laplace's equation

There's something intuitively and aesthetically gratifying about a simple equation that can be applied to so many different things in nature. For instance, mathematical figures such as the golden ratio or the Fibonacci sequence have captivated scientists, philosophers and artists alike.

Here's another, less heralded example: Laplace's equation (shown above). This elegant equation contains only five symbols, and yet it can be applied to everything from electromagnetism to the gravitational field to fluid dynamics to heat conduction.

Laplace's equation and gravitational field Laplace's equation can be applied to everything from electromagnetism to the gravitational field. (Photo: AuntSpray/Shutterstock)

It's virtually everywhere, in everything. Some have referred to it as one of the "Rosetta Stones of physics" (Wired), while others have called it "the outward sign" of one of the a priori forms in philosopher Immanuel Kant's renowned theory of perception, which essentially means that it may represent a fundamental description of how nature must be perceived and understood.

The equation is named after Pierre-Simon Laplace, a French mathematician often listed as one of the greatest scientists of all time, occasionally called the "French Newton" or "Newton of France." In fact, he's responsible for ostensibly solving at least one major problem that Newton never could: how the solar system remains relatively stable over astronomical timescales (though the solar system is today considered to be a chaotic system.)

Laplace even came close to being the first to consider the existence of black holes, having suggested that there could be massive stars with gravity is so great that not even light could escape from their surface.

Solutions derived from Laplace's equation are essentially what make up "potential theory," or the study of harmonic functions. In nature, harmonics occur in mediums that experience periodic motion or oscillation motion, which might help to intuit why this equation can have such a wide application in our understanding of things like electrical fields, gravitational fields or fluid dynamics.

In fact, the term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials that satisfy Laplace's equation.

Interestingly, the equation is so applicable that, unlike many other thinkers of his time, Laplace famously claimed to have no need for the hypothesis of God's existence. To describe the motions of the heavenly bodies in the solar system, he claimed to only need to know the arrangement of the planets and the laws of motion, thus eliminating the need for postulating the intervention of a "supreme intelligence."

That's a powerful equation. And yet, such a simple one. Whether the equation is truly written into the functions of our own minds, or a thought in the mind of God, or some essential sequence that's woven into the fabric of nature, it's nevertheless beautiful. One might even say harmonious.