Beautiful Equations I

One of the things that I love the most about mathematics and the sciences are their ability to describe the world around us through elegant equations and ideas. In light of this, I’ve decided to start a series on beautiful equations.

General Relativity

einstein-eq

This equation was developed by Einstein as part of his general theory of relativity in 1915. The theory revolutionized the way scientists understood gravity, as it described that what we perceive as the force of gravity in fact arises from the curvature of space and time.

The right-hand side of the equation describes the energy content of our universe, including ‘dark energy’ that causes the current cosmic acceleration, whilst the left-hand side describes the geometry of space-time.

This equation shows how mass and energy determine the geometry, and hence curvature of space-time, which results in gravity. Additionally, it can tell us how the universe has evolved since the Big Bang, and predicts that there should be black holes.

Standard Model

The Standard Model Lagrangian

The Standard Model Lagrangian

The standard model is the description of all the fundamental particles in our universe. It’s encapsulated into one main equation – called The Standard Model Lagrangian – which was developed by French mathematician Lagrange in the 18th century.

This model has successfully described all elementary particles and forces that we have observed to date. However it has not yet been united with general relativity, and so cannot describe gravity.

Our standard model currently divides into two groups: bosons and fermions.

The fermions are the ‘matter’ particles that make up the elements around us and are divided into quarks and leptons. The quarks are constituents of the proton and neutron, while the electron is an example of a lepton.

The bosons are the ‘force carriers’ and are responsible fore the forces between particles. The three forces described in the standard model are

  • Electromagnetismsmparticles
    • Carried by photons
    • Effects particles with an electric charge and is responsible for electricity and magnetism
    • Photons bing electrons to atoms, and atoms to atoms to form molecules
  • The strong nuclear force
    • Carried by Gluons
    • Gluons bind quarks to form hadrons and hadrons to form atoms
  • The weak nuclear force
    • Carried by W and Z bosons
    • The W boson is involved in natural radioactivity such as beta decay

The final particle is the Higgs boson, which is what gives particles their mass – all particles have no inherent mass, gaining it by interacting with the Higgs field.

Pythagorean Theorem

Probably one of the most famous equations used in secondary school mathematics, it describes how for any right-angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

pythgrn

The proof of this equation is beautifully simple, and is therefore an ‘entrance level’ proof for any budding mathematicians (in fact it was one of the first proofs I learnt!)Area of the whole square:

Area of the whole square:pythagorean-theorem-proof

A = (a + b)(a + b)

Area of the pieces:

Yellow: c2

Blue: 4(0.5 * ab) = 2ab

A = c2 + 2ab

Equating the two areas:

c2 + 2ab = (a + b)(a + b)

c2 + 2ab = a2 + b2 + 2ab

c2 = a2 + b2

1 = 0.999999999….

Strikingly simple, but extraordinarily provocative.

Let X = 0.999…

Then 10X = 9.999…

Subtracting one from the other gives us:

9X = 9

  X = 1

Therefore 0.999… = 1

“It’s beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity,” says Steven Strogatz of Cornell University.

 

Euler’s Polyhedron Formula

This equation encapsulates something about the nature of spheres: if you cut a sphere into faces, edges and vertices and then subtract the number of edges from the sum of the number of vertices and faces, the answer will always be 2. This is the Euler characteristic of the sphere.

EC = 2

This is a topological constant, thus any object with an Euler characteristic of 2 is homeomorphic to the sphere.

The general equation for Euler’s characteristic is:

EC

For example, a torus has an Euler characteristic of 0.

Euler–Lagrange equations and Noether’s theorem

Although they’re abstract, they’re incredibly powerful and have survived major revolutions in physics, such as quantum mechanics and relativity.

Euler - Lagrange

In the equation L stands for Lagrangian, which is a measure of the energy in a physical system, such as springs or fundamental particles. When you solve the equation, it will allow you to see how the system will evolve over time.

Noether’s Theorem, formalised by Emmy Noether, is closely related to the Lagrangian equation. It is fundamental to physics and the role of symmetry. The theorem states that if a system has symmetry, then there is a corresponding conservation law. For example, the idea that the fundamental laws of physics are the same today as tomorrow (time symmetry) implies that energy is conserved.

Symmetry is perhaps the driving concept in fundamental physics, primarily due to Noether’s contribution.

The Callan-Symanzik equation

The Callan-Symanzik equation is a differential equation that has numerous applications. For example, it allows physicist to estimate the mass and size of a proton and neutron.

The Callan-Symanzik equation

Basic physics can easily show how the gravitational and electric force between two objects is proportional to the inverse of the distance between them squared.

u8l3b3

On a simple level, this is also true of the strong nuclear force. However, tiny quantum fluctuations can slightly alter a force’s dependence on distance, thus having dramatic consequences on the strong nuclear force.

This prevents the force from decreasing at long distances, and causes it to trap quarks and combine them to form protons and neutrons. What the Callan-Symanzik equation does is relate this effect, which is difficult to calculate when the distance is roughly the size of a proton, to more effects that are easier to calculate and that can be measured when the distance is much smaller than a proton.

Let me know which one from these is your favourite! x

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