25 September 2026

Making a Silver-Halide Photogram — When Chemistry Becomes a Photograph


 

Making a Silver-Halide Photogram — When Chemistry Becomes a Photograph

There is something almost magical about watching a photograph appear in a tray of developer.

A blank sheet of photographic paper goes into the liquid. For a few seconds, apparently nothing happens. Then faint grey shapes begin to emerge. Shadows deepen. Edges become clearer. Within a minute or two, an image that simply was not visible before is sitting in front of you.

Of course, it is not magic.

It is chemistry.

And for students who have grown up in a world where taking a photograph means tapping a screen and seeing the result instantly, traditional black-and-white photography provides a wonderful opportunity to connect chemistry, physics, history and technology in one very memorable practical investigation.

I would begin with something extremely simple: a silver-halide photogram.

Then I would take the experiment much further.

I can get an old 35 mm camera out of the cupboard, load it with black-and-white film, let students take photographs and then actually process the film. Instead of an image appearing instantly on a screen, they can watch the entire photographic process unfold in front of them.

The final moment — unrolling the processed film and seeing those tiny negatives for the first time — is particularly satisfying.

For much of the twentieth century, this was how an enormous proportion of family photographs, school photographs, newspaper photographs and scientific images were made.

The smartphone camera may be astonishingly sophisticated, but there is a great deal of science hidden by that convenience.

Traditional photography allows us to uncover it again.


What Is a Photogram?

A photogram is one of the simplest photographs it is possible to make because you do not actually need a camera.

Instead, objects are placed directly onto light-sensitive photographic paper.

The paper is briefly exposed to light and then processed using photographic chemicals.

Where light reaches the paper strongly, the finished image becomes dark.

Where an opaque object blocks the light, the paper remains comparatively light.

Translucent objects produce shades of grey.

The result can be surprisingly beautiful.

Leaves, feathers, mesh, keys, electronic components, pieces of lace, laboratory glassware and even water droplets can create fascinating images.

But behind those shapes lies some excellent chemistry.


The Chemistry Begins with a Precipitate

Traditional photographic materials contain microscopic crystals of silver halides suspended in a gelatin emulsion.

Common examples include silver bromide and silver chloride.

Silver bromide can be formed by precipitation when silver ions meet bromide ions:

Ag+ + Br- -> AgBr(s)

Silver bromide forms a pale cream precipitate.

This gives us a useful introductory experiment before we even touch the photographic paper.

Under appropriate laboratory conditions, students can observe a small-scale precipitation reaction involving silver ions and halide ions and then investigate what happens when the precipitate is exposed to light.

That gives us our first clue about why silver compounds became so important in photography.

They are light sensitive.


Why Does Silver Bromide Respond to Light?

Silver bromide crystals consist of silver ions and bromide ions arranged within a crystal lattice.

When light of sufficient energy is absorbed by the crystal, electrons can become available and ultimately allow tiny quantities of silver ions to be converted into metallic silver.

In simplified terms:

Ag+ + e- -> Ag

Only a tiny amount of metallic silver is initially produced.

There is not enough for us to see a photograph.

Instead, the exposure creates what photographers call a latent image.

The photograph is already encoded into the photographic material, but it remains invisible.

This is one of the ideas I particularly like discussing with students.

An image can exist even though you cannot yet see it.

That is a wonderfully strange concept.


The Developer Reveals the Hidden Image

The job of the photographic developer is to turn that invisible latent image into something we can see.

A developer contains reducing agents.

The tiny regions created by exposure to light encourage further reduction of silver ions within the exposed silver-halide crystals.

More metallic silver is produced.

And metallic silver is dark.

The more exposure a region receives, the greater the eventual density of metallic silver produced during development.

So the image begins to appear.

ILFORD describes conventional black-and-white film in essentially these terms: light creates a latent image within silver-halide crystals, and development amplifies that extremely small initial change into visible grains of metallic silver. Ilford Photo

This is a lovely example of chemistry being used as an amplification process.

A very small photochemical change eventually produces a macroscopic image that our eyes can see.


Watching the Photograph Appear

This is the moment students tend to remember.

The exposed paper goes into the developer.

Initially it looks blank.

Then something happens.

Perhaps the outline of a key appears.

Then the veins of a leaf.

Then areas that were completely exposed become increasingly dark.

It is particularly effective if students have previously only experienced digital photography.

A digital photograph appears instantly because enormously complicated electronics are doing the processing invisibly.

With silver-halide photography, much of the process happens physically in front of you.

You can watch chemistry creating the image.


But Development Cannot Simply Continue

If we left the photographic paper in active developer indefinitely, the image would continue changing.

Traditional processing therefore separates the stages carefully.

A typical black-and-white sequence is:

  1. development;
  2. stop bath or rinse;
  3. fixing;
  4. washing;
  5. drying.

ILFORD's own technical guidance describes film processing as development, stop bath, fixing, washing, wetting-agent rinse and drying, with temperature, timing and agitation important for consistent results. Ilford Photo

This gives students another useful lesson.

Good experimental science is not simply about putting chemicals together.

It is about controlling variables.

Time matters.

Temperature matters.

Concentration matters.

Agitation matters.

And reproducibility matters.


The Fixer Performs a Completely Different Job

Development produces metallic silver in those crystals associated with the latent image.

But a problem remains.

Large amounts of unexposed silver halide are still present.

If we simply took the photograph into daylight at this stage, those remaining light-sensitive crystals would react and the photograph would gradually darken.

We therefore need to remove them.

That is the job of the fixer.

Traditional fixing agents contain thiosulphate ions, usually using sodium thiosulphate or ammonium thiosulphate.

A simplified representation is:

AgBr + 2S2O3^2- -> [Ag(S2O3)2]^3- + Br-

The important point is that the insoluble silver halide is converted into a soluble silver-thiosulphate complex which can be removed from the photographic material.

The metallic silver forming the image remains.

ILFORD similarly describes fixation as the removal of residual silver halide while leaving metallic silver behind to form the permanent image. Ilford Photo

Now the photograph can safely be exposed to ordinary light.

The image has been fixed.

That familiar photographic word therefore has a very literal chemical meaning.


A First Practical: Make a Photogram

For a first session I would keep things deliberately simple.

Students can choose several objects with different optical properties.

For example:

  • a metal key;
  • a leaf;
  • a feather;
  • a piece of netting;
  • a glass slide;
  • a translucent plastic object;
  • a spring;
  • wire;
  • a small electronic circuit board;
  • pieces of laboratory apparatus.

Under suitable darkroom or safelight conditions, arrange them directly on black-and-white photographic paper.

Expose the arrangement to light for a controlled period.

Then develop, stop, fix and wash the paper according to the photographic paper and chemical manufacturers' instructions.

Suddenly we have something worth investigating rather than merely demonstrating.


Turn the Photogram into a Proper Experiment

There are many variables that students could investigate.

For example, keep everything else constant and change the exposure time.

Try a series such as:

1 second

2 seconds

4 seconds

8 seconds

16 seconds

The exact useful times will depend on the light source, distance and photographic material, so an initial test strip is much better science than simply guessing the "correct" exposure.

Students can compare the resulting density.

Does doubling exposure time double the apparent darkness?

Probably not in any simple visual sense.

And that opens another discussion about photographic response, density and logarithmic scales.


Investigating Distance

Move the light source farther from the photographic paper.

What happens?

Now photography connects with physics.

For an approximately point-like source, illumination is related to distance through the inverse-square relationship:

intensity proportional to 1 / distance^2

Double the distance and, under idealised conditions, the intensity falls to approximately one quarter.

The experiment suddenly connects photochemistry with GCSE and A-level physics.


Opaque, Transparent and Translucent

A photogram can also become a simple investigation of light transmission.

An opaque metal washer may block almost all the light.

Clear glass might transmit most of it.

Frosted plastic scatters the light.

Coloured transparent materials may behave differently depending upon the spectral sensitivity of the photographic paper.

Rather than treating the photograph simply as artwork, students can ask:

What can we infer about an object's interaction with light from the image it produces?

That is a much more scientific question.


Then Bring Out the 35 mm Camera

The photogram establishes the basic principle.

Now comes the part I particularly enjoy.

I can take an old 35 mm camera out of the cupboard.

For many students, even loading the film may be unfamiliar.

Open the back.

Place in the film cassette.

Pull the film leader across.

Engage it with the take-up mechanism.

Close the camera.

Advance the film.

Suddenly photography becomes mechanical as well as chemical.

There is no LCD screen.

There is no instant review.

There is no delete button.

You have perhaps 24 or 36 exposures.

You actually have to think before pressing the shutter.


A Camera Is Really Just Controlling Light

Once the mystery is removed, the basic photographic camera is beautifully simple.

The lens forms an image.

The aperture controls how much light can enter.

The shutter controls how long that light reaches the film.

The film records the image.

That allows us to introduce another basic photographic relationship:

exposure approximately depends on light intensity x exposure time

Photography becomes a practical demonstration of optics.

Students can investigate:

  • aperture;
  • shutter speed;
  • focus;
  • depth of field;
  • motion blur;
  • focal length;
  • exposure.

Things that are largely automated by a smartphone become visible decisions again.


The Strange Experience of Not Knowing Whether the Photograph Worked

This is something younger photographers have almost completely lost.

Take a photograph digitally and you immediately inspect it.

Was it sharp?

Was the exposure correct?

Did somebody blink?

With film, you do not know.

You press the shutter and move on.

The image exists as a latent chemical change inside the film cassette, but there is nothing yet to inspect.

Only when the film has been developed do you discover what you actually captured.

That delay changes the way you take photographs.

It encourages thought before exposure rather than correction afterwards.


Processing the Film

After the photographs have been taken, the film must be processed.

This provides another wonderful moment.

The film has to be removed from its cassette and loaded onto a processing reel in complete darkness or inside a changing bag.

Once the light-tight processing tank is closed, the rest of the process can normally be carried out in ordinary room light.

Developer is added.

The film is agitated according to the chosen process.

Development is stopped.

The film is fixed.

Then it is washed.

Temperature control matters because photographic development is a chemical reaction and its rate varies with temperature. ILFORD specifically notes the dependence of development on temperature and pH. Ilford Photo

Again, what looks like photography has become experimental chemistry.


And Then Comes the Reveal

Eventually the tank opens.

The film is carefully removed from the spiral.

And there they are.

Tiny photographs.

Except that they look completely wrong.

The bright sky appears dark.

Dark clothing may appear pale.

Light and dark are reversed.

The students are looking at a negative.

For somebody who has only known digital photography, physically holding a strip of 35 mm negatives can be surprisingly fascinating.

The negative is not merely an old-fashioned curiosity.

It reveals how the entire photographic system works.

A bright part of the original scene exposes the film strongly and eventually produces a dense region containing more metallic silver.

A dark part of the scene gives less exposure and produces a more transparent region.

When we subsequently use the negative to make a print, that relationship is reversed again.

The final photograph looks normal.


From Negative to Positive Print

Now we can complete the journey.

Place the negative in an enlarger.

Project its image onto photographic paper.

Adjust focus and enlargement.

Expose the paper.

Then once again:

developer...

the image appears...

stop...

fix...

wash...

dry.

We have gone from:

real scene -> camera -> latent image -> negative -> projected image -> photographic paper -> positive print

That entire chain is enormously instructive.

A modern phone compresses all of this into perhaps a fraction of a second.

The old process allows students to see every stage.


A Contact Sheet Makes an Excellent Teaching Tool

Before making individual enlargements, I would also show students how a contact sheet works.

Place strips of negatives directly against photographic paper beneath glass.

Expose the whole sheet.

Process it.

You then have miniature positive versions of every frame.

This was once an important part of photographic workflow.

The photographer could inspect the contact sheet and decide which frames were worth enlarging.

It also teaches something interesting about selection.

A photographer might take 36 photographs but print only three.

That is another contrast with the modern habit of accumulating thousands of nearly identical digital images.


Why Black-and-White Photography Is Perfect for Teaching Chemistry

Colour photography is scientifically fascinating, but it introduces several additional layers of complexity.

Black-and-white silver photography exposes the essential chemistry far more clearly.

We can follow silver through the whole process.

Begin with:

Ag+

Form:

AgBr

Expose it to light.

Create a latent image.

Develop exposed crystals.

Produce:

Ag metal

Remove unwanted AgBr during fixation.

What remains is an image made largely from microscopic particles of metallic silver.

The photograph is therefore not simply a picture.

It is a chemical object.


There Is Also a Valuable Environmental Discussion

Once we start processing photographic materials, another question becomes important:

What happens to the chemistry afterwards?

Used photographic fixer can contain silver compounds and should not simply be treated as though it were harmless water. ILFORD technical information for photographic processing notes silver in waste fixer streams. Ilford Photo

That creates an opportunity to discuss:

  • chemical waste;
  • heavy-metal recovery;
  • laboratory responsibility;
  • recycling;
  • safe storage;
  • why disposal instructions matter.

This is exactly the sort of broader scientific thinking I want students to develop.

An experiment does not finish simply because we have obtained our result.

We are also responsible for the materials we have used.


Safety Matters

Traditional photography is perfectly capable of being an excellent teaching practical, but it should still be treated as laboratory chemistry.

I would use commercial photographic developer and fixer according to their current instructions and safety data, with appropriate eye protection, gloves where specified, good ventilation and separate labelled equipment.

Silver nitrate used for introductory precipitation experiments requires particular care because it can damage eyes, irritate tissue and produce persistent stains.

Photographic chemistry should never be placed in drinks bottles or unlabelled containers.

And silver-containing waste should be collected and disposed of appropriately rather than casually poured away.

Students should see good chemical practice as part of the experiment, not as an inconvenience added to it.


What Could Students Actually Investigate?

Once the basic technique is working, the possibilities expand enormously.

A photogram could become an investigation into exposure time.

Film could be used to explore shutter speed and motion.

Different apertures could demonstrate depth of field.

A test strip could investigate photographic-paper exposure.

Negatives could be compared for different camera settings.

Students could measure optical density.

They could investigate developer temperature.

They could compare fresh and ageing chemistry.

They could examine film grain under magnification.

They could even compare a film image with a modern digital sensor image of exactly the same scene.

Suddenly one old camera has become a gateway into:

chemistry, optics, electronics, materials science, imaging, measurement, art and technological history.


From Silver Grains to Silicon Pixels

The final stage of the lesson should probably bring us back to the device almost every student has in their pocket.

A smartphone camera does not normally use silver halide.

Its sensor contains millions of photosensitive semiconductor elements.

Photons still have to be detected.

Light still carries the information.

Lenses still have to form an image.

Exposure still matters.

But the method of recording that information has changed dramatically.

In the traditional camera:

light -> chemical change

In a modern digital camera:

light -> electrical signal -> numerical data

That is an extraordinary technological transition.

And it happened within living memory.


"Was Every Photograph Really Made Like This?"

I sometimes tell students that this is how photographs used to be made.

It is worth adding a little historical precision.

Photography has used many processes during its history: daguerreotypes, wet-plate collodion, glass negatives, silver-gelatin materials, colour films and numerous specialist processes.

For example, the daguerreotype used a silver-coated copper plate sensitised with silver halides rather than modern roll film. National Science and Media Museum blog

But for a large part of the twentieth century, silver-halide film and photographic paper were the dominant technology behind ordinary photography.

The family snapshot.

The school photograph.

The holiday photograph.

The wedding album.

News photography.

Scientific photography.

The rolls of film sent away in envelopes and returned days later as prints.

For students who have never known anything except instant digital images, that world can seem surprisingly remote.

Yet it is not ancient history at all.


One Photograph, Several Sciences

This is exactly the sort of experiment I enjoy because it refuses to stay neatly inside one subject.

The precipitation reaction is chemistry.

The photosensitivity is photochemistry.

Development involves reduction.

Fixing involves complex ions and solubility.

The lens introduces optics.

Exposure brings in intensity and time.

The camera introduces engineering.

Film grain leads into materials science.

The history of photography introduces technological change.

And the final print becomes art.

That is much closer to real science than treating every topic as though it belongs in its own isolated chapter of a textbook.


The Photograph Appearing in the Tray Is Still Special

There are faster ways to make a photograph.

There are easier ways.

There are certainly cheaper ways if you already own a smartphone.

But very few are as educational.

I can explain silver ions, precipitation, reduction, lenses, shutter speeds and negatives on a whiteboard.

Students may understand them perfectly well.

But putting a supposedly blank sheet of photographic paper into developer and watching an image slowly emerge is different.

Then taking an old 35 mm camera, processing the film and holding the still-wet strip of negatives up to the light completes the story.

For a few moments, students experience photography not as an app, but as a scientific process.

And perhaps the most important question is no longer:

"What photograph did we take?"

It becomes:

"How did light and chemistry manage to make an image at all?"

That is a much more interesting question.

24 September 2026

What Can a Hole in the Moon Tell Us About Something That Happened Billions of Years Ago?


 

What Can a Hole in the Moon Tell Us About Something That Happened Billions of Years Ago?

Look at the surface of the Moon through even a modest telescope and one feature immediately dominates the view.

Craters.

Some are tiny. Others are hundreds of kilometres across. Some overlap older craters. Some have bright rays extending across the lunar surface. Some have relatively smooth floors, while larger examples can contain terraces, collapsed walls and mountains rising from their centres.

They are not simply holes.

They are records of events.

Nobody watched most of these impacts happen. There were no cameras, seismographs or written observations. Yet planetary scientists can examine the crater that remains and work backwards, asking questions such as:

  • How large was the impacting object?

  • How energetic was the collision?

  • At what angle did it arrive?

  • What was the surface made from?

  • Which event happened first?

  • How old might this part of the landscape be?

That makes impact craters a wonderful example of one of the most important ideas in science:

We can investigate events we never actually witnessed by studying the evidence they left behind.

And we can explore some of that science with a surprisingly simple experiment.

Making a miniature impact landscape

The basic experiment needs very little specialised equipment.

I would start with a shallow tray containing a fairly deep layer of fine material such as flour.

On top of the flour, add a very thin layer of contrasting material. Cocoa powder works particularly well, although anything fine and visibly different from the underlying material can be used.

The result represents a very simplified planetary surface.

Then drop an object into it.

A marble or small ball bearing produces an immediate and rather dramatic result.

There is a crater.

There is a raised rim.

Material has been thrown outwards.

The coloured surface layer has been disturbed.

And suddenly there is far more to investigate than simply measuring the diameter of a hole.

Start by changing just one variable

As with any worthwhile scientific investigation, the temptation is to change everything at once.

Resist it.

Choose one variable and investigate it systematically.

For example, keep the impactor the same but release it from heights of:

20 cm

40 cm

60 cm

80 cm

100 cm

After each impact, carefully measure the crater diameter.

Students can record something like:

Drop heightCrater diameter
20 cm...
40 cm...
60 cm...
80 cm...
100 cm...

They can then plot crater diameter against drop height.

Immediately the experiment has moved beyond merely producing an impressive photograph.

We are looking for a relationship.

Why should height make a difference?

Before the object is released, it has gravitational potential energy.

For a simple vertical drop:

GPE = mgh

where:

m = mass of the impactor
g = gravitational field strength
h = height above the surface

As it falls, much of that gravitational potential energy becomes kinetic energy.

Immediately before impact:

KE = 1/2 mv^2

The higher the starting point, the greater the energy available when the impactor reaches the surface.

That energy has to go somewhere.

It can:

  • move surface material;

  • break or deform material;

  • eject particles;

  • heat the impactor and target;

  • produce sound;

  • generate vibrations;

  • and create the crater itself.

Our flour experiment is extremely low-energy compared with a real asteroid impact, but the important principle is there:

An impact is an energy-transfer event.

Mass is another obvious variable

Keep the drop height constant but change the mass of the impactor.

Perhaps students could use objects with similar diameters but different masses.

That is experimentally more interesting than simply changing to a bigger object because it helps separate two different variables:

mass and size.

If the speed is approximately the same, kinetic energy depends directly upon mass:

KE = 1/2 mv^2

Double the mass and, at the same speed, the kinetic energy doubles.

But does the crater diameter double?

Probably not.

And that is where the investigation begins to become much more interesting.

Science is full of relationships that are not simply proportional

Students often meet simple proportional relationships:

double one quantity and another doubles.

Nature frequently refuses to be that cooperative.

Crater dimensions depend on many interacting factors, including:

  • impact energy;

  • impactor size;

  • impactor density;

  • impact speed;

  • impact angle;

  • surface density;

  • surface strength;

  • gravity.

Scientists therefore use scaling relationships to connect laboratory experiments, computer simulations and enormous planetary impacts.

A marble falling into flour is obviously not a meteorite hitting the Moon at many kilometres per second.

The experiment is an analogue.

That distinction is important.

We are investigating some of the principles involved in crater formation, not claiming that a tray of flour perfectly reproduces a lunar impact.

That itself is an excellent scientific discussion.

When is a model useful even though it is not completely realistic?

Change the diameter of the impactor

Another investigation is to use spheres of different diameters.

Students might initially predict:

Bigger object = bigger crater.

That is probably true in broad terms, but it raises another question.

Why?

A larger object may also have:

  • greater mass;

  • greater surface area;

  • different density;

  • different aerodynamic behaviour.

It becomes a nice introduction to experimental design.

If we genuinely want to investigate diameter alone, how do we control the other variables?

This is often more scientifically valuable than producing a perfectly neat graph.

Students begin discovering that designing a fair experiment can be harder than carrying one out.

What happens if the impactor arrives at an angle?

Dropping objects vertically is easy.

Real objects in the Solar System are not obliged to cooperate.

Asteroids and meteoroids can approach a planetary surface at different angles.

A simple classroom experiment can investigate this by arranging for the projectile to enter the material obliquely rather than vertically.

Students can investigate:

  • crater shape;

  • crater length and width;

  • direction of ejecta;

  • distribution of disturbed surface material.

At the relatively low velocities of a classroom experiment, changing the angle may produce noticeably asymmetric results.

Real planetary impacts are considerably more complicated because they usually occur at enormous speeds. Hypervelocity impacts can behave rather differently from a slowly dropped ball.

Again, that difference provides an excellent opportunity to discuss the limitations of models.

Look at the ejecta, not just the crater

This is one reason I particularly like using a thin contrasting surface layer.

When the impact occurs, material is thrown outwards.

This material is called ejecta.

Instead of simply measuring crater diameter, students can look at:

  • maximum ejecta distance;

  • direction;

  • symmetry;

  • thickness;

  • streaks or rays;

  • distribution around the crater.

Photographing the tray directly from above makes these patterns much easier to compare.

A ruler included in each photograph gives a scale.

Students could even analyse the images digitally rather than measuring the crater directly.

Suddenly we have moved into scientific imaging and quantitative image analysis.

High-speed video could make this even better

This is one experiment where a camera can reveal something the eye easily misses.

Film the impact at the highest useful frame rate available.

Played back slowly, students may see:

  1. the impactor entering the surface;

  2. material beginning to move outwards;

  3. the developing cavity;

  4. ejecta travelling away from the impact;

  5. material falling back around the crater.

What appears to be an instantaneous event becomes a sequence.

A side view can show the ejecta rising.

A top view reveals its distribution.

Using two cameras simultaneously would make an especially effective demonstration because the same event could be examined from two completely different perspectives.

A crater is much more than a hole

Now we can return to the Moon.

Planetary scientists do not simply measure crater diameters.

The morphology of a crater — its shape and structure — contains information.

Depending on its size and the conditions under which it formed, an impact crater can contain features such as:

  • a raised rim;

  • an ejecta blanket;

  • rays extending across the surrounding terrain;

  • slumped or terraced walls;

  • a relatively flat floor;

  • central peaks in larger complex craters.

These structures tell us something about the extraordinary forces involved.

For a sufficiently large impact, the ground does not simply behave like a rigid solid being struck with a hammer. Under the enormous pressures produced during a hypervelocity impact, rock can fracture, flow and rebound on a huge scale.

That is why enormous impact structures can be far more complicated than simple bowl-shaped holes.

Why does the Moon have so many craters?

The Moon provides an almost perfect place to introduce another geological idea.

A crater can only tell us its history if the evidence survives.

On Earth, landscapes are continually being modified.

We have:

  • wind;

  • rain;

  • rivers;

  • glaciers;

  • vegetation;

  • weathering;

  • erosion;

  • sedimentation;

  • plate tectonics.

Earth's surface is extraordinarily active.

The Moon has no rivers washing craters away, no vegetation covering them and no active plate tectonic system recycling its surface in the way Earth's crust is recycled.

Its landscape can therefore preserve extremely old evidence.

Looking at the Moon is rather like looking at an ancient astronomical archive.

Counting craters can even tell us something about age

Imagine two neighbouring lunar surfaces.

One is covered with craters.

The other contains relatively few.

Which is probably older?

The heavily cratered surface has generally been exposed to impacts for longer, whereas a younger surface may have been resurfaced more recently.

Planetary scientists therefore use crater counting as one technique for comparing the relative ages of surfaces.

It is not simply:

more craters = exact age.

Scientists must consider crater sizes, resurfacing events, overlapping structures and models of impact frequency.

But the central principle is wonderfully accessible.

If impacts accumulate with time, the number and distribution of craters can help reconstruct a landscape's history.

A geological detective story

Overlapping craters introduce another beautifully simple idea.

Suppose crater A cuts across crater B.

Which formed first?

Crater B must already have existed before crater A could have disrupted it.

Students have just used relative dating.

The same reasoning is used throughout geology.

A feature that cuts another feature must generally be younger than the feature it cuts.

A tray of flour has now taken us into stratigraphy and geological history.

Mars adds another layer to the story

The same reasoning can be applied to Mars.

But Mars has had a different geological and atmospheric history from the Moon.

Its surface shows:

  • impact craters;

  • enormous volcanoes;

  • valleys;

  • sedimentary structures;

  • evidence of erosion;

  • ancient surfaces;

  • younger resurfaced areas.

Comparing cratered landscapes on Mars with those on the Moon therefore becomes much more than identifying holes.

Students can ask:

What has happened to this landscape since the crater formed?

Has material filled the crater?

Has erosion modified it?

Has volcanic activity covered older structures?

Has wind moved sediment across it?

This is planetary geology becoming a genuine investigation rather than simply learning the names of planets.

And then there is Earth

Impact craters exist here too.

They are simply harder to preserve.

One of the most famous impact structures is associated with the event about 66 million years ago at the end of the Cretaceous Period.

The Chicxulub impact structure in what is now Mexico is roughly 180 km across.

Its significance reaches far beyond geology because it is associated with one of the greatest mass-extinction events in Earth's history.

Suddenly our tray of flour connects:

physics
to astronomy
to geology
to palaeontology
to evolution.

That is exactly why I enjoy experiments that sit outside the formal syllabus.

Individual school subjects suddenly stop looking quite so separate.

Could students calculate the impact energy?

Yes — and this could make an excellent A-level extension.

For the falling object, begin with:

GPE = mgh

If losses are ignored, immediately before impact:

KE approximately equals mgh

Students could calculate the approximate impact energy for each drop.

They could then plot:

crater diameter against impact energy

rather than simply crater diameter against height.

This is scientifically much more meaningful because different combinations of mass and height can produce the same gravitational potential energy.

For example, students could deliberately choose different masses and heights designed to give approximately equal values of mgh.

Would they produce identical craters?

That becomes a much more sophisticated investigation.

A useful challenge: equal energy, different impactor

Suppose we arrange two impacts with approximately the same calculated energy.

One uses:

a lighter object dropped from higher up.

The other uses:

a heavier object dropped from a lower height.

If KE is approximately the same, will the craters be identical?

That question is far more interesting than merely confirming that higher drops make larger holes.

Students may discover that impactor geometry, momentum, contact area and the behaviour of the target material also matter.

The experiment begins to reveal the danger of reducing a complicated physical event to a single number.

Momentum gives us another way of looking at it

Kinetic energy is not the only useful quantity.

Momentum is:

p = mv

Two objects can have the same kinetic energy but different momenta.

This gives A-level students another possible investigation.

Which quantity appears to correlate more strongly with the crater dimensions in our particular experimental setup?

Energy?

Momentum?

Impactor diameter?

Perhaps no single variable completely explains the result.

That is much closer to real experimental science.

An investigation students could genuinely design themselves

I would be tempted not to give students a complete method.

Instead I might provide the question:

What determines the size of an impact crater?

Then allow them to decide:

  • what variable to change;

  • what quantities to measure;

  • what controls are necessary;

  • how many repeats are needed;

  • how uncertainty should be handled;

  • what graph should be plotted.

Different students might investigate entirely different aspects of the same phenomenon.

One group could investigate mass.

Another could investigate height.

Another could investigate projectile diameter.

Another could concentrate on impact angle.

Another could analyse ejecta.

At the end, the class could combine its evidence.

That begins to resemble the way scientific research actually develops.

Repeats matter

Flour does not behave perfectly.

Neither do students dropping marbles.

Two apparently identical impacts may produce slightly different crater diameters.

That is not experimental failure.

It is experimental reality.

Repeat each condition several times and calculate a mean crater diameter.

Students can then discuss:

  • random variation;

  • anomalous results;

  • measurement uncertainty;

  • repeatability;

  • how many repeats are sufficient.

A very visually dramatic experiment has quietly become an exercise in serious experimental technique.

One practical problem: how do you measure a crater?

Even this apparently simple question deserves thought.

Where exactly does the crater end?

Do we measure:

  • the inner depression?

  • the outer rim?

  • the maximum diameter?

  • two perpendicular diameters and take a mean?

If the crater is elliptical, one measurement is clearly inadequate.

For an angled impact, students might record:

major axis = ...

minor axis = ...

and calculate their ratio.

Experimental definitions matter.

Two groups cannot meaningfully compare their data unless they have agreed what they mean by "crater diameter".

That is a lesson extending far beyond planetary science.

Improve the experiment with photography

A particularly good method would be to create a permanent visual record of every impact.

Mount a camera above the tray.

Keep:

  • camera position;

  • focal length;

  • lighting;

  • tray position;

  • scale ruler

constant.

Photograph every crater before resetting the surface.

The photographs can then be compared later.

Students could measure crater dimensions directly from the image and perhaps investigate the area covered by ejecta.

A numbered card beside the tray could identify each experimental condition.

That turns a messy practical experiment into a much better documented investigation.

Safety and practical organisation

The experiment is straightforward, but a little organisation helps.

Use relatively small, manageable impactors and sensible drop heights.

Protect the surrounding area because fine powders can travel surprisingly far.

Avoid throwing hard objects or launching high-speed projectiles.

The aim is to investigate impact processes, not to reproduce genuine asteroid velocities in the laboratory.

A large tray or shallow container also makes resetting the surface much easier.

After each test:

  1. recover the impactor;

  2. level the flour;

  3. recreate the thin contrasting layer;

  4. check the scale;

  5. repeat the experiment.

Consistency here will greatly improve the results.

The experiment I would like students to remember

The best science practicals are not necessarily those involving the most complicated equipment.

Sometimes the best experiment begins with a question that becomes larger the longer you investigate it.

Drop a marble into flour and initially the question is:

How big is the hole?

A few minutes later it becomes:

How does crater diameter depend on impact energy?

Then:

Can we infer the properties of an impactor from the crater it leaves behind?

And eventually:

How can scientists reconstruct an event that happened billions of years before human beings existed?

That is a remarkable journey from a baking ingredient and a marble.

Science is the art of reading evidence

Perhaps that is the most important idea behind this experiment.

Science is not restricted to events we can watch happening.

We cannot travel back to observe the formation of every lunar crater.

We cannot stand beside an asteroid as it strikes ancient Mars.

We were not present for the enormous impacts that shaped the early Solar System.

But those events left evidence.

Crater dimensions.

Ejecta.

Fractured rocks.

Overlapping structures.

Chemical signatures.

Altered landscapes.

Scientists learn to read those clues.

And from them, we reconstruct a history.

So the next time you look through a telescope and see the battered surface of the Moon, it is worth remembering:

You are not simply looking at holes in the ground.

You are looking at billions of years of Solar System history, written into the landscape.

23 September 2026

Chaos — When Tiny Differences Become Enormous

 


Chaos — When Tiny Differences Become Enormous

If mathematics tells us exactly what happens next, why can't we always predict the future?

One of the most surprising ideas students can meet beyond the normal A-level Mathematics syllabus is chaos.

At first, the word sounds distinctly unmathematical.

We normally use "chaos" to mean disorder, randomness or complete confusion. Mathematics, on the other hand, is supposed to be precise. Give a mathematician an equation and some starting values, and surely the answer should be completely predictable.

But that is not always what happens.

Some mathematical systems obey perfectly definite rules and contain no randomness whatsoever, yet their long-term behaviour can become effectively impossible to predict.

Even more remarkably, two systems starting in almost exactly the same state can eventually behave completely differently.

That is the central idea of chaos theory.

And we can investigate it with an equation simple enough to put into a spreadsheet.

A Surprisingly Simple Equation

Consider the rule:

x(n+1) = rx(n)(1 - x(n))

This is known as the logistic map.

It was originally developed from ideas about population growth, although today it is also one of the classic examples used to introduce chaotic behaviour.

There are only two important quantities.

x(n) represents the current value of the population, expressed as a fraction of some maximum possible population.

r is a parameter controlling how rapidly the population reproduces.

The equation then tells us the next value:

x(n+1).

Suppose:

x = 0.4

and:

r = 2.5

Then the next value is:

x(next) = 2.50.4(1 - 0.4)

x(next) = 2.50.40.6

x(next) = 0.6

We then feed 0.6 back into exactly the same equation to obtain the next value.

And we keep going.

This process is called iteration.

Nothing random has been introduced. Every answer is determined completely by the answer before it.

Yet something very strange is about to happen.

Experiment 1 — Build Chaos in a Spreadsheet

This makes an excellent computer-based mathematical investigation because students do not need specialised software.

Excel, Google Sheets or almost any spreadsheet will do.

Create three columns:

Iteration | x value | Second x value

In the first x column begin with:

0.5000

In the second begin with:

0.5001

The two starting conditions therefore differ by only:

0.0001

Now choose:

r = 3.9

For each new row calculate:

x(next) = 3.9x(1 - x)

Copy the formula down perhaps 50 or 100 rows.

At first, the two columns appear almost identical.

That is exactly what we would expect.

Their starting values were almost identical.

But keep going.

Something remarkable happens.

Around iteration 20, the difference is becoming noticeable.

By about iteration 25, the two values can already differ by several hundredths.

By iteration 30, one calculation can give approximately:

0.973

while the other gives approximately:

0.284

We started with:

0.5000

and:

0.5001

A difference of just one ten-thousandth.

Thirty iterations later the two systems can be in completely different places.

Nothing random was added.

Both calculations followed precisely the same mathematical rule.

Only their initial conditions were slightly different.

That is one of the defining characteristics of chaos:

sensitive dependence on initial conditions.

Draw the Two Curves

The effect becomes even clearer if the spreadsheet results are plotted.

Put iteration number on the horizontal axis and x on the vertical axis.

Plot both calculations on the same graph.

For the first few iterations, the lines may appear to lie almost exactly on top of one another.

Then they begin to separate.

Soon afterwards they appear to have almost no relationship at all.

This is far more powerful than simply telling students that chaotic systems are sensitive to starting conditions.

They can actually watch predictability disappear.

But the Equation Hasn't Changed

This raises a fascinating question.

Why should prediction become difficult?

At iteration 30, we are still doing exactly the same calculation:

x(n+1) = 3.9x(n)(1 - x(n))

There is no dice throw.

There is no random-number generator.

There is no hidden choice being made by the computer.

If we know x(n) exactly, we can calculate x(n+1) exactly.

The system is therefore deterministic.

Yet long-term prediction becomes extraordinarily difficult.

This leads to one of the most important distinctions students can meet:

Deterministic does not necessarily mean predictable.

Those two words are not synonyms.

Now Change r

There is another wonderful feature of the logistic map.

Instead of changing the starting value, keep the starting value the same and slowly change r.

The character of the entire system changes.

For relatively small values of r, the population settles down.

Increase r and oscillation begins.

Increase it again and the oscillation becomes more complicated.

Eventually the behaviour becomes chaotic.

A rough journey looks like this.

At r = 2.5 — Stability

Begin with almost any sensible starting value between 0 and 1.

After several iterations, the values settle towards a fixed value.

The population reaches an equilibrium.

The next generation is approximately the same size as the previous generation.

Everything appears reassuringly predictable.

At r = 3.2 — Oscillation

Increase r and the fixed equilibrium loses its stability.

Instead of settling at one value, the system begins to alternate between two values.

High.

Low.

High.

Low.

The population has entered a period-2 cycle.

Increase r Again — Period Doubling

Increase r further and something even stranger happens.

Instead of cycling through two values, the system begins cycling through four.

Then eight.

Then sixteen.

The cycles double again and again.

This is called period doubling.

The intervals between these changes become progressively smaller.

Eventually, at around:

r = 3.57

the behaviour becomes predominantly chaotic.

A very simple nonlinear equation has travelled from stability to oscillation to apparently irregular behaviour.

The Bifurcation Diagram

One of the most beautiful pictures in modern mathematics emerges if we repeat this experiment for thousands of different values of r.

For each r value, we discard the early iterations and plot the values that remain.

The resulting picture is called a bifurcation diagram.

At first there is a single branch.

Then it divides into two.

Those two divide into four.

Four divide into eight.

The branches become increasingly dense until a complicated region of chaotic behaviour appears.

But even inside the chaos there are unexpected windows of order.

Stable cycles suddenly reappear.

Then they split again and return to chaos.

The result looks almost like a mathematical tree growing out of a single line.

It is a spectacular reminder that very complicated structures can emerge from extremely simple rules.

Where Does the Complexity Come From?

The logistic equation contains a crucial feature:

x*(1 - x)

This makes the equation nonlinear.

Nonlinear systems behave differently from the straight-line relationships students meet early in mathematics.

For example:

y = 3x + 2

is linear.

Double x and the effect on y is straightforward.

But in the logistic map, x is multiplied by another expression containing x.

The system also feeds its own output back into itself.

Today's result becomes tomorrow's input.

A small difference can therefore alter the next result.

That altered result then creates another difference.

That difference affects the following calculation.

And the process continues.

In some chaotic systems, uncertainty effectively grows with each iteration.

Eventually the uncertainty can dominate the prediction.

The Butterfly Effect

Chaos theory is frequently associated with the phrase:

the butterfly effect.

It is sometimes exaggerated into the claim that a butterfly flapping its wings directly causes a hurricane.

That is not really the point.

The important idea is that in a sufficiently sensitive system, an extremely small difference in the initial conditions can eventually contribute to a very large difference in the final state.

Imagine trying to measure the atmosphere.

We might measure:

  • temperature;

  • pressure;

  • humidity;

  • wind speed;

  • wind direction.

But we can never measure every quantity at every location with infinite precision.

Suppose the true temperature somewhere is:

17.263847... degrees C

but our instrument records:

17.26 degrees C.

For many calculations that difference is irrelevant.

In a chaotic dynamical system, however, tiny differences can grow.

This places a fundamental limit on how far ahead some systems can realistically be predicted.

Why Weather Forecasts Become More Difficult

Weather is an excellent real-world connection, although the real atmosphere is vastly more complicated than the logistic map.

Modern forecasting begins with observations of the atmosphere and uses mathematical models to calculate how conditions are likely to evolve.

But we never know the exact state of the entire atmosphere.

There are always uncertainties.

Because atmospheric dynamics can display chaotic behaviour, forecasts that begin with very slightly different initial conditions can eventually diverge.

This is one reason meteorologists use ensemble forecasts.

Instead of running only one simulation, computers run many forecasts beginning from slightly different plausible starting conditions.

If nearly all the simulations produce a similar outcome, confidence may be relatively high.

If the simulations spread widely, uncertainty is greater.

Chaos therefore does not mean:

"weather forecasting is impossible."

It means there are limits to how precisely some aspects of weather can be predicted far into the future.

Populations — Where the Logistic Map Began

The logistic map is particularly interesting because it can be interpreted as a simplified population model.

Imagine a species in an environment with limited resources.

If the population is small, there is plenty of food and space.

The population can grow rapidly.

But as the population increases, competition becomes stronger.

Growth is restricted.

This is represented by the factor:

(1 - x)

When x is small, this factor is large.

When x approaches 1, it becomes small.

The equation therefore contains both reproduction and limitation.

Of course, real ecosystems involve predators, disease, migration, climate, age structure and countless other factors.

The logistic map is not a realistic complete model of an ecosystem.

Its importance is that even a drastically simplified deterministic population model can produce extraordinarily complicated behaviour.

A Double Pendulum — Chaos You Can See

There is also a wonderful physical demonstration of chaos.

Take an ordinary pendulum and attach a second pendulum to its end.

The result is a double pendulum.

Release it from one position and record the motion.

Then reset it as accurately as possible and release it again from almost the same position.

Initially the two motions may look similar.

Soon they can become completely different.

The movement can become spectacular: swinging, rotating and reversing direction in ways that are extremely difficult to anticipate.

Again, the motion is governed by physical laws.

The pendulum is not deciding randomly where to move.

But the system is highly sensitive to its starting conditions.

It would make an excellent companion practical to the spreadsheet experiment.

One is mathematical.

One is physical.

Both demonstrate the same underlying idea.

Turbulence and Fluid Motion

Another connection appears in moving fluids.

Water flowing slowly through a pipe can display relatively orderly behaviour.

Increase the speed and the motion may become turbulent.

Eddies form within eddies.

Structures appear, change and disappear.

Air flowing around buildings, aircraft wings, sails and vehicles can show similarly complicated behaviour.

Turbulence is a far more complicated subject than the logistic map, and the two should not simply be treated as the same thing.

But both belong to the wider mathematical world of nonlinear dynamical systems, where simple expectations about cause and effect can fail.

Does Chaos Mean Everything Is Random?

No.

This is perhaps the most important misconception to challenge.

Random behaviour involves genuine unpredictability in the process or a probabilistic description of outcomes.

A chaotic deterministic system follows fixed rules.

If we somehow knew the starting conditions with infinite accuracy and could perform the calculations with infinite precision, the future state would be determined.

The problem is that real measurements do not contain infinite information.

Neither do computers.

Numbers have to be stored to finite precision.

And if tiny errors grow rapidly, eventually those tiny uncertainties matter.

Chaos therefore creates a fascinating middle ground.

The system is not random.

But its long-term behaviour can become practically unpredictable.

A Very Good Student Challenge

Once students have created the spreadsheet, I would encourage them not simply to accept the standard values for r.

Explore.

Try:

r = 2.0

r = 2.8

r = 3.1

r = 3.4

r = 3.5

r = 3.55

r = 3.6

r = 3.8

r = 3.9

r = 4.0

For each value, ask:

  • Does the system approach a single value?

  • Does it oscillate?

  • How many values appear in the cycle?

  • Does it appear chaotic?

  • How much does changing the initial value matter?

  • How many iterations are required before two nearby starting conditions noticeably separate?

Then try changing r in much smaller steps.

Students will begin discovering the bifurcations for themselves.

That turns the exercise from a demonstration into a genuine mathematical investigation.

Go Further — Can You Find Order Inside Chaos?

There is an additional surprise for students who want to explore further.

Chaotic behaviour does not simply begin and then continue uniformly.

Within chaotic regions there are windows of periodic behaviour.

For certain values of r, an apparently chaotic system suddenly settles into a repeating cycle again.

Then that cycle undergoes its own sequence of period doubling.

So even inside apparent disorder, mathematical structure remains.

That is one of the reasons the bifurcation diagram is so fascinating.

It is not simply a picture of increasing messiness.

It contains extraordinary organisation.

An Even Deeper Result — The Feigenbaum Constant

There is another beautiful piece of mathematics hiding here.

As the period doublings occur, the spacing between successive bifurcations decreases in a systematic way.

The ratio between these intervals approaches approximately:

4.669...

This number is known as the Feigenbaum constant.

The remarkable thing is that the same constant appears in many completely different nonlinear systems undergoing period doubling.

So a pattern first explored through a simple population equation reveals something much deeper.

Different mathematical and physical systems can approach chaos in remarkably similar ways.

That idea — that apparently unrelated systems can share universal mathematical behaviour — is one of the great attractions of mathematics beyond the examination syllabus.

Why I Like This as an A-Level Investigation

There is very little difficult calculation here.

An A-level student can understand the equation.

A spreadsheet can perform the repeated arithmetic.

Yet the ideas lead rapidly into university-level mathematics, physics, meteorology and computational modelling.

That makes it exactly the sort of topic I enjoy exploring beyond the syllabus.

Students sometimes assume that more advanced mathematics must mean longer equations, more complicated algebra and increasingly difficult manipulation.

Chaos theory demonstrates something much more interesting.

A simple equation does not necessarily produce simple behaviour.

In fact, one of the deepest questions becomes:

How much can we know about the future even when we know the rules?

Mathematics, Measurement and Prediction

There is also a useful scientific lesson here.

Whenever we make a prediction, three things matter:

  1. the mathematical model;

  2. the starting information;

  3. the sensitivity of the system to errors in that information.

If a system is not particularly sensitive, small measurement errors may remain small.

Prediction can remain useful for a long time.

If the system is chaotic, an apparently insignificant uncertainty can eventually grow until two possible futures bear little resemblance to one another.

That distinction matters in fields ranging from weather forecasting to orbital dynamics, fluid mechanics and biological populations.

The Bigger Lesson

Students are often introduced to mathematics as a subject in which every problem has a definite answer.

And in one sense, the logistic map reinforces that idea.

At every stage we can calculate exactly what the next number should be.

But it also reveals something deeper.

Knowing the rule is not always enough to make useful long-term predictions.

A system can be:

  • deterministic but unpredictable;

  • simple in its rule but complicated in its behaviour;

  • orderly in one region and chaotic in another;

  • extraordinarily sensitive to differences too small to notice initially.

That is a much richer view of mathematics.

Conclusion — Can We Predict the Future?

Start with:

x = 0.5000

and:

x = 0.5001.

The difference appears insignificant.

Apply exactly the same deterministic mathematical rule repeatedly.

At first the answers stay close.

Then they separate.

Eventually they can describe entirely different states of the system.

No randomness was introduced.

No rules were changed.

Nothing went wrong with the mathematics.

The unpredictability emerged from the mathematics itself.

That is the extraordinary lesson of chaos theory.

We often imagine that if we know the laws governing a system, we should be able to predict its future.

Chaos tells us something more subtle:

Knowing what happens next does not necessarily mean we can know what happens much later.

And all of that can begin with one surprisingly simple equation:

x(n+1) = rx(n)(1 - x(n))

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