8 Beautiful Equations in Science and What Each One Means

Which equations do scientists call beautiful? Usually the ones that pack a huge idea into a few symbols: Einstein's field equations, Maxwell's equations of electromagnetism, Boltzmann's formula for entropy and Euler's identity are regular winners when physicists and mathematicians are asked to choose. Einstein's E = mc² gets most of the public attention, but many less famous formulas have devoted fans.

The eight images below come from a well-known gallery that paired famous equations with striking pictures. For each one we explain, in plain language, what the symbols say, who found it and why people find it elegant. You do not need to solve anything; the aim is to read each equation as a sentence about how the world works.

What makes an equation beautiful?

Beauty in mathematics is mostly about economy and surprise. A beautiful equation connects things nobody expected to be related, it is short, and it keeps proving useful far beyond the problem it was written for. In a 1990 reader poll by The Mathematical Intelligencer, Euler's identity was voted the most beautiful theorem in mathematics, and in a 2004 Physics World poll it tied with Maxwell's equations as the greatest equation ever.

A good way to approach any of the formulas below is to ask three questions: what does each symbol stand for, what does the equals sign connect, and what would change if one quantity doubled?

Averages and the shape of the universe

The weighted average (first image). The opening picture shows rows of composite faces, each blended from photos that received similar attractiveness ratings on a 1 to 10 scale, next to the formula for a weighted mean: each value is multiplied by its weight, the results are added, and the total is divided by the sum of the weights. It is one of the most used formulas in everyday statistics, from school grades to price indexes. The precise origin of these particular composite images is not confirmed.

The Friedmann equations. Set against a NASA-style timeline of the universe from the Big Bang to today's accelerating expansion, these two equations describe how the scale of the universe changes over time. Alexander Friedmann derived them from Einstein's general relativity in 1922. They link the expansion rate to the density of matter and energy, the curvature of space (k) and the cosmological constant (Λ), and they underpin the standard Big Bang model used by cosmologists today.

Einstein's field equations. Published by Albert Einstein in 1915, these equations state that the curvature of spacetime (the left side) is set by the matter and energy within it (the right side). The extra term with Λ is the cosmological constant, which Einstein introduced in 1917 and which is now used to describe dark energy. In a sense, this compact line is the whole theory of gravity that explains black holes, gravitational waves and the bending of light.

Time, light and electricity

Time dilation in special relativity. This formula says that a clock moving at speed v runs slower, as seen by a stationary observer, by a factor that depends on v compared with the speed of light, c. At everyday speeds the effect is tiny, but it is real and measurable: fast-moving particles from cosmic rays survive longer than they would at rest, and satellite navigation systems must account for relativity to stay accurate. The maths needs nothing beyond a square root, which is part of its charm.

Maxwell's equations. These four lines, worked out by James Clerk Maxwell in the 1860s and condensed into their modern vector form later, describe all classical electricity, magnetism and light. They show that a changing magnetic field creates an electric field and vice versa, which is how electromagnetic waves travel. The background photo appears to show the Z machine at Sandia National Laboratories in Albuquerque, New Mexico, a giant pulsed-power facility used for fusion and extreme-materials research, with electrical arcs flashing across it.

Entropy, quantum particles and Euler's identity

Boltzmann's entropy formula. S = k log W says that the entropy of a system (S) is proportional to the logarithm of the number of microscopic arrangements (W) that look the same from the outside; k is Boltzmann's constant. Ludwig Boltzmann developed the idea in the 1870s and Max Planck later gave it this exact form. The photo shows Boltzmann's grave at Vienna's Central Cemetery (Zentralfriedhof), where the formula is carved above his bust.

The Schrödinger equation. Published by Erwin Schrödinger in 1926, this equation describes how the quantum state of a particle, its wave function Ψ, changes over time. It is the quantum counterpart of Newton's laws and underlies chemistry, semiconductors and lasers. The background is a bubble-chamber photograph: a device invented by Donald Glaser in 1952, for which he received the 1960 Nobel Prize in Physics, where charged particles leave trails of tiny bubbles that curl into spirals inside a magnetic field.

Euler's identity. e^iπ + 1 = 0 links five of the most important numbers in mathematics, 0, 1, π, e and the imaginary unit i, using addition, multiplication and exponentiation exactly once each. It follows from Euler's formula, which connects exponential growth with rotation around a circle. The photo shows the identity as a raised scar design on someone's back, perhaps the most permanent tribute any equation has received.

How to read a famous equation

• Name the quantities first. Before worrying about the symbols, find out what each letter measures and in what units.
• Read the equals sign as a link. Ask which two ideas the equation says are the same thing, such as geometry and matter in Einstein's case.
• Try the extremes. Set a speed to zero or make a number huge and see what the equation predicts.
• Remember the assumptions. A short formula hides conditions about where it applies; elegance alone does not make an equation true, experiment does.

Read this way, a gallery of formulas stops being decoration and becomes a quick tour of some of the biggest discoveries in science.

Explore the related article: A Flat-Earth Argument and the Role of Physical Evidence.

Sources: Images via Wired and Science Dump; Wikipedia: Euler's identity; Wikipedia: Boltzmann's entropy formula; Wikipedia: Friedmann equations; Wikipedia: Z Pulsed Power Facility; Wikipedia: Bubble chamber; Live Science: The World's Most Beautiful Equations.