04 September 2026

Chemiluminescence — Creating Light From a Chemical Reaction

 


Chemiluminescence — Creating Light From a Chemical Reaction

We are used to making light with electricity, flames or something that is already extremely hot. Chemiluminescence is different. The chemistry itself produces the light.

There are some practical demonstrations that immediately change the atmosphere in a laboratory.

Chemiluminescence is one of them.

Turn down the room lights, start the reaction, and suddenly a solution begins to glow blue. There is no electrical connection to it. There is no bulb hidden underneath it. Nothing is burning.

The light is being produced by a chemical reaction.

That makes chemiluminescence spectacular to watch, but it also makes it an excellent piece of science. Behind that glow are several important ideas: chemical energy, electron excitation, photon emission, reaction rates, catalysts and activation energy.

And, importantly, it gives us something that can be investigated rather than simply admired.


What Actually Is Chemiluminescence?

Chemiluminescence is the production of light as a result of a chemical reaction.

Normally, when an exothermic reaction releases chemical energy, much of that energy eventually appears as thermal energy.

We notice the mixture getting warmer.

Chemiluminescent reactions take a rather more interesting route.

Part of the energy released by the reaction is used to produce molecules in an electronically excited state.

Those excited molecules are unstable.

They eventually return to a lower-energy state and release the excess energy as a photon of light.

In simplified form:

chemical reactants -> excited product -> product + light

We can represent the final stage as:

excited molecule -> ground-state molecule + photon

The energy of the photon is related to its frequency:

E = hf

where:

  • E is photon energy;
  • h is Planck's constant;
  • f is frequency.

So the colour we observe is directly connected to the amount of energy being released when the molecule returns to its lower-energy state.

That is a remarkable connection.

A bottle glowing blue in a dark laboratory is ultimately demonstrating quantum behaviour.


Light Without Becoming Red Hot

This is perhaps the first surprising feature.

If I want a piece of metal to glow, I normally have to make it extremely hot.

The filament in a traditional incandescent lamp works because electricity heats the filament to a very high temperature.

A flame emits light because energetic particles and hot gases are involved.

Chemiluminescence does not require the material to reach anything approaching those temperatures.

For that reason it is sometimes described as a form of cold light.

That does not necessarily mean absolutely no heat is generated. The chemical reaction can still release thermal energy.

The important point is that the visible light is not being generated simply because the material has become incandescent.

The mechanism is different.

And that immediately raises a good question for students:

If it isn't hot enough to glow, where is the light coming from?

That question takes us straight into energy levels.


The Famous Example: Luminol

Probably the best-known chemiluminescent substance is luminol.

Under suitable chemical conditions, luminol undergoes an oxidation reaction that ultimately produces an electronically excited product.

As that product returns to its lower-energy state, blue light is emitted.

It is a wonderfully striking demonstration.

In a sufficiently dark laboratory, the glow can appear almost unreal.

For teaching purposes I prefer to concentrate on the science rather than merely trying to create the brightest possible reaction.

A good chemiluminescence demonstration should lead to questions such as:

  • Why is light produced?
  • Why is it blue?
  • Why does the brightness change?
  • Why does the reaction eventually stop?
  • What determines how quickly the light fades?
  • Can we measure it?

The moment students start asking those questions, the demonstration has become an experiment.


Why Luminol Appears in Crime Programmes

Luminol is also famous because of forensic science.

The iron associated with haemoglobin in blood can help catalyse the oxidation chemistry involved in the luminol reaction.

Investigators can therefore spray an appropriate luminol reagent over a suspected area in darkness and look for chemiluminescence.

Very small traces of blood may sometimes become visible.

Television crime programmes have understandably made this look enormously dramatic.

But the real chemistry is more interesting than the television version.

A positive glow does not automatically prove:

"There is definitely human blood here."

Other materials can interfere with or catalyse similar reactions.

In forensic science, luminol is therefore useful as a presumptive test, rather than being the final word on what a sample contains.

That distinction is also a useful lesson in experimental science.

A result can provide evidence without providing complete proof.


Glow Sticks Are Chemiluminescence Too

Students may have encountered chemiluminescence long before entering a laboratory.

A glow stick is essentially a small chemical reactor.

Inside are substances kept separate until the stick is bent or activated.

Once mixed, a sequence of chemical reactions transfers energy to a fluorescent dye.

The dye becomes electronically excited.

When it relaxes:

excited dye -> dye + light

Different fluorescent dyes can therefore produce different colours.

This makes glow sticks particularly interesting because they show how chemistry can be used to control the colour of emitted light.

And they provide us with an extremely simple investigation.


A Very Accessible Experiment: Hot and Cold Glow Sticks

Take identical glow sticks and activate them at approximately the same time.

Keep one at room temperature.

Place another, still sealed, in cold water.

Place another in comfortably warm water.

Do not use very hot water and do not cut the glow sticks open.

Now compare their brightness.

The warm glow stick will normally appear brighter.

The cold one will normally be dimmer.

But wait.

Leave them for longer and another difference becomes apparent.

The warmer glow stick tends to use up its reactants more quickly.

It may initially be bright but fade sooner.

The colder glow stick generally reacts more slowly.

It is dimmer, but the glow can persist for longer.

This is a lovely demonstration of reaction kinetics.

Higher temperature -> faster reaction -> brighter light initially -> shorter useful lifetime.

Lower temperature -> slower reaction -> dimmer light -> longer useful lifetime.

Suddenly the familiar statement that "increasing temperature increases the rate of reaction" becomes something students can actually see.


From Demonstration to Investigation

The next stage is to stop relying on our eyes.

Can we measure the light?

A light sensor or data logger can transform the experiment.

Place the chemiluminescent source in a darkened enclosure with a light sensor positioned at a fixed distance.

Record light intensity against time.

Instead of simply saying:

"It gets dimmer."

we can produce a graph.

That immediately opens the door to much more interesting questions.

For example:

How quickly does intensity decrease?

Record intensity every few seconds or use continuous data logging.

Plot:

light intensity against time

and compare different conditions.

How does temperature affect maximum brightness?

Repeat the experiment at several controlled temperatures.

How does temperature affect duration?

Define a threshold intensity and measure how long the glow remains above it.

Is the relationship linear?

Probably not.

And that gives students another important lesson.

Natural systems do not have to produce nice straight-line graphs simply because straight lines are convenient to analyse.


Measuring Chemiluminescence With a Camera

A camera can also be turned into a surprisingly useful scientific instrument.

Keep:

  • exposure time constant;
  • aperture constant;
  • ISO constant;
  • camera position fixed;
  • background lighting constant.

Photograph the chemiluminescent reaction at regular intervals.

You can then compare image brightness.

Even simple image-analysis software can allow students to extract approximate intensity values from the photographs.

This brings together chemistry, physics and computing.

The result is no longer simply:

"The reaction glowed."

It becomes:

"The recorded light intensity decreased with time, and the rate depended upon temperature."

That is a much stronger scientific statement.


The Luminol Demonstration in the Laboratory

For a more advanced laboratory demonstration, luminol provides the classic blue chemiluminescence experiment.

Because luminol demonstrations involve oxidising reagents and alkaline conditions, this is something I would carry out using a recognised educational procedure or commercial demonstration system, with appropriate eye protection, gloves and laboratory controls rather than improvising concentrations simply to obtain a stronger glow.

But once the reaction is running, there are several excellent investigations that can be built around it.

Rather than merely turning off the lights and watching, students might investigate:

  • light intensity against time;
  • the effect of temperature;
  • the effect of changing concentrations using an approved procedure;
  • the role of a catalyst;
  • how long measurable light emission continues;
  • the colour or wavelength of the emitted light.

That last possibility takes us into spectroscopy.


Can We Measure the Colour?

To our eyes, luminol appears blue.

But "blue" is not a particularly scientific measurement.

If suitable equipment is available, the emitted light can be examined using a spectrometer.

Now we can investigate the wavelength distribution of the emitted radiation.

This creates a beautiful connection between chemistry and physics.

The reaction determines the electronic state of the product.

The electronic transition determines the energy released.

The energy determines the photon frequency.

The frequency determines the wavelength.

And the wavelength determines the colour we perceive.

So one glowing solution can connect:

chemical reactions -> molecular energy -> photons -> spectroscopy -> human vision

That is exactly the sort of connection that makes practical science so powerful.


Chemiluminescence, Fluorescence and Phosphorescence Are Not the Same Thing

These terms are easily confused.

They all involve molecules releasing light, but the source of the excitation is different.

Fluorescence

A molecule absorbs electromagnetic radiation, often ultraviolet light.

It becomes excited and then rapidly emits light.

Turn off the exciting radiation and the fluorescence normally disappears almost immediately.

Phosphorescence

Energy is again absorbed first, but the return to the lower-energy state can be much slower.

The material can therefore continue glowing after the original light source has been removed.

That is why some "glow in the dark" materials remain visible.

Chemiluminescence

The initial energy comes from a chemical reaction.

No external ultraviolet lamp is required to excite the molecules.

The chemistry itself supplies the energy.

That difference is crucial.


Biology Has Its Own Version: Bioluminescence

Nature discovered the same principle long before chemists did.

Fireflies, some fungi, marine organisms and many deep-sea creatures can produce light through biochemical reactions.

This is called bioluminescence.

Bioluminescence is essentially chemiluminescence occurring within a biological system.

Luciferin molecules undergo enzyme-controlled reactions involving luciferase.

The result is light.

Organisms use it for remarkably different purposes:

  • attracting mates;
  • communication;
  • camouflage;
  • attracting prey;
  • startling predators;
  • signalling.

Think about the evolutionary significance.

In the deep ocean, sunlight may be virtually absent.

An organism capable of producing light has suddenly gained an entirely new method of communication.

Chemistry has become biology.


Why Does the Reaction Eventually Go Dark?

This is another deceptively useful question.

Students sometimes think of the glow as if it were a property of the substance.

But chemiluminescence exists only while the necessary chemical reaction continues.

Reactants are being consumed.

Eventually one or more reactants becomes sufficiently depleted that the reaction rate falls.

Fewer excited molecules are being produced each second.

So fewer photons are emitted.

The glow becomes dimmer.

Eventually it disappears.

This reinforces a very basic but important chemical idea:

A reaction cannot continue indefinitely if its reactants are being consumed.

A glow stick is therefore also a tiny practical demonstration of limiting reactants.


Brightness and Duration Present an Interesting Trade-Off

This leads to one of my favourite aspects of the glow-stick investigation.

Suppose your goal is simply:

Make the glow stick as bright as possible.

Warm it.

But suppose your goal changes:

Make it remain visible for as long as possible.

Now cooling it may be advantageous.

Neither condition is universally "best".

The best condition depends upon what you are trying to achieve.

That is a very useful scientific and engineering lesson.

Optimization almost always requires deciding what we actually want to optimise.

Maximum brightness?

Maximum duration?

Total light output?

Minimum chemical use?

Performance at a particular temperature?

The science gives us the data.

The engineering problem determines how we use it.


A Possible Student Investigation

A very manageable investigation would be:

Question

How does temperature affect the intensity and duration of chemiluminescence?

Use identical sealed glow sticks from the same batch.

Test them under several temperature conditions.

Measure:

  • temperature;
  • initial light intensity;
  • maximum light intensity;
  • intensity at regular time intervals;
  • time taken to fall below a chosen brightness level.

Then plot graphs of:

light intensity against time

for each temperature.

Students could then discuss:

  • reaction rate;
  • collision theory;
  • energy transfer;
  • repeatability;
  • control variables;
  • uncertainty;
  • limitations of the measuring equipment.

Suddenly a relatively inexpensive glow stick has become an experiment touching several areas of GCSE and A-level science.


What Would We Need to Control?

This is where the practical becomes particularly valuable.

If we want to compare the results scientifically, we need to think about controls.

Keep constant, as far as possible:

  • glow-stick type;
  • manufacturing batch;
  • activation method;
  • time between activation and first measurement;
  • distance from sensor;
  • sensor orientation;
  • ambient lighting;
  • temperature throughout the experiment.

That last point is particularly interesting.

Putting a glow stick into water at 10 C does not necessarily mean that the reacting chemicals instantly become 10 C.

There will be a period of thermal equilibration.

That provides another opportunity to discuss the difference between the condition we think we have created and the condition actually experienced by the experimental system.


What About Experimental Error?

Imagine two glow sticks apparently behaving differently.

Is that because of temperature?

Perhaps.

But there could also be manufacturing variation.

One stick might contain slightly different quantities of reactants.

They might not have been activated in exactly the same way.

One could be positioned slightly closer to the sensor.

External light might interfere.

That means a better experiment would use repeats.

For each temperature:

repeat the measurement several times

and calculate a mean.

Students can then begin thinking about spread and uncertainty rather than treating every individual measurement as perfectly reliable.

Again, chemiluminescence has become much more than a colourful demonstration.


Why I Like Experiments Like This

One reason I enjoy practical science is that a relatively simple observation can lead surprisingly far.

You begin with:

"Look — it glows."

Five minutes later you can be discussing:

  • activation energy;
  • molecular collisions;
  • electron energy levels;
  • photons;
  • spectroscopy;
  • catalysts;
  • rate equations;
  • forensic science;
  • biological evolution;
  • experimental uncertainty.

That is what good practical science should do.

The experiment is not an interruption to the theory.

It creates reasons to want to understand the theory.

A student who has watched the brightness of a chemiluminescent reaction change with temperature has a much more concrete reason to care about reaction rates.

A student who has seen blue light appear without a lamp has a reason to ask what a photon actually represents.

Those questions matter.


The Moment When the Lights Go Out

There is also something valuable about the sheer theatre of chemiluminescence.

Science teaching does not have to be dull in order to be rigorous.

Turn down the laboratory lights.

Start the reaction.

Watch blue light appear where there was darkness.

For a moment, students simply watch.

Then comes the question:

"How is it doing that?"

That is exactly the question we want.

The spectacle gets their attention.

The science keeps it.


Conclusion — Sometimes Chemistry Really Does Glow

Chemiluminescence is a wonderful example of why science becomes so much more interesting when we move beyond simply learning definitions.

A chemical reaction releases energy.

Some of that energy creates electronically excited molecules.

Those molecules return to lower-energy states.

Photons are released.

We see light.

But from that simple chain of events we can investigate reaction rates, temperature, catalysts, spectroscopy, forensic chemistry, biochemistry and experimental design.

We can measure the changing brightness.

We can produce graphs.

We can test hypotheses.

And we can ask whether making a reaction brighter necessarily makes it better.

So the next time somebody cracks a glow stick at a party, perhaps look at it slightly differently.

Inside that little plastic tube is a chemical reaction producing excited molecular states and releasing photons into the room.

And if we take it into the laboratory, measure what is happening and start asking questions, that glowing stick becomes a surprisingly sophisticated scientific experiment.

Sometimes the best way to illuminate a scientific idea is quite literally to make the chemistry produce the light.

03 September 2026

How Do You Discover a Planet You Cannot See?


 

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:

4%

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 diameterMinimum brightnessPercentage 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


 

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:

StageNumber of edgesRelative edge lengthRelative perimeter
0313
1121/34
2481/916/3
31921/2764/9

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:

  1. Draw something.
  2. Replace each part according to a rule.
  3. Apply the same rule to the new parts.
  4. 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.

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