16 September 2026

Can a Triangle Roll Like a Wheel? — Curves of Constant Width

Surely a wheel has to be circular? Mathematics says otherwise.

Ask almost any student to draw something that could roll smoothly and they will draw a circle.

That seems perfectly reasonable.

After all, the defining feature of a circle is that every point on its circumference is the same distance from its centre. Put an axle through the centre and, as the wheel turns, the axle remains at exactly the same height above the ground.

So perhaps we could turn that observation around:

If a shape rolls while maintaining a constant height, must it be a circle?

Surprisingly, no.

There is an entire family of shapes known as curves of constant width, and one of the simplest looks suspiciously like a triangle.

It is called a Reuleaux triangle.

And it provides one of those wonderful mathematical demonstrations where you can put an object into a student's hands and watch their intuition collide with geometry.

First: What Does "Width" Actually Mean?

We need to be careful with the word width.

With a rectangle, width seems obvious.

With an irregular curved object, it is less obvious.

Imagine placing the object between two perfectly parallel rulers and bringing the rulers together until they just touch it.

The distance between the rulers is the width of the object in that particular direction.

Now rotate the object slightly.

For most shapes, the distance between the rulers changes.

Take an ordinary equilateral triangle. Depending upon how you orient it, the distance from one supporting line to the other changes considerably.

Take a circle, however, and nothing changes.

Turn it through 1 degree, 20 degrees or 173 degrees and the distance remains its diameter.

That is why we call a circle a shape of constant width.

The surprising part is this:

The circle is not the only one.

Meet the Reuleaux Triangle

A Reuleaux triangle looks rather like an equilateral triangle whose sides have bulged outwards.

But those curved sides have been constructed very carefully.

Start with an equilateral triangle of side length, say, 100 mm.

Now:

Put the point of a compass on the first vertex.
Set its radius to 100 mm.
Draw an arc joining the other two vertices.
Move the compass to the second vertex and repeat.
Repeat again from the third vertex.

The resulting three arcs form a Reuleaux triangle.

Every curved side therefore has a radius equal to the side length of the original equilateral triangle.

If the original triangle had side length 100 mm, the resulting Reuleaux triangle has constant width:

Width = 100 mm

Whichever way you rotate it.

That deserves testing rather than simply believing.

A Superb Practical Investigation

This is exactly the sort of mathematics I like students to investigate physically.

It could be made very simply from stiff card or foamboard, but I would be tempted to make several more accurate versions using a laser cutter or 3D printer.

For example, produce:

one circle;
one ordinary equilateral triangle;
one square;
one Reuleaux triangle.

Make them all approximately the same maximum width.

Then construct two parallel rails.

The bottom one stays fixed.

The upper one should be able to rest on the shape.

Now roll each shape between them.

The square clearly fails.

The ordinary triangle certainly fails.

The circle works exactly as expected.

Then try the Reuleaux triangle.

The upper rail remains the same distance above the lower one.

That is the moment when the mathematics becomes memorable.

It is not circular, yet its width remains constant as it rotates.

Measure It Rather Than Just Watching It

A GCSE investigation could stop at the visual demonstration.

But it becomes much more interesting if students collect measurements.

Make a Reuleaux triangle with a nominal width of 100 mm.

Place it between two parallel surfaces and measure their separation using:

a ruler;
vernier callipers;
a digital calliper;
or perhaps a displacement sensor.

Rotate the shape through successive angles.

For example:

0 degrees
10 degrees
20 degrees
30 degrees
40 degrees
50 degrees
60 degrees

Record the measured width at each position.

Students could compare this with an ordinary equilateral triangle of similar dimensions.

Immediately we have moved beyond merely looking at an unusual shape.

We now have an experiment involving:

measurement;
precision;
geometrical reasoning;
experimental uncertainty;
graphical representation;
and mathematical proof.

That crossover between practical investigation and mathematics is extremely powerful.

But Can It Really Replace a Wheel?

This is where the story becomes even more interesting.

I deliberately began by suggesting that a Reuleaux triangle could roll like a wheel.

That statement needs qualification.

Suppose we put an axle through the centre of a circular wheel.

As the wheel rotates, the centre stays exactly the same distance from the ground.

That is why your bicycle does not repeatedly rise and fall as its wheels turn.

Now find the centre of a Reuleaux triangle and roll it along a flat surface.

The centre does not remain at a constant height.

It moves up and down.

So if we built a bicycle with Reuleaux-triangle wheels and conventional central axles, it would give an extremely uncomfortable ride.

The constant-width property tells us something different.

It tells us that the distance between two parallel tangent lines remains constant.

That is not the same as saying that a particular point inside the shape remains a fixed distance above the ground.

This is an excellent mathematical lesson in itself:

Two statements that sound almost identical can describe very different geometrical properties.

Try the "Rolling Platform" Demonstration

There is another entertaining experiment.

Cut several identical constant-width shapes and place them between two parallel boards.

The lower board is fixed.

Rest another board across the top.

Because each roller has constant width, the separation between the boards can remain constant as the shapes rotate.

Students expecting the upper board to bounce dramatically are often surprised.

Then ask the important question:

Where are the centres of the rollers going?

Now the apparent paradox becomes much more interesting.

The centres can move up and down even though the supporting surfaces remain a constant distance apart.

That distinction is much easier to appreciate when you can physically see it.

Why Does the Reuleaux Triangle Have Constant Width?

At GCSE level, I would initially investigate this visually.

At A level, I would want students to think about why it works.

Take one vertex of the Reuleaux triangle.

The curved side opposite that vertex was drawn using that vertex as the centre of a circle.

The distance from that vertex to every point on the opposite arc is therefore exactly the same.

It is equal to the side length of the original equilateral triangle.

As the shape rotates, the pair of points touching two parallel supporting lines changes.

But the geometry of the arcs ensures that the perpendicular distance between those supporting lines remains constant.

The constant width has not appeared by accident.

It is built into the construction.

And It Isn't the Only Shape

Once students accept that a Reuleaux triangle exists, the next question almost asks itself:

Are there more?

Yes.

There are infinitely many curves of constant width.

Some are symmetrical.

Some are surprisingly irregular looking.

Reuleaux polygons can be constructed using an odd number of suitably arranged circular arcs.

And curves of constant width do not even have to be made from circular arcs.

The circle is therefore only one member of a much larger mathematical family.

This is where a very familiar GCSE concept — measuring width — suddenly opens into a much deeper area of geometry.

You May Already Have a Constant-Width Shape in Your Pocket

For students in Britain, there is an especially good real-world connection.

Take out a 20p or 50p coin.

Neither is circular.

Both use what is known as an equilateral curve heptagon: a seven-sided curved form designed with a constant rolling diameter.

That allows the coin to pass consistently through mechanisms such as vending machines while still being easily distinguishable from ordinary round coins.

The Royal Mint specifically identifies the constant-width geometry as the reason these coins can roll successfully through such mechanisms.

That gives students an immediate practical investigation.

Take a 50p coin and place it between two parallel rulers.

Carefully rotate it.

At first sight, its seven-sided shape makes you expect the distance between the rulers to change.

It doesn't in the way intuition suggests.

A piece of advanced geometry has been sitting in millions of pockets for decades.

Can You Drill a Square Hole?

Now comes my favourite question.

Can you drill a square hole using something that rotates?

A conventional circular drill bit obviously produces a circular hole.

That seems almost inevitable.

But cutters based on the geometry of the Reuleaux triangle can be used to produce holes that are remarkably close to square.

There is an important qualification.

The cutter cannot simply rotate about a fixed central axis.

Its centre has to move around a small path while it rotates, usually controlled by a guide or special mechanism.

As the cutter rotates and moves, its corners sweep out almost all of a square.

The resulting hole has slightly rounded corners rather than being a mathematically perfect square, but the result is astonishingly close.

So the answer is:

Yes — a rotating tool can make an approximately square hole, provided the rotation is combined with the correct motion.

Once again, the interesting mathematics is hidden in what we mean by rotate.

A Challenge for Students: Design the Experiment

Rather than immediately giving students the explanation, I would set them several challenges.

Challenge 1 — Construct one

Using only a ruler and compass, construct a Reuleaux triangle starting with an equilateral triangle.

Explain why all three arcs have the same radius.

Challenge 2 — Test constant width

Devise an experiment to determine whether the width really remains constant.

What equipment would give the best measurement?

How would you estimate uncertainty?

Challenge 3 — Compare shapes

Compare:

a circle;
an equilateral triangle;
a Reuleaux triangle;
a 20p or 50p coin.

Which have constant width?

Challenge 4 — Investigate the centre

Mark the approximate centre of a Reuleaux triangle.

Roll it along a horizontal line.

Trace the movement of the centre.

Does it travel horizontally?

Challenge 5 — Build a rolling platform

Make three or four identical Reuleaux triangles.

Place a board on top and investigate whether the board remains at constant height as they roll.

Challenge 6 — Square-hole geometry

Research or model the path needed for a Reuleaux-style cutter to reach the four corners of an approximately square hole.

At this point we are well beyond routine textbook geometry.

An A-Level Extension: A Very Strange Perimeter Result

There is an even more remarkable result for students wanting to go further.

Suppose a convex curve has constant width w.

A theorem called Barbier's theorem tells us that its perimeter is:

P = pi x w

That is exactly the same perimeter as a circle of diameter w.

So if we make:

a circle of constant width 100 mm;
a Reuleaux triangle of constant width 100 mm;
another convex constant-width curve of width 100 mm;

they all have the same perimeter:

P = pi x 100

P is approximately 314.16 mm.

Their areas, however, do not have to be the same.

That is a wonderful result because it challenges another piece of intuition.

The shapes can look dramatically different and enclose different areas, yet their boundaries have exactly the same total length.

From GCSE Geometry to Real Mathematical Thinking

None of this is necessary for answering a standard GCSE geometry question.

That is precisely why I think it is valuable.

Students can sometimes come away from school mathematics with the impression that geometry consists of remembering a collection of facts:

Angles on a straight line add to 180 degrees.

The area of a triangle is half base times height.

The circumference of a circle is pi times the diameter.

Pythagoras applies to right-angled triangles.

All useful.

But mathematics becomes much more interesting when we start asking:

Must that always be true?

Is that the only possible shape?

What happens if I change one of the assumptions?

The Reuleaux triangle begins with an apparently childish question — can a triangle roll? — and leads very quickly into genuine mathematical investigation.

Why I Like This Experiment

One of the reasons I enjoy demonstrations like this is that the apparatus can be extremely simple while the mathematics behind it is surprisingly deep.

A student can hold a Reuleaux triangle.

They can turn it.

They can measure it.

They can make one from card.

They can compare it with a coin from their pocket.

And yet the same investigation can eventually lead towards:

mathematical proof;
loci;
tangents;
circular arcs;
optimisation;
mechanical engineering;
machining;
and the geometry of constant-width curves.

It works just as well as an intriguing GCSE enrichment activity as it does as the starting point for a much more sophisticated A-level investigation.

That is exactly the sort of mathematics I want students to encounter beyond the examination syllabus.

Conclusion — Perhaps a Wheel Doesn't Have to Be Round

So, can a triangle roll like a wheel?

In one fascinating sense, yes.

A Reuleaux triangle has constant width, so it can rotate between parallel supporting surfaces while maintaining the same separation between them.

But in another sense, no.

Its centre does not remain at a constant height, so putting a conventional axle through it does not produce the wonderfully smooth motion of a circular wheel.

And that distinction is what makes the problem so good.

It begins with something every child thinks they understand — a wheel — and reveals that even the apparently obvious can hide much deeper mathematics.

Then, just when students think the Reuleaux triangle is merely a mathematical curiosity, we can take out a 20p or 50p coin and discover that constant-width geometry is already being used in everyday engineering.

And finally we can ask the question that usually gets the best reaction of all:

Could you use one to drill a square hole?

Suddenly geometry is no longer a page of angle calculations.

It has become something to construct, measure, test, question and explore.

Surely a wheel has to be circular?

Mathematics says otherwise.

15 September 2026

Young's Modulus of a Wire — Measuring Something You Cannot See

One of the difficulties with teaching elasticity is that the most important changes are often almost invisible.

Take a long metal wire, hang a mass from it and the wire stretches.

In principle, that sounds like an excellent experiment.

In practice, a student may look at the wire and think:

"Has it actually stretched at all?"

The extension may only be a fraction of a millimetre.

That is why the traditional A-level Young's modulus experiment is scientifically important, but I like to accompany it with a second, much more visual experiment.

Alongside the metal wire, I use a thin strip cut from a plastic bag and hold it securely between two specially made clips.

Now the deformation can be seen.

The strip stretches.

It becomes noticeably longer.

Eventually it passes its elastic limit.

Remove the force and it does not return completely to its original dimensions.

Continue stretching and the material begins to change dramatically before finally breaking.

Suddenly ideas such as elastic deformation, plastic deformation, elastic limit, strain and failure are not simply words in a textbook.

Students have watched them happen.

And once they have seen that, the much more precise wire experiment begins to make considerably more sense.

What Is Young's Modulus?

Young's modulus describes the stiffness of a material when it is stretched or compressed.

It compares the stress applied to a material with the strain that results.

Young's modulus is defined as:

E = stress / strain

Stress is:

stress = force / cross-sectional area

or:

stress = F / A

Strain is:

strain = extension / original length

or:

strain = ΔL / L

Therefore:

E = (F / A) / (ΔL / L)

which can also be written as:

E = FL / AΔL

where:

E = Young's modulus
F = applied force
L = original length of the wire
A = cross-sectional area of the wire
ΔL = extension of the wire

Stress is measured in pascals, Pa.

Strain has no units because it is a ratio of two lengths.

Young's modulus therefore also has units of pascals.

For metals, the values are usually very large, so they are often quoted in gigapascals, GPa.

Young's Modulus Is Really a Measure of Stiffness

Students sometimes describe Young's modulus as the "strength" of a material.

That is not quite correct.

A material with a high Young's modulus is stiff.

It does not change length very much for a given stress.

That does not necessarily mean that it is the material most difficult to break.

Strength, stiffness, toughness, hardness and brittleness describe different properties.

This distinction becomes much easier to appreciate when students can physically handle different materials.

A thin plastic strip may stretch enormously before breaking.

A metal wire may extend by only a tiny amount.

Yet the fact that the plastic stretches further does not simply mean that it is "stronger".

The two materials behave differently.

The Traditional A-Level Wire Experiment

The formal experiment requires much more careful measurement.

A long piece of metal wire is fixed securely.

Its original length is measured.

Known loads are then applied and the resulting extensions recorded.

There are several variations of the apparatus, but the underlying principle is the same.

We need to determine:

the original length of the wire;
the diameter of the wire;
the force applied;
the resulting extension.

From these measurements we can calculate Young's modulus.

Step 1 — Measuring the Original Length

Suppose the original length of the wire is:

L = 2.00 m

Using a relatively long wire is useful.

If the wire were only 10 cm long, its extension might be extremely small.

A 2 m wire gives twenty times the extension of an otherwise identical 10 cm wire under the same stress.

This makes the measurement easier.

It is a good example of experimental design.

Scientists do not simply ask:

"Can this quantity be measured?"

They also ask:

"How can I arrange the experiment so that the quantity is easier to measure accurately?"

Step 2 — Measuring the Diameter

The diameter of the wire must be measured carefully, normally using a micrometer screw gauge.

This measurement is particularly important because the diameter is used to calculate the cross-sectional area.

For a circular wire:

A = πd^2 / 4

Suppose the diameter is:

d = 0.50 mm

First convert this to metres:

d = 0.00050 m

Then:

A = π(0.00050)^2 / 4

The cross-sectional area is therefore very small.

And that matters enormously.

Why Measuring the Diameter Carefully Matters

There is an important experimental lesson hidden inside this calculation.

Because:

A is proportional to d^2

a small error in the measurement of diameter can produce a larger percentage error in the calculated area.

That is why I encourage students to measure the diameter at several positions along the wire and, ideally, in different orientations.

The wire may not be perfectly uniform.

We then calculate a mean diameter.

This is not unnecessary repetition.

It is part of good experimental science.

Step 3 — Adding Known Loads

Masses are gradually added to the wire.

The force produced by a hanging mass is:

F = mg

For example, a mass of 1.00 kg produces approximately:

F = 1.00 x 9.81

F = 9.81 N

At A level, it is useful to make students distinguish carefully between mass and force.

The balance or slotted masses may be labelled in kilograms or grams.

Young's modulus requires the force in newtons.

Step 4 — Measuring the Extension

This is where the experiment becomes demanding.

The extension of a metal wire may be very small.

Depending upon the apparatus available, it might be measured using a vernier scale, travelling microscope, pointer arrangement or another sensitive displacement measurement system.

What matters is that students appreciate the difference between:

the total length of the wire

and

the extension of the wire.

If a wire changes from:

2.0000 m

to:

2.0008 m

then the extension is:

ΔL = 0.0008 m

not 2.0008 m.

That sounds obvious when written down.

Under exam pressure, it is an easy mistake to make.

A Sample Young's Modulus Calculation

Suppose we have:

L = 2.00 m

d = 0.50 mm

F = 20.0 N

ΔL = 0.80 mm

Convert everything into SI units:

d = 0.00050 m

ΔL = 0.00080 m

Calculate the area:

A = πd^2 / 4

A = π(0.00050)^2 / 4

A approximately = 1.96 x 10^-7 m^2

Now:

E = FL / AΔL

Therefore:

E = (20.0 x 2.00) / ((1.96 x 10^-7) x 0.00080)

This produces a value of the order expected for a stiff engineering material.

The precise numerical answer is less important educationally than understanding how each measured quantity enters the calculation.

An Even Better Approach — Plot a Graph

A single measurement can be used to calculate Young's modulus.

A series of measurements is usually better.

Students can add a sequence of loads and record the extension produced by each one.

They might plot:

force against extension

or:

stress against strain.

Within the elastic region, the graph should be approximately linear.

For a stress-strain graph:

Young's modulus = gradient

provided the graph is in the region where stress is proportional to strain.

That makes Young's modulus much more than a number inserted into a formula.

It becomes a property visible in the shape of experimental data.

A steeper stress-strain graph represents a stiffer material.

But What Does "Elastic" Actually Look Like?

This is where I find the plastic-strip experiment particularly useful.

The wire experiment is excellent for measurement.

It is less successful as a visual demonstration.

The student may see a pointer move by a fraction of a millimetre, but the material itself appears almost unchanged.

So I take a strip cut from an ordinary thin plastic bag.

The strip is placed between two specially made clips so that the load is spread across the material rather than concentrated at a single point.

Then I begin to stretch it.

The result is completely different from the metal wire.

Stage 1 — Elastic Deformation

Initially the plastic strip stretches.

If the force is removed soon enough, much of the deformation disappears.

The material attempts to return towards its original shape.

This introduces the idea of elastic deformation.

An elastic material returns to its original dimensions once the deforming force is removed.

Students can see the difference between:

a material being deformed

and

a material being permanently changed.

Stage 2 — Passing the Elastic Limit

Stretch the plastic further and something changes.

The material no longer returns completely to its original length.

It has undergone permanent deformation.

We have moved into the region of plastic deformation.

This is a confusing piece of terminology because we are using a plastic material to demonstrate plastic deformation.

But the word "plastic" in "plastic deformation" does not mean the material must be plastic.

A metal can undergo plastic deformation as well.

Plastic deformation simply means that the change in shape remains after the force has been removed.

This is one of the reasons I like the demonstration.

Students can see a before-and-after difference with their own eyes.

Stage 3 — Necking and Localised Deformation

Depending upon the type of plastic used, parts of the strip may begin to become thinner.

The deformation is no longer perfectly uniform.

One section may stretch considerably more than another.

The material may appear to whiten or change texture.

This raises another important point.

Simple Young's modulus calculations normally assume that the material is behaving uniformly.

Once large-scale permanent deformation begins, those assumptions become increasingly inappropriate.

Stage 4 — Failure

Eventually the strip breaks.

This tends to be the moment students remember.

There is a temptation to think of breaking as a completely separate event.

In reality, it is part of the material's entire response to stress.

We can consider a sequence:

elastic deformation → permanent deformation → major structural change → failure

The precise sequence differs between materials.

But the important idea is that materials do not simply fall into two categories:

not broken

and

broken.

A great deal can happen in between.

Why I Would Not Use the Plastic Strip to Calculate a Precise Young's Modulus

The plastic demonstration is extremely useful, but it is important not to pretend that it is doing something it is not.

A thin strip cut from a plastic bag is not necessarily a convenient material for obtaining a highly accurate Young's modulus.

Its thickness may be difficult to measure accurately.

Its behaviour may depend upon the direction in which it was manufactured.

It may show time-dependent deformation.

Its width and thickness can change significantly as it stretches.

Its behaviour may become non-linear quite quickly.

So I use the two experiments for different purposes.

The metal wire provides the measurement.

The plastic strip provides the understanding.

Together they are much more powerful than either experiment on its own.

The Difference Between Seeing a Number and Seeing the Physics

This illustrates a wider problem in science teaching.

A practical can technically satisfy every requirement of a specification and still fail to make the underlying physics memorable.

Students might:

measure the diameter;
add the masses;
read the scale;
fill in the table;
plot the graph;
calculate Young's modulus.

They may obtain an excellent result.

But if we ask them two weeks later:

"What was actually happening to the material?"

the answer may be much less certain.

The visual demonstration helps provide the missing mental picture.

What Is Happening Inside the Material?

At the microscopic level, stretching a solid changes the spacing between its atoms, molecules or structural chains.

In a metal operating within its elastic region, the atoms move slightly from their equilibrium positions.

Remove the force and the interatomic forces restore the original structure.

If the material is pushed beyond its elastic behaviour, structural changes can occur that cannot simply reverse when the force is removed.

In polymers, the behaviour can be even more dramatic because long molecular chains may uncoil, rotate, slide and realign.

This is why the visible behaviour of the plastic strip can be so striking.

Hooke's Law and Young's Modulus Are Related — But Not Identical

Students sometimes confuse Hooke's law with Young's modulus.

Hooke's law is often written as:

F = kx

where:

F = force
k = spring constant
x = extension

This describes a particular spring or object.

Young's modulus describes a material.

A thick steel wire and a thin steel wire do not have the same spring constant.

The thicker wire is much harder to stretch.

But if they are made from the same steel, they should have approximately the same Young's modulus.

That is an extremely important distinction.

The spring constant depends upon the dimensions of the object.

Young's modulus is intended to characterise the material itself.

Why Engineers Care About Young's Modulus

Young's modulus is not an abstract examination quantity.

Engineers constantly need to know how much structures will deform when loaded.

Consider:

bridges;
aircraft wings;
cranes;
bicycle frames;
sailing masts;
suspension systems;
buildings;
cables;
medical implants.

It is not enough to know that something will not immediately break.

It may also need to remain sufficiently stiff.

A bridge that bends dramatically every time a lorry crosses it would not be acceptable even if it technically remained intact.

An aircraft wing must flex, but its deformation must remain controlled.

A sailing mast must bend sufficiently to respond to loads and sail forces, while still maintaining the required structural behaviour.

Young's modulus helps engineers predict that deformation.

A Useful Classroom Comparison

I sometimes ask students to imagine two rods of exactly the same dimensions.

One is made from rubber.

The other is made from steel.

Apply the same force.

Which extends more?

Almost everyone immediately says the rubber.

Then ask:

Which has the larger Young's modulus?

Now the answer is steel.

The material that produces the smaller strain for the same stress has the higher Young's modulus.

That simple comparison often makes the meaning of the quantity much clearer.

Common Examination Mistakes

The Young's modulus practical brings together a surprising number of A-level skills.

Typical mistakes include:

1. Forgetting to convert millimetres into metres

0.50 mm is:

0.00050 m

not:

0.050 m.

2. Using diameter instead of cross-sectional area

The formula requires A, not d.

For a circular wire:

A = πd^2 / 4

3. Using mass rather than force

The mass must be converted using:

F = mg

4. Using total length instead of extension

Strain is:

ΔL / L

not:

final length / original length.

5. Giving strain a unit

Strain is dimensionless.

6. Calling Young's modulus "strength"

It is primarily a measure of stiffness.

7. Using data beyond the elastic region

Young's modulus is normally determined from the initial linear region of the stress-strain relationship.

Turning It Into a Better Investigation

There are several ways the basic experiment can be developed.

Students could investigate:

different wire diameters;
different metals;
different original lengths;
loading and unloading;
repeat measurements;
uncertainty in diameter;
whether extension remains proportional to load;
what happens when the elastic region is exceeded.

They can also compare the metal-wire experiment with materials that behave very differently.

This is where demonstrations using plastics, elastic bands or polymer fibres become particularly useful.

Students begin to realise that "stretching something" can produce a remarkably wide range of material behaviours.

Safety Matters

Any experiment involving tension and suspended masses deserves careful thought.

Loads should be added gradually.

The wire and supports must be securely fixed.

The region beneath suspended masses should be kept clear.

Eye protection can be appropriate where there is a possibility of wire or material failure.

When deliberately taking a material towards breaking point, students should not place their faces or hands close to the stretched sample.

The dramatic part of an experiment should never come at the expense of sensible laboratory practice.

Why I Use Both Experiments

I have taught enough physics over the years to know that students often remember the experiment that gave them the strongest mental picture.

The precision experiment gives them the science.

The visual experiment gives them the memory.

With the wire, we can calculate:

E = stress / strain

With the plastic strip, we can ask:

What does deformation actually look like?

When does the material return to its original shape?

When does the change become permanent?

What happens immediately before failure?

And perhaps most importantly:

Is stretching always the same thing?

The answer is very clearly no.

The Bigger Lesson — Measurement and Understanding Need Each Other

Young's modulus is an excellent example of why practical science should be more than simply following instructions.

The formal experiment teaches careful measurement.

Students use a micrometer.

They calculate cross-sectional area.

They convert units.

They calculate force.

They measure tiny extensions.

They plot graphs.

They analyse uncertainty.

All of those skills matter.

But the plastic-strip demonstration adds something the equations cannot provide by themselves.

It allows students to see the material changing.

That combination is what good practical science should achieve.

Measure precisely.

Calculate carefully.

But also look closely at what nature is actually doing.

Because sometimes the difference between remembering a formula until the examination and genuinely understanding the physics is simply this:

one experiment lets you calculate the effect — and another lets you see it happen.

14 September 2026

Mendel's Experiments — But Actually Grow the Generations

Most biology students can draw a Punnett square.

They know that Gregor Mendel worked with pea plants. They have probably learned the words dominant, recessive, homozygous, heterozygous, genotype and phenotype.

Many can confidently predict a 3:1 ratio.

But there is something slightly strange about the way Mendelian genetics is usually taught.

Very few students ever do anything resembling Mendel's actual experiment.

Instead, genetics can become a paper exercise:

Parent A has genotype AA.
Parent B has genotype aa.
What proportion of the offspring will show the dominant characteristic?

The answer is useful, but something important has disappeared.

Mendel did not begin with a Punnett square.

He began with plants.

He grew them. He selected parents. He controlled pollination. He waited for seeds. He planted the next generation. He counted hundreds and sometimes thousands of offspring.

Most importantly, he collected real biological data.

That suggests a wonderful experiment for students who want to explore biology beyond the normal school practical syllabus:

repeat a small version of Mendel's investigation and actually grow the generations.

Genetics Before Genetics Had a Name

Gregor Mendel carried out his famous pea experiments during the nineteenth century, long before anybody knew about DNA, chromosomes or genes in the modern sense.

He selected pea plants because they offered several useful features.

They could be grown relatively easily.

They produced large numbers of offspring.

Their flowers normally self-pollinated, but pollination could also be controlled experimentally.

Most importantly, Mendel identified characteristics that could be separated into clearly recognisable forms.

These included characteristics associated with:

seed shape;
seed colour;
flower colour;
pod shape;
pod colour;
flower position;
plant height.

Mendel began with plants that reliably produced the same characteristic generation after generation.

Today we would describe these as true-breeding lines.

He then crossed contrasting plants and followed what happened through several generations.

This is the part students often hear about.

It is also the part worth actually doing.

The Three Generations That Matter

The terminology initially sounds more complicated than the experiment.

P generation

The parental generation, or P generation, contains the original parents selected for crossing.

Imagine, for simplicity, that we have a characteristic controlled by two alleles.

A = dominant allele
a = recessive allele

Suppose our original parents are:

AA x aa

One parent is homozygous dominant and the other homozygous recessive.

F1 generation

Their offspring form the first filial generation, usually written F1.

Every offspring receives:

A from one parent
a from the other

Therefore all the offspring are:

If A is completely dominant, all the F1 plants show the dominant phenotype.

This result alone is interesting.

The recessive characteristic appears to have disappeared.

But it has not disappeared genetically.

The allele is still there.

F2 generation

Now allow F1 individuals to produce another generation.

The cross becomes:

Aa x Aa

The possible offspring genotypes are:

AA
Aa
Aa
aa

The predicted genotype ratio is therefore:

1 AA : 2 Aa : 1 aa

But because AA and Aa show the same dominant phenotype, the predicted phenotype ratio becomes:

3 dominant : 1 recessive

That familiar classroom ratio suddenly becomes much more interesting when the four possibilities are replaced by 100, 200 or 500 living organisms.

The Practical Challenge — Can We Actually Grow It?

Using traditional garden peas is possible, but it is not necessarily the best choice for a teaching investigation.

Mendel had patience.

School students generally have timetables.

A better approach is to look for fast-growing plants with clearly identifiable inherited characteristics.

Certain varieties of fast-growing Brassica, for example, have been developed specifically for education and can complete their life cycles surprisingly quickly.

Depending upon the variety being used, potential characteristics might include differences in:

pigmentation;
stem characteristics;
leaf characteristics;
hairiness;
colour;
other easily scored phenotypes.

The essential requirement is not that the organism happens to be a pea.

The important thing is that:

the characteristic is genetically determined;
the alternative phenotypes can be distinguished reliably;
the inheritance pattern is known;
generation time is reasonably short;
enough offspring can be produced to make meaningful comparisons.

That creates a genuine experimental genetics project rather than simply a demonstration.

Start with the Parental Generation

The first stage is careful observation.

Students should photograph and describe the parental plants.

For each parent record:

plant identification number;
phenotype;
assumed genotype if known;
date planted;
date flowering began;
height;
relevant physical characteristics.

Labelling is extremely important.

A surprisingly useful lesson from multiday biology experiments is that memory is a terrible laboratory notebook.

Plant P1 may seem unmistakable today.

Three weeks later, surrounded by twenty similar plants, it may not be quite so obvious.

Every plant should therefore have an identification code from the beginning.

Controlling Pollination

This is where the experiment starts feeling much more like real biology.

Rather than simply allowing random pollination, selected parents can be crossed.

The exact method depends upon the species being used, but students may transfer pollen between chosen flowers using a small brush or another suitable technique.

Flowers can then be labelled so that the resulting seeds can be traced to a particular cross.

With some species it may also be necessary to prevent unwanted pollination.

The objective is simple:

Know who the parents were.

That is fundamental to any breeding experiment.

A Punnett square assumes we know the parental genotypes.

The practical investigation shows how much work may be required before we are justified in making that assumption.

Grow the F1 Generation

Seeds produced by the selected cross can then be planted.

This produces the F1 generation.

Now comes the first prediction.

If the parental generation consisted of true-breeding contrasting forms under a simple dominant-recessive inheritance model, students might predict that all F1 offspring will show the dominant phenotype.

But instead of simply writing:

100% dominant

they can test it.

Suppose 36 F1 seedlings germinate.

Students might find:

Dominant phenotype = 36
Recessive phenotype = 0

That is certainly consistent with the prediction.

But suppose they find:

Dominant phenotype = 35
Recessive phenotype = 1

Now the experiment becomes more interesting.

Was the plant classified incorrectly?

Was one parent not actually true-breeding?

Was there accidental pollination?

Was the supposed characteristic more complicated than expected?

Good experiments do not merely confirm theories.

They make us ask better questions when observations do not agree with predictions.

Then Produce the F2 Generation

The next step is the really satisfying one.

Cross suitable F1 plants, or allow self-pollination where appropriate, and collect the next group of seeds.

Grow those seeds.

Now look for the characteristic that apparently vanished during the F1 generation.

If the simple Mendelian model applies, the recessive phenotype should reappear.

The theoretical expectation is:

3 dominant : 1 recessive

But there is an important word in that statement.

Expectation.

It does not mean every four plants will consist of exactly three dominant plants and one recessive plant.

Mendelian Ratios Are Probabilities, Not Instructions

This is one of the most valuable lessons in the entire experiment.

Consider tossing a coin.

The probability of heads is 1/2.

If I toss the coin four times, that does not guarantee:

2 heads
2 tails

I might obtain:

3 heads
1 tail

or even:

4 heads
0 tails

The same principle applies to inheritance.

For an Aa x Aa cross, each offspring independently has a 3/4 probability of showing the dominant phenotype and a 1/4 probability of showing the recessive phenotype.

With only eight plants, the observed ratio might look rather unlike 3:1.

With 200 plants, it is likely to be considerably closer.

This gives us an immediate connection between genetics and statistics.

A Numerical Example

Suppose we grow 160 F2 plants.

The expected numbers for a 3:1 ratio are:

Dominant phenotype:

160 x 3/4 = 120

Recessive phenotype:

160 x 1/4 = 40

But perhaps our actual results are:

Dominant = 118
Recessive = 42

The observed ratio is:

118 : 42

or approximately:

2.81 : 1

Does that mean Mendelian genetics has failed?

Of course not.

The result is extremely close to the expected pattern.

Biological data contains variation.

That is precisely why collecting the data is more educational than simply completing a Punnett square.

The Faster Version — Genetic Maize

There is another excellent way to investigate Mendelian ratios that requires much less waiting.

Count maize kernels.

Specially prepared genetic maize ears can contain kernels displaying easily distinguished inherited phenotypes.

Depending upon the particular educational material being used, kernels may differ in characteristics such as colour or texture.

Instead of growing two generations of plants, students can inspect hundreds of individual kernels.

That turns an ear of maize into a remarkably compact genetics experiment.

Imagine counting 200 kernels and finding:

Dominant phenotype = 146
Recessive phenotype = 54

The theoretical 3:1 expectation would be:

Dominant = 150
Recessive = 50

Again, the experimental result is not exactly 3:1.

Nor should we necessarily expect it to be.

The interesting question becomes:

Is the difference small enough to be explained by chance?

That takes us into another extremely important area of biology.

Adding a Chi-Squared Test

For A-level students, the investigation can be extended using a chi-squared test.

The basic calculation is:

X^2 = sum((O - E)^2 / E)

where:

O = observed frequency
E = expected frequency

Take an example with 160 individuals:

Observed dominant = 118
Expected dominant = 120

Observed recessive = 42
Expected recessive = 40

For the dominant phenotype:

(118 - 120)^2 / 120 = 4 / 120

For the recessive phenotype:

(42 - 40)^2 / 40 = 4 / 40

Therefore:

X^2 = 4/120 + 4/40

X^2 = approximately 0.133

Students can then compare their value with an appropriate critical value.

Suddenly several areas of the biology course have come together:

genetics;
probability;
experimental design;
sampling;
mathematical analysis;
hypothesis testing.

And all because we counted real organisms rather than simply filling four boxes in a Punnett square.

Try Changing the Sample Size

There is another experiment hidden inside the experiment.

Suppose you have an ear containing several hundred kernels.

Count only 20 randomly selected kernels.

Calculate the ratio.

Then count 50.

Then 100.

Then 200.

Finally count as many as practical.

You will probably find that the estimated ratio jumps around dramatically with small samples and tends to become more stable as the sample becomes larger.

That is a powerful demonstration of sampling error.

Students often encounter the instruction:

"Use a large sample size to improve reliability."

This experiment shows them why.

Two Groups Can Get Different Answers

An even better activity is to give several groups different samples from the same population.

Imagine four groups each count 40 kernels.

They might obtain:

Group 1 — 32:8
Group 2 — 28:12
Group 3 — 31:9
Group 4 — 29:11

None is exactly the same.

Combine the results, however:

Dominant = 120
Recessive = 40

And suddenly the overall result is exactly 3:1.

That leads naturally to discussions of:

replication;
sample size;
random variation;
pooled data;
reliability.

These are scientific ideas that extend far beyond genetics.

Why Not Every Characteristic Behaves Like Mendel's Peas

There is also an important warning to include.

Students can sometimes leave school believing that every characteristic works like:

A = dominant
a = recessive

Biology is considerably more interesting than that.

Some characteristics involve:

incomplete dominance;
codominance;
multiple alleles;
linked genes;
sex-linked inheritance;
polygenic inheritance;
interactions between different genes;
environmental influences on phenotype.

Human height, for example, cannot sensibly be explained using one simple dominant and one recessive allele.

Neither can intelligence, body mass, skin pigmentation or many other complex characteristics.

Mendel's model is enormously important because it reveals fundamental principles of inheritance.

But it is a starting point, not a description of every biological characteristic.

A real breeding experiment provides an excellent opportunity to make that distinction.

Phenotype Is Not the Same as Genotype

Another useful question is:

If a plant shows the dominant phenotype, can we tell whether it is AA or Aa just by looking at it?

No.

Both genotypes produce the same phenotype under complete dominance.

That creates the possibility of another classic genetic technique: the test cross.

An individual showing the dominant phenotype but having an unknown genotype can be crossed with a homozygous recessive individual.

If the unknown plant is:

all offspring should show the dominant phenotype.

But if it is:

approximately half the offspring should show the dominant phenotype and half the recessive phenotype.

Again, the genotype is not observed directly.

It is inferred from experimental evidence.

That is an important scientific distinction.

Make the Investigation a Proper Research Project

Rather than treating this as a one-hour practical, I would make it a continuing investigation.

Students could maintain a genetics notebook containing:

Week 1

Plant or examine parental generation.

Week 2 onward

Measure growth and record phenotypes.

Flowering

Carry out selected crosses.

Seed production

Collect and label offspring.

Next generation

Germinate and record F1 phenotypes.

Later

Produce F2 offspring where practical.

Analysis

Compare observed and predicted ratios.

Evaluation

Consider experimental errors and alternative explanations.

Photography can make this particularly effective.

Photograph each generation under similar conditions and build a visual family history:

P -> F1 -> F2

You could even create a simple digital pedigree showing which plants produced which offspring.

That begins to resemble the type of record-keeping required in genuine biological research.

What Could Go Wrong?

Quite a lot.

And that is part of the value of the experiment.

Seeds may fail to germinate.

Plants may die.

Pollination may fail.

Labels may become detached.

Phenotypes may not be as obvious as expected.

A supposedly true-breeding line might not behave as anticipated.

Sample sizes may be too small.

Environmental differences may influence the appearance of plants.

And occasionally the results may simply refuse to give a beautiful textbook ratio.

None of this makes the experiment unsuccessful.

It makes it real biology.

A perfectly clean 3:1 result printed in a textbook teaches Mendelian inheritance.

An experimental ratio of 73:27 that students have actually produced teaches Mendelian inheritance and science.

From Punnett Squares to Evidence

This is why I particularly like experiments of this type.

There is nothing wrong with Punnett squares. They are extremely useful models.

But there is a danger when students spend too long manipulating letters on paper that they forget what those letters represent.

A represents biological information carried by chromosomes inside real cells.

Those cells produce gametes.

Gametes combine.

Seeds develop.

Plants grow.

Phenotypes appear.

And somewhere among a tray of F2 seedlings, a characteristic that apparently disappeared a generation earlier suddenly returns.

That is a far more memorable way to understand the meaning of a recessive allele.

Mendel Was Counting — Not Drawing Squares

One of the most revealing things about Mendel's work is the sheer importance of counting.

He did not simply notice that some offspring had one characteristic and some another.

He recorded how many.

That changed inheritance from a collection of observations into something that could be investigated mathematically.

Modern genetics has travelled an extraordinary distance since then.

Today we can sequence DNA, identify mutations and examine individual genes.

But the fundamental scientific approach remains recognisable:

make a prediction, carry out a cross, observe the offspring, count them and compare the evidence with the prediction.

That is why repeating even a modest version of Mendel's experiment can be so valuable.

Students stop being told that F2 offspring should produce a 3:1 ratio.

They grow them.

They count them.

They discover that nature rarely gives perfectly tidy numbers.

And then they have to decide whether their evidence supports the model.

At that moment, Mendelian genetics stops being a Punnett square.

It becomes experimental biology.