What Are The Hidden Rules Of The Universe?

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Imagine a glorious future where humans have overcome our present troubles,

And we eventually leave this tiny planet behind to live amongst the stars,

In time, our descendants span the entire Milky Way,

Countless trillions living in harmony from the galactic centre to the most distant spiral

arm.

One day, a message arrives from outside, from another distant galaxy, another distant cluster.

There is other life, other intelligence, out there in the universe.

Ideas are exchanged and science, art and philosophy prosper.

And eventually, plans are made - in the darkness of intergalactic space, the two great civilizations

will meet.

Many generations labour to build the immense craft capable of traversing great distances.

And many generations are born, live their lives and die on the immense journey.

But after countless millennia, in the inky blackness, the emissaries of the two civilizations

approach.

Two individuals float across the void, hands reaching out.

They touch.

And both vanish in a blinding flash.

The great ships stand off from each other, staring in amazement.

The civilizations are fundamentally different.

And it is only now it becomes clear - one is made of matter, the other anti-matter,

and any contact leads to annihilation.

The ships retreat into the darkness, knowing that true contact will always be impossible.

But how did it come to this?

How did they not know about the difference in their fundamental make-up?

They had shared their science, their mathematics, and their engineering

Surely this difference would have been obvious?

But the scientists knew better, they knew that nature holds onto its secrets tightly.

They knew that written into the laws of the universe were rules that cloaked these secrets.

Rules fundamental to the functioning of the universe.

They knew that these rules were built into many of the universe´s processes, from the

supermassive to the microscopic.

They knew that it was these rules that had helped reality freeze in the first billionth

of a second of time.

And behind them all - they knew there was always one common factor.

Symmetry.

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Two great civilizations Inhabiting two distinct and distant regions

of the universe, And yet we know their laws of physics must

have been the same.

Then how could they have failed to tell matter from anti-matter?

Two particles, two atoms, two intergalactic beings with opposite electric charge?

In our patch of the universe, matter is everywhere, and anti-matter is rare.

But in their patch of the universe, the inverse must have been true.

Whilst we live by the light of a star, they lived by the light of anti-stars.

Whilst our ships were built from atoms, theirs were constructed from anti-atoms.

It would be understandable to think that the laws of physics do not care about the difference

between matter and anti-matter.

But that is not quite true.

We now know that nature does treat matter and anti-matter slightly differently.

Though these differences are subtle and unexpected, And they are related to one of the deepest

concepts in all of physics.

What is symmetry?

We all know our first perception of symmetry, a mirror reflection from left to right.

Armed with such a mirror, a child can sketch a symmetrical shape,

With what is on the left also seen on the right, and vice versa.

But looking in a mirror reveals that this reflection runs deeply in nature.

Our faces and bodies are symmetrical, a right eye with a left, a left arm with a right.

Nature makes many uses of this mirror, or bilateral, symmetry.

Most mammals, reptiles, birds, and fish share this simple mirroring.

And when the rules of natural symmetry are broken, such as the oversized claw of a Fiddler

Crab, The appearance can be quite jarring.

But symmetry is more than just reflection.

A starfish builds itself using rotational symmetry, so when rotated its appearance does

not change.

and a millipede uses translational symmetry, constructing its body with identical chunks.

Through symmetries, nature is being economical in design.

Symmetry can also spark a deep emotional response.

In human concepts of beauty, symmetry of appearance is pleasing,

even though beauty is only skin deep, and inside we can be quite asymmetrical.

And this appreciation has made its way into art and design.

The great temples of the ancients in Egypt and Greek show exquisite symmetry

As do more modern buildings like the Taj Mahal and Arc de Triomphe

and many great artworks, such as DaVinci’s Last Supper.

And art and design call on more than just simple mirror symmetry, as rotational symmetry

can be powerful.

From the rotating disks of the pre-Columbian Americans, to the exquisite Islamic patterns

adorning many mosques across the globe.

And indeed the massive Pentagon, home of the US Department of Defence, bears its geometry

in its name.

Symmetry is everywhere.

But you might be wondering how this relates to the fundamental rules of the universe?

Well, our story starts, as all good stories do - with a duel.

At dawn on the 30th May 1832, two men met in a field outside of Paris,

The cause of the conflagration is unclear, but rumour goes it was over the affections

of a young woman.

By the time the sun was fully up that day, 20-year-old mathematician, Everiste Galois,

lay dying in the grass.

The night before Galois had had a premonition of his upcoming death.

Into the early hours, he had scribbled, with letters to friends and colleagues,

And amongst these missives, he had poured out the last of his mathematical ideas.

For in his short life, Galois had brought many new ideas to the world of maths, and

central to these was the idea of symmetry.

The mathematics of symmetry was not a new idea.

Ancient geometers had realised that a rotated circle remained the same circle.

And triangles, squares and pentagons appeared the same after fixed rotations.

But Galois saw deeper.

His focus was algebra, and more specifically - polynomials.

Polynomials are equations with quantities to a power, squares, cubes, and higher.

For example “one plus x plus x squared” is a simple polynomial.

As is three plus five times x to the power of five minus x to the power of seven.

Such polynomials are found across all of science, engineering, economics, and many other fields.

Other mathematicians had scrambled to try and find algebraic solutions to higher-order

polynomials, But Galois wondered if such an equation existed

for all polynomials - one equation to unlock them all.

Yet instead of discovering this ultimate equation, he actually showed it was a fool’s errand

- the solutions just didn’t exist.

And he didn’t do this by simply fighting through the algebra,

but by looking into the symmetries of the solutions.

He realised the solutions to polynomial equations were complex numbers,

each being a point on the infinite complex plane.

These points could appear as shapes, triangles, pentagons or others.

And shapes, of course, have symmetries – they can be reflected or rotated.

This was a radical way of doing mathematics - translating equations into a geometric picture

and thinking about their symmetries.

And since his work in the early nineteenth century, mathematical insights into symmetries

have grown, It has blossomed into group theory, the study

of groups of things that share properties and symmetries.

Modern cryptology is underpinned by group theory, but it is symmetry´s impact on science

which has been the most startling.

And so as he lay dying in the morning light, Galois didn’t know that his scribbles would

revolutionize our universe.

Long before they met, our two great civilizations came into contact.

Radio beams were fired across the universe, taking many millions of years to reach their

destination.

It was a slow and cumbersome way of talking, But this new conversation with our distant

neighbours did work.

Both civilizations knew the laws of electromagnetism.

On Earth, the jiggling of electrons formed the radio beam that carried the messages.

And in the instruments of the aliens, charges jiggled as the messages arrived.

But unlike on Earth, these were anti-electrons, positrons, dancing to the beat of the radio

waves.

Electromagnetism has a fundamental symmetry.

If you swap all the positive and negative charges, the outcomes remain the same.

The exchange of radio waves alone between the two civilizations could never tell matter

from anti-matter.

Something else was needed, some crack in the symmetry of physics.

To understand why, we need to start our story almost a century ago.

We need to start with a funeral.

On the first of May in 1935, an obituary appeared in the New York Times,

This was not unusual, but the author, famous physics professor Albert Einstein, certainly

was.

Emmy Noether, Einstein wrote, had been the “most significant creative [female] mathematical

genius produced so far”, And her insights had been necessary for “the

deeper penetration into the laws of nature”.

But what did he mean?

Emmy Noether was only fifty-three when cancer struck her.

And whilst her name might not be as iconic today as Einstein and the other old fathers

of physics, Her work was just as important - and Noether’s

theorem has everything to do with symmetry.

And key to this was one simple fact - the universe, nature itself - is lazy.

With the coming of modern science, mathematics had steadily replaced mysticism,

and geometric symmetries had emerged from the equations.

The spherical pull of Newton’s gravity resulted in spherical planets and stars,

Whilst the magnetic field of an electrical current was found to be shaped like a cylinder.

These geometric symmetries, symmetries of shape, were pleasing to the eye,

But they were really only skin-deep.

There were other, much deeper symmetries lurking in the mathematics of the universe.

As with any story, there were many potential players, but here we will focus on just two.

The first is Italian-born Giuseppe Lodovico Lagrangia, better known as Joseph-Louis Lagrange.

And Irish mathematician William Rowan Hamilton.

They share few similarities in their lives - Lagrangia lived through the French revolution

and was instrumental in bringing in the metric system - Hamilton grew up as a wonderkid in

Dublin almost a century later.

But despite their separation in time, their work dovetailed on one key thing - reformulating

the equations of Newton.

And key to their insight was the idea that the universe was lazy - the curious concept

of "least action".

Indeed, long before, the French mathematician Pierre de Fermat had proposed that light always

took the quickest path through any optical system.

Imagine you are at the beach, with waves crashing into the shore.

Off in the distance, somewhere to the left, you spot something in the water.

A person is waving.

No, a person is drowning!

You have to rush to save them.

But which way do you go?

You can run fast on the sand, or you swim slower in the water.

Dashing straight to the shore, and then swimming out and left will take too long.

Should you run along the shore and minimize the time you spend swimming?

As you race the calculations in your head, you realise there is a better path.

An optimal path!