These Are the Most Beautiful Equations in Mathematics | Scientific American

June 20, 2024<br>8 min read<br>Add Us On GoogleAdd SciAm<br>These Are the Most Beautiful Equations, according to Mathematicians

Mathematicians picked the most dazzling, thought-provoking and compelling equations they know

By Rachel Crowell edited by Clara Moskowitz

Ailana Fraser

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To mathematicians, equations are art. Just as many are moved by a painting or piece of music, to those who appreciate and understand math, expressions of numbers, variables, operations and relations between quantities can be just as compelling.<br>As is the case with artistic beauty, mathematical beauty is in the eye of the beholder. One mathematician may prize simple-to-state, succinct equations, while another may favor the opposite. And just as some favor modern art while others prefer medieval works, both ancient and contemporary equations are admired for their cleverness, power and possibility.<br>Here are some mathematicians&rsquo; selections of the most beautiful math equations, as well as their own words on what factors make them so exquisite.<br>On supporting science journalism<br>If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<br>Loewner Differential Equation<br>Ewain Gwynee

Some equations are beautiful because they reveal unexpected relationships between different subjects. The Loewner differential equation, introduced by Charles Loewner in 1923, describes the time evolution of a family of conformal (angle-preserving) functions defined on subsets of the complex plane. The input for the equation is a continuous function W(t), called the driving function.<br>Nearly 80 years later, in 1999, Oded Schramm discovered that the solution to the Loewner equation has special symmetries when the driving function is taken to be Brownian motion, a random function that is a central object of study in probability theory. Building on Schramm&rsquo;s discovery, it was proven that the solution to the Loewner equation for this choice of driving function, called Schramm-Loewner evolution, describes the large-scale behavior of various critical models in two-dimensional statistical mechanics. This application of the Loewner equation in probability theory was completely unexpected prior to Schramm&rsquo;s work. It revolutionized the mathematical study of statistical mechanics and has led to some of the most exciting breakthroughs in mathematics in the past two decades. —Ewain Gwynne, University of Chicago<br>Gauss-Bonnet Equation<br>Ailana Fraser

The Gauss-Bonnet formula is a beautiful equation in differential geometry that asserts the equality of the integral of the Gauss curvature over a surface and a constant multiplied by the Euler characteristic of the surface. The equation is remarkable because it relates two very different quantities: the curvature of the surface on the one hand and the topology of the surface on the other hand. The curvature of a surface at a point is a measure of the bending of the surface at that point, or how much the surface deviates from being a plane. The Euler characteristic is a global topological invariant of the surface that describes the topological structure of the surface regardless of how it is bent. The equation is surprising because, for example, it implies that if you continuously deform a surface, the total curvature will remain unchanged. Versions of the Gauss-Bonnet equation were first formulated by Carl Friedrich Gauss and Pierre Ossian Bonnet in the first half of the 19th century, and it remains one of the most beautiful and striking equations in geometry. Some beautiful aspects of this equation are the simplicity of its expression and the profoundness of what it says. —Ailana Fraser, University of British Columbia<br>Sobolev Inequality<br>Karen Uhlenbeck

I love inequalities. The Sobolev and related inequalities estimate functions in terms of derivatives and form the basis for our understanding of partial differential equations. The failure of the inequality for n = 2 relates to properties special to n = 2. —Karen Uhlenbeck, Institute for Advanced Study<br>Riemann-Roch Equality<br>S&aacute;ndor Kov&aacute;cs

A Riemann surface is a compact orientable surface. Riemann surfaces are distinguished by the number of holes or handles they admit. This number is called the genus, and it is denoted by g. For instance, a Riemann surface of g=0 is a sphere, and a Riemann surface of g=1 is...