Science's ongoing quest to find the smallest possible things remains tantalisingly incomplete, as physicist Prof Andy Parker explains.

Physics has a problem with small things. Or, to be more precise, with infinitely small things.

We imagine that we can move any distance we like, no matter how small.

This perception was exploited by Zeno in one of his famous paradoxes. Achilles could never actually get anywhere since the distance he would have to cover could be halved an infinite number of times - halfway there, halfway again, and so on. He would have to take an infinite number of ever-smaller steps to reach his goal.

Mathematicians have explained this apparent paradox, and are completely comfortable with infinite numbers, as well as infinitely small distances and objects. Their answers are used in physics to describe the world inside the atom.

But nature is not so comfortable with this. When we try to describe something as a "point" - an infinitely small object, that throws up some of the most intractable problems in physics.

Since all of particle physics relies on "point-like" particles, reacting to forces in tiny spaces, one can anticipate trouble.

This duly appears in the form of nonsense answers when the equations are used at the smallest distances.

Physicists are therefore increasingly suspicious of points, and asking whether in fact Nature has a limit for the smallest possible object, or even whether there is a smallest possible space.

Russian Dolls

The quest for the smallest building blocks of Nature probably stretches back to the first caveman who tried to put a sharp edge on a flint.

Horizon: How small is the Universe? is on BBC Two at 21:00 BST on Monday 3 Sept

Catch up afterwards via iPlayer (UK only) at the above link

Andy Parker is a professor of high-energy physics at Cambridge University

The Greeks gave us the concept of billiard-ball shaped atoms which stick together to make up the materials we see, and this picture is still in most peoples' minds today.

Over a century ago, JJ Thomson managed to extract electrons from atoms in Cambridge, and he was followed in 1932 by Cockcroft and Walton, who split the atomic nucleus with a cleverly designed particle accelerator.

These turned out to be only the first Russian Dolls.

Successive experiments, using more and more powerful accelerators, revealed that the nucleus was composed of protons and neutrons, and that they in turn were made of quarks.

The evidence for the Higgs boson recently produced at the Large Hadron Collider at Cern is the latest of these.

But all attempts to split quarks or electrons, even using the awesome power of the LHC have failed.

The basic building blocks seem to be points, certainly smaller than 0.0000000000000000001 metres across.

To infinity

One can see where the problem comes from. All the forces in nature get stronger at short distances.

Newton's famous "inverse-square law" of gravity, for example, says that the force of gravity gets four times stronger if you halve your distance from an object.

If we imagine particles as points, you can make the distance between two of them as small as you like, so the force becomes infinite. Ultimately this would break up the fabric of space, creating a foam of black holes. That would certainly slow Achilles down!

Physicists can normally sidestep this problem, using the fuzziness built into quantum mechanics which allows matter to behave as particles or waves.

You may also have heard of Heisenberg's Uncertainty Principle which does not allow us to know exactly where anything is. So even though a particle might be a point, its location is uncertain, and in the equations it looks like a fuzzy ball - problem solved!

Well almost - we don't actually know how to apply quantum mechanics to gravity, and so we still get stuck with nonsensical predictions such as the complete collapse of space if we try to describe strong gravitational fields, like those inside black holes.

Find out more Newton's Law of Gravity laid the foundations of modern physics.

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How the Higgs boson completed the predicted Standard Model of elementary particles

It turns out that quantum mechanics and Einstein's theory of gravity just don't mix.

Various ingenious solutions have been proposed to this problem.

The most obvious is that there is another Russian Doll, and the smallest particles are tiny billiard balls. If so, one day, perhaps with the Hadron Collider, we will see the size of the smallest objects.

But theoretical physicists prefer the idea that the particles are not in fact round, but tiny "strings", like bits of elastic.

They have a finite length, but an infinitely small width. This solves the problem, since you can never be at the same distance from all of the string.

You may have guessed that is what we call String Theory.

Strings can vibrate, and this allows us to explain all of the strange fundamental particles which we see as different vibrations of the strings - different notes from a cosmic violin.

So far, so simple - but to explain the particles we know about, the strings have to vibrate in lots of different ways.

Superstring Theory allows them to vibrate in a bizarre space with 11 dimensions - up, down, sideways, "crossways" and 7 other ways!

Experiments at the LHC are looking for evidence that you can move "crossways". If we can, there could be whole universes, as big and marvellous as our own, sitting just down the road "crossways".

We can go even further - perhaps we should not be looking for the smallest object, but the smallest distance.

If space is composed of lots of small grains, then our problem can be solved, since no two particles can ever be closer together than the size of a grain.

Achilles would move along in a series of small, but finite, steps. By looking at particles travelling over huge distances across the cosmos, we hope to see the accumulated effect of bumping along lots of tiny grains, rather than gliding through the smooth space which we imagine.

In the end, the answers will be found in experiments, not in our imaginations. Perhaps the most amazing thing we have discovered is the scientific method, which allows us to pose and answer questions like "How small is the Universe?". Not bad for slightly evolved cavemen!

Andy Parker is Professor of High Energy Physics at Cambridge University and a founder of the ATLAS experiment for the Large Hadron Collider.

Horizon: How Small is the Universe? is on BBC Two at 21:00 BST on Monday 3 September. Or catch up afterwards via iPlayer (UK only) at the above link.