Paul Erdős, the famously eccentric, peripatetic and prolific 20th-century mathematician, was fond of the idea that God has a celestial volume containing the perfect proof of every mathematical theorem. “This one is from The Book,” he would declare when he wanted to bestow his highest praise on a beautiful proof.

Never mind that Erdős doubted God’s very existence. “You don’t have to believe in God, but you should believe in The Book,” Erdős explained to other mathematicians.

In 1994, during conversations with Erdős at the Oberwolfach Research Institute for Mathematics in Germany, the mathematician Martin Aigner came up with an idea: Why not actually try to make God’s Book — or at least an earthly shadow of it? Aigner enlisted fellow mathematician Günter Ziegler, and the two started collecting examples of exceptionally beautiful proofs, with enthusiastic contributions from Erdős himself. The resulting volume, Proofs From THE BOOK, was published in 1998, sadly too late for Erdős to see it — he had died about two years after the project commenced, at age 83.

“Many of the proofs trace directly back to him, or were initiated by his supreme insight in asking the right question or in making the right conjecture,” Aigner and Ziegler, who are now both professors at the Free University of Berlin, write in the preface.

The book, which has been called “a glimpse of mathematical heaven,” presents proofs of dozens of theorems from number theory, geometry, analysis, combinatorics and graph theory. Over the two decades since it first appeared, it has gone through five editions, each with new proofs added, and has been translated into 13 languages.

In January, Ziegler traveled to San Diego for the Joint Mathematics Meetings, where he received (on his and Aigner’s behalf) the 2018 Steele Prize for Mathematical Exposition. “The density of elegant ideas per page [in the book] is extraordinarily high,” the prize citation reads.

Quanta Magazine sat down with Ziegler at the meeting to discuss beautiful (and ugly) mathematics. The interview has been edited and condensed for clarity.

You’ve said that you and Martin Aigner have a similar sense of which proofs are worthy of inclusion in THE BOOK. What goes into your aesthetic?

We’ve always shied away from trying to define what is a perfect proof. And I think that’s not only shyness, but actually, there is no definition and no uniform criterion. Of course, there are all these components of a beautiful proof. It can’t be too long; it has to be clear; there has to be a special idea; it might connect things that usually one wouldn’t think of as having any connection.

For some theorems, there are different perfect proofs for different types of readers. I mean, what is a proof? A proof, in the end, is something that convinces the reader of things being true. And whether the proof is understandable and beautiful depends not only on the proof but also on the reader: What do you know? What do you like? What do you find obvious?

You noted in the fifth edition that mathematicians have come up with at least 196 different proofs of the “quadratic reciprocity” theorem (concerning which numbers in “clock” arithmetics are perfect squares) and nearly 100 proofs of the fundamental theorem of algebra (concerning solutions to polynomial equations). Why do you think mathematicians keep devising new proofs for certain theorems, when they already know the theorems are true?

These are things that are central in mathematics, so it’s important to understand them from many different angles. There are theorems that have several genuinely different proofs, and each proof tells you something different about the theorem and the structures. So, it’s really valuable to explore these proofs to understand how you can go beyond the original statement of the theorem.

An example comes to mind — which is not in our book but is very fundamental — Steinitz’s theorem for polyhedra. This says that if you have a planar graph (a network of vertices and edges in the plane) that stays connected if you remove one or two vertices, then there is a convex polyhedron that has exactly the same connectivity pattern. This is a theorem that has three entirely different types of proof — the “Steinitz-type” proof, the “rubber band” proof and the “circle packing” proof. And each of these three has variations.

Any of the Steinitz-type proofs will tell you not only that there is a polyhedron but also that there’s a polyhedron with integers for the coordinates of the vertices. And the circle packing proof tells you that there’s a polyhedron that has all its edges tangent to a sphere. You don’t get that from the Steinitz-type proof, or the other way around — the circle packing proof will not prove that you can do it with integer coordinates. So, having several proofs leads you to several ways to understand the situation beyond the original basic theorem.