Birthday Celebration Fit for a Renaissance Man

The year 2014 marks 500 years since the creation  of Albrecht Dürer’s Melencolia I, one of his three “Master Prints” widely considered the pinnacle of classical printmaking. Dürer, a Renaissance man, incorporated into his prints his world view and his deep interest in science, especially mathematics. Among the prints, Melencolia I holds a special place, influencing many generations of artists, philosophers, scientists, mathematicians, and students of science.

Two events in New York City on the weekend closest to his birthday (May 21) will take stock of the progress of science influenced by Dürer and reexamine his contribution to the arts.

On Saturday, May 17, from 10:15 a.m. until 6:30 p.m., a scientific conference —aimed at mathematics, computer science, engineering, and science appreciators as young as high school age as well as scholars—will feature prominent mathematicians, engineers, and scientists working in fields pioneered by Dürer. Internationally recognized mathematicians David and Gregory Chudnovsky, Distinguished Industry Professors at the New York University Polytechnic School of Engineering, have organized the conference, sponsored by the Alfred P. Sloan Foundation. It will be held in the Pfizer Auditorium of the NYU Polytechnic School of Engineering’s Dibner Building, 5 MetroTech Center, in Downtown Brooklyn. The event is free and open to the public. Video will be streamed at engineering.nyu.edu/live. Registration is optional here.

The conference, “500 Years of Melancholia in Mathematics,” will present achievements of the late 20th and early 21st centuries in areas of mathematics inspired by Dürer. These include three-dimensional geometry and topology, combinatorics of two-dimensional arrays, and the more recent area of computer science, whose roots reach back 500 years. Modern wireless technology, molecular biology, and encryption also trace to Dürer and will be highlighted during the conference.

On Sunday, May 18, from 3 to 4:30 p.m., The Metropolitan Museum of Art will host a series of lectures, “Spotlight on a Masterpiece: Albrecht Dürer's Melencolia I.”  Nadine M. Orenstein, the museum’s Curator of the Department of Drawings and Prints, will welcome and introduce speakers Angela B. Campbell, the museum’s Assistant Conservator of the Sherman Fairchild Center for Works on Paper and Photograph Conservation; Susan Dackerman, the Carl A. Weyerhaeuser Curator of Prints at the Harvard Art Museums; Laurinda Dixon, Professor of Art History, Department of Art and Music Histories at Syracuse University; and Mitchell B. Merback, Associate Professor, Department of the History of Art, The Johns Hopkins University. The event is free with admission to the museum.

Dürer deeply influenced scientists, particularly in Northern Europe in the 16th through 18th centuries. The images in Melencolia I have been scrutinized for 500 years.  The unusual “Dürer’s polyhedron” on the left side of the print became more relevant in the late 20th century in applied sciences and mathematics (in relation to quasicrystals, discovered by Nobel laureate Dan Shechtman).  An allusion to the impossible problem of squaring the circle, occupying the center of the print, plays a role in modern number theory. 

Dürer’s magic square, on the other hand, branched in many mathematical directions and continues to be the source of difficult problems and new fields within the area of mathematics called combinatorics.

Dürer’s interest in 3D solids influenced the astronomer and mathematician Johannes Kepler.  Dürer’s polyhedra were followed by Kepler polyhedra.  There is an interesting tradition in mathematical education, going back to Dürer, of teaching geometry by folding polygons into 3D polyhedra.  Dürer was the first to publish this approach in 1525. The area related to Dürer’s polyhedra is still an active subject of research.

The early years of the 21st century saw remarkable progress in solving the last outstanding problems in 3D geometry and 3D topology. The most remarkable achievement was the solution of the Kepler conjecture on the densest sphere packing in 3D space.  This conjecture, outstanding since 1611, was finally solved through a computer-assisted proof. 

Sphere packing in dimensions higher than 3 is still largely unsolved.  Many variations of the sphere packing problem became crucial in modern information and communication theory and molecular biology.  These include error-correcting codes, which underpin all modern communication and information theory.  The ubiquitous cell phone would not be possible without good solutions to a general sphere packing problems.

No mathematical table has inspired the interest of the general public like the Dürer magic square, which is prominently displayed in Melencolia I.  Magic squares continue to be an inspiration for mathematicians and even for mathematical enthusiasts, carving a large area in recreational mathematics.  Even though Dürer had not invented magic squares, he is probably the first to widely disseminate the concept in Europe, via Melencolia I, and therefore responsible for their rigorous study to this day. 

As late as the 20th century, the problem of enumeration of magic squares became a part of a more general set of problems in number theory. It turned out that the exponential cost of enumeration required new approaches. New algorithms allowed Richard Schroeppel—a speaker at the May 17 conference—to determine the number of (non isomorphic) magic squares of order 5 to be 275,305,224. Attempts to solve magic squares of order 6 introduce new concepts related to multidimensional hypergeometric identities and commutative algebra—Stanley’s theory of magic squares. Richard Stanley, too, will speak at the conference.

One of the most important extensions of magic squares began with a paper of the great mathematician Leonhard Euler published in 1776, which was then refined in the 19th century, and applied recently for the efficient design of statistical experiments as well as for frequency-hopping wireless communications.

These and more trace their roots to Dürer and Melencolia I.

“The twin Art and Mathematics conferences will provide a unique opportunity to reflect upon the benefits and disadvantages of the separation of ‘Two Cultures,’ which seemed to be in perfect harmony in the person of Albrecht Dürer,” said NYU Professor David Chudnovsky. Added Gregory Chudnovsky: “Continuing interest in the symbolism of Dürer’s art among the engineers and scientists is one of the remaining links that still connects divergent areas of human ingenuity.”

Lectures will include:

• John Conway, Princeton University—“Magic Squares, Including Frenicle's 880 of Order Four”
• Sergiu Klainerman, Princeton University—“Are Black Holes Real?”
• Jeffrey Lagarias, University of Michigan—“Dürer, Polyhedra and Shadows”
• John W. Morgan, Stony Brook University, "Geometry and Topology: From Gauss and Riemann to the Modern Day”
• Richard Schroeppel, Sandia National Lab—“Magic Cubes and Hypercubes”
• Richard Stanley, Massachusetts Institute of Technology—“Magic Squares and Syzygies”
• Günter M. Ziegler, Free University of Berlin—“Three Giants, Five Stars, Some Mistakes: Leonardo, Dürer, Kepler, and Their Polyhedra”