curves:

The shape of a graph can tell us more than the minimum value does.

Astronomers use the timing of transits to learn about planetary systems. Repeated transits reveal a planet's orbital period, while transit duration and shape contribute further information about the system's geometry.

Investigation 3: Central or Grazing Transit?

This is one of my favourite variations because it shows why the graph needs interpreting rather than simply reading.

First send the planet straight across the centre of the star.

Then repeat the experiment with the planet just clipping the upper edge.

The second is a grazing transit.

The planet never completely crosses the stellar disc, so it never blocks as much light.

The resulting light curve may therefore be:

shallower;
shorter;
differently shaped.

Now ask:

Did we use a smaller planet?

No.

But if we looked only at the depth of the graph without considering the geometry, we might draw the wrong conclusion.

This is real science.

Measurements are rarely interpreted in isolation. Scientists construct models and ask which combination of variables could have generated the data.

Investigation 4: Add Measurement Noise

Real astronomical data do not form beautifully smooth textbook curves.

So perhaps ours should not either.

Try introducing small disturbances.

Move somebody near the apparatus.

Allow a little ambient light into the room.

Introduce a tiny variation in lamp brightness.

Move the detector slightly.

The graph becomes noisier.

Now hide a transit somewhere within the results and ask students to identify it.

This changes the question from:

"Can you see the dip?"

to:

"Are you sufficiently confident that this dip represents a planet?"

That is much closer to the real problem.

NASA's own citizen-science projects invite people to examine actual stellar light curves for the tell-tale patterns of planetary transits.

One Dip Is Not Necessarily a Planet

This is an important addition to the experiment.

Suppose our star becomes slightly dimmer once.

Have we discovered a planet?

Not necessarily.

There could be other explanations.

Astronomers therefore look for evidence that supports the planetary interpretation — particularly repeated transits occurring at regular intervals.

If a similar dip appears every 5.2 days, for example, that becomes much more interesting.

The interval gives us the orbital period.

Our laboratory version could mimic this by mounting the planet on a rotating arm so that it repeatedly passes in front of the star.

Students could be given a long data trace containing several transits and asked:

What is the orbital period of this planet?

Measure the time from one transit centre to the next.

If dips occur at:

10 s, 25 s, 40 s, 55 s...

the model orbital period is approximately:

15 seconds

The same reasoning can be applied to astronomical observations collected over days, months or years.

Could There Be More Than One Planet?

Now things become considerably more entertaining.

Introduce two different-sized planets travelling with different periods.

One produces a deep dip every 20 seconds.

The other produces a shallower dip every 13 seconds.

Record for long enough and the light curve becomes much more complicated.

Students then have to identify two repeating patterns.

NASA notes that light curves become more complicated when several planets transit the same star, but astronomers can disentangle the different signals.

You have effectively turned a lamp, two balls and a light sensor into a simplified planetary-system discovery problem.

From a School Experiment to TESS

This is where I think demonstrations like this become especially valuable.

We have not merely constructed an analogy for something astronomers used to do.

The basic technique remains enormously important.

NASA's TESS — the Transiting Exoplanet Survey Satellite — searches stars for periodic changes in brightness associated with planetary transits. NASA reported in May 2026 that TESS had identified more than 7,900 candidates and 885 confirmed exoplanets at that point.

There is something rather satisfying about showing a student a graph generated using a ball and a light sensor and then explaining:

Space telescopes are looking for essentially the same signature.

The instrumentation is vastly more sophisticated.

The mathematics is much more sophisticated.

The data processing is vastly more sophisticated.

But the underlying observation is recognisable.

Something crossed the star.

The star became dimmer.

Measure that change carefully enough and you may have discovered another world.

And a Transit Can Tell Us Even More

The story does not end with finding the planet.

Modern astronomers can study starlight passing through a planet's atmosphere during a transit.

Different gases absorb particular wavelengths of light.

Instead of measuring only:

How much light disappeared?

astronomers can ask:

Which wavelengths disappeared slightly more than others?

That opens the door to studying exoplanet atmospheres.

NASA's James Webb Space Telescope, for example, records extremely detailed transit observations. Its measurements of LHS 475 b included more than a thousand individual brightness measurements over an observation lasting almost three hours.

Our laboratory experiment has therefore taken us from a simple shadow all the way to spectroscopy of the atmospheres of planets orbiting other stars.

Can Students Work With Real Data?

Yes — and this would make an excellent extension.

Once students understand the model experiment, show them a genuine exoplanet light curve and ask them to identify:

normal stellar brightness;
start of transit;
minimum brightness;
end of transit;
transit depth;
transit duration;
uncertainty and scatter.

They can then compare the real curve with the one obtained experimentally.

NASA's Planet Hunters TESS citizen-science project goes a stage further: participants can examine actual TESS light curves looking for possible transits. No specialist astronomy knowledge is required to begin.

NASA also runs Exoplanet Watch, where observers can collect telescope images and turn them into transit light curves using its EXOTIC analysis software.

That creates an extraordinary progression:

Model planet -> model light curve -> real astronomical data -> citizen science.

The Experiment Is Really About Evidence

There is a much broader lesson here than exoplanets.

We often teach science using objects that can conveniently be seen.

Here is the cell.

Here is the circuit.

Here is the spring.

Here is the reaction.

But much of science deals with things that cannot be observed directly.

We discovered the internal structure of atoms from scattering.

We infer the presence of dark matter from gravitational effects.

We determine the composition of distant stars from their spectra.

We study Earth's interior using seismic waves.

And we discover planets by watching stars become fractionally dimmer.

The ability to reason from an effect to an unseen cause is one of the most powerful forms of scientific thinking.

A Small Shadow From Another World

What I particularly like about the exoplanet transit experiment is that it begins with equipment that looks almost trivial.

A lamp.

A ball.

A sensor.

A graph.

But the question behind it is enormous:

Are there planets orbiting other stars?

For centuries that was largely speculation.

Today we can measure them.

A tiny repeated decrease in a distant star's brightness can tell us that a planet exists, estimate how large it is, determine how frequently it orbits and, with considerably more sophisticated observations, begin investigating its atmosphere.

So perhaps the most important lesson is not really about exoplanets at all.

It is about what scientists mean by evidence.

Sometimes discovery does not begin by seeing the thing you are searching for.

Sometimes it begins by noticing that something else has changed.

And asking why.

Practical challenge

Try building your own transit experiment.

Start with one planet and see whether you can produce a convincing light curve.

Then make it progressively harder:

different planet sizes -> different speeds -> grazing transits -> measurement noise -> repeated transits -> two planets

Finally, compare your graph with a genuine exoplanet light curve.

You may be surprised by how recognisable it looks.

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.