Johannes Kepler and COVID-19: 400 Years of Mathematical Modeling

This is a guest post by John Hessler, a specialist in computational geography and GIS in the Geography and Maps Division.

Johann Kepler’s sketches of polyhedrals, detail. Rare Books & Special Collections Division.

Viruses are strange creatures. They are not really alive. They are only fragments of genetic code floating about, or marking time in the cells of an unsuspecting host until they begin replicating. Then they make their presence felt in the world. All of us have become more familiar with these facts as the SARS-CoV-2 virus, which causes COVID-19, has come into our lives.

The size of the COVID-19 virus is small by genomic standards. It is made up of 29,000 base pairs of the genetic material that is at the root of all life on earth. By comparison, you would need 3 billion of these base pairs to make a mouse from scratch.

When viruses like COVID-19 enter humans, they begin to mutate as their genetic material comes and goes through human cells. In this way the virus evolves, sometimes becoming better at infecting its hosts, sometimes not. The geometry and shape of the each of the individual virus particles — known as virions — is critical to the viruses’ success. It affects how, in the case of COVID-19, the virus jumped from its animal host to humans, and then how it spreads among humans.

Because viruses have such small genetic codes they have to be very efficient when it comes to building their shapes. Many, such as COVID-19, have polyhedral designs. A polyhedron is a three-dimensional shape with flat faces that are made of two-dimensional polygons. They have straight edges and sharp corners. As geometric figures, they have been studied since the time of the ancient Greeks.

In the early 17th century the astronomer Johannes Kepler, perhaps best known for his derivation of the laws of planetary motion, was fascinated by these shapes. In 1619, exactly 400 years before the outbreak of COVID-19, he produced a book called “Harmonices Mundi” (“Harmony of the Worlds”) that peered into and tried to understand these simple yet mysterious forms. The Library has copies of this work in the Rare Books and Special Collections Division.

Kepler, a genius polymath with an esoteric streak, thought that many of these shapes were universal and critically important to understanding the structure of the world. He wondered how they participated in the functioning and mechanics of the heavens. He thought they were at the foundation of the harmonies we hear in music. Determined to get a handle on this, he unfolded these complex shapes to see what simpler geometric forms, like triangles and squares, made them up, searching for some hidden geometric code.

Tilings of polyhedra from Kepler’s “Harmonices Mundi.” Rare Book and Special Collections Division.

A virus with the polyhedral forms studied by Kepler. Courtesy of Reidun Twarock, Nature Communications.

Today, these are the same shapes that mathematicians and theoretical biologists are using to understand the evolution of viruses. The outside case of a virus is known as a capsid. It is made up of the proteins that hold, in the case of COVID-19, the short strands of RNA that enter cells and infect their hosts. The shape of many viral capsids are built up from the simple forms Kepler studied. Capsids are also highly symmetrical, and because viruses have so little genetic code to work with, they need to conserve the information they have to take advantage of whatever shortcut nature and mathematics allows.

Scholars today are using these same shapes to ponder how viruses function and how these lifeless fragments of genetic material spring into being. In the mysteries of the polyhedral, we see modern epidemiology meeting 17th-century mathematics.

In this simple geometry, the past truly meets our present moment; rare books meet COVID-19. Kepler would have loved this mystery.

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