Even for those of us who finished high school algebra on a wing and a prayer, there's something compelling about equations. The world's complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.
For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.
We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.
Above:The Beautiful Equation: Euler’s Identity
Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.
Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math -- 0, 1, i, pi and e -- that are essential tools in scientific work.
Theoretical physicist Richard Feynman was a huge fan and called it a "jewel" and a “remarkable” formula. Fans today refer to it as “the most beautiful equation."
Image: Cory Doctorow/Wikipedia
The Entire Universe in Figures: Friedmann Equations
Derived from Einstein’s theory of General Relativity, the two Friedmann equations describe the life of the entire universe, from fiery Big Bang birth to chilly accelerated expansion death.
The formulas contain an odd term known as the cosmological constant (the triangle thing with no bottom), initially inserted by Einstein to counteract gravity and keep the universe eternally unchanging. When observations showed that the cosmos was actually expanding, Einstein called this insertion his biggest mistake. Recent experiments have vindicated Einstein, showing that there is a great and mysterious force known as dark energy accelerating the expansion of the universe. Its discovery was the subject of the most recent Nobel Prize in physics, though understanding how it works has thus far eluded scientists.
Image: NASA / WMAP Science Team
Boltzmann’s Entropy Formula
Nature loves chaos when it pushes systems toward equilibrium, and geeks call this universal property entropy.
Austrian physicist Ludwig Boltzmann laid entropy’s statistical foundations; his work was so important that the great physicist Max Planck suggested that his version of Boltzmann’s formula* be engraved on Boltzmann’s tombstone in Vienna (above).
The equation describes the tight relationship between entropy (S), and the myriad ways particles in a system can be arranged (k log W). The last part is tricky. k is Boltzmann's constant and W is the number of microscopic elements of a system (e.g. the momentum and position of individual atoms of gas) in a macroscopic system in a state of balance (e.g., gas sealed in a bottle).
Note: Not to be confused with the other Boltzmann equation, which describes how gases or fluids move energy around.
Image: Tom Schneider/NCI-Frederick
Electricity and Magnetism: Maxwell’s Equations
Without these four equations, every lolcat on the Internet couldn’t exist. First put together by James Clerk Maxwell in 1861, the formulas describe all known behaviors of electricity and magnetism and show the relationship between the two forces. They state that a moving electric charge will generate a magnetic field while a shifting magnetic field similarly creates an electric field.
The second equation, Gauss’ law for magnetism, also demonstrates a profound difference between electricity and magnetism. While electricity exists as separate charges, like the plus and minus of a battery, magnets always come in a joined pair; you can never break the ‘north’ part of a magnet from the ‘south’ side. Some recent physical models posit that north- or south-less magnets (known as magnetic monopoles) might actually be present in small numbers in the universe, and several experiments are busy searching for their existence.
Image: The Z machine, largest x-ray generator in the world. Courtesy Sandia National Laboratories
Certain Uncertainty: Schrödinger Equation
Erwin Schrödinger’s famous equation reigns supreme over the smallest objects in the universe. It illustrates how subatomic particles change with time when under the influence of a force. Any particular atom or molecule is described by its wavefunction, the probability of where and when the particle appears, represented by the Greek letter psi.
Unfortunately, since the early days of quantum mechanics, physicists have been at odds as to how exactly to interpret Schrödinger’s equation. Some favor the idea that the wavefunction is merely a useful calculation tool but doesn't correspond to anything real. Others say it puts a limit on the amount we can know about the universe, since we only know what state a particle is in once it is measured.
Schrödinger himself argued that the wavefunction represented a real, physical object. He disagreed with the a-particle-only-collapses-when-measured interpretation, and his famous cat experiment was actually intended to demonstrate that interpretation's shortcomings.
Image: The ghostly tracks of subatomic particles in a hydrogen bubble chamber. Courtesy of CERN
All Life Is an Island: Island Biogeography
Though physicists can describe the universe's expansion in a few lines, the basic properties of life on Earth are far harder to quantify. During the latter half of the 20th century, biologists arrived at the theory of island biogeography, which described the dynamics of animal populations on islands. At left in this equation is the number of species a given island can support; at right, animal abundances, available areas, and rates of immigration and emigration. The theory has expanded beyond oceanic islands to include many types of ecologies, especially those isolated by human activity. Outside the polar regions, almost all nature now exists in human-defined islands -- and the biggest island of all, of course, is Earth.
Image: The Cache River wildlife preserve in Arksansas, an isolated patch of bottomwoods swamp where the extinct Ivory-Billed Woodpecker was reportedly seen. (NASA)
The Essence of Evolution: Nowak's Evolvability
At its most basic level, life is what replicates itself -- but how did it begin? It's the ultimate chicken-and-egg problem, and one that scientists studying what's called pre-life try to answer. On the left side of this equation, proposed by Harvard University mathematical biologist Martin Nowak, is a symbol representing all possible strings of molecules; at right are the speed of chemical reactions, the tendency of shorter strings to be more common than longer strings, selection pressures and fitness ratings. As Nowak has shown, all that's necessary for life to emerge are molecules subject to forces of selection and mutation. If those conditions are met, self-replication will emerge with the inexorability of gravity.
Image: Sunrise over California's Mono Lake, a hot, oxygen-deprived, arsenic-rich lake that scientists think mimics conditions on early Earth. (NASA)
The Razor's Edge of Outbreak: R-Nought
Brought to mainstream attention by the thriller Contagion, R0, pronounced R-nought, is a very simple figure: It refers to the average number of people an individual infected with a pathogen will go on to infect. If it's less than one, the disease will burn itself out; if greater than one, it will spread. In a world where a flu virus from Mexico can infect millions of people around the world in a matter of months, this equation is as symbolic as it is straightforward.
Image: Subway riders in Mexico City during the 2009 swine flu outbreak. (Eneas de Troya/Flickr)
Hot or Not: The (Limited) Mathematics of Beauty
Not everything can be quantified, especially when it comes to matters of the human heart and mind. For decades, psychologists and biologists have tried to represent physical beauty in formula form; but even if some tendencies emerge when hundreds of individual preferences are measured, what any one individual considers beautiful is impossible to predict.
At right is an equation from an unpublished attempty by Israeli computer scientists to design a program capable of quantifying the attractiveness of a face. "Y" is the empirical beauty score; at right, various measurements of how different features in a face compared to a baseline face. The program was brilliantly coded, but it didn't work very well.
Image: Pierre Tourigny/Flickr
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