The Architect of Discrete Geometry: A Profile of Jacob E. Goodman (1933–2021)
Jacob E. Goodman was a seminal figure in modern mathematics who played a pivotal role in establishing Discrete and Computational Geometry as a recognized, cohesive field of study. While many mathematicians labor in established niches, Goodman was a "field-builder," creating the journals, handbooks, and theoretical frameworks that allowed a disparate collection of geometric problems to become a rigorous academic discipline.
1. Biography: From New York to the Frontiers of Geometry
Jacob Eli Goodman was born on November 15, 1933, in New York City. A product of the city’s rich intellectual environment, he pursued his higher education entirely within the prestigious institutions of Manhattan. He earned his B.A. from New York University and moved to Columbia University for his graduate studies, where he earned his Ph.D. in 1962. His doctoral dissertation, On the Cohomology of Algebraic Varieties, was supervised by the eminent mathematician Heisuke Hironaka (a future Fields Medalist).
Goodman joined the faculty of the City College of New York (CCNY), part of the City University of New York (CUNY) system, in the early 1960s. He remained there for his entire professional career, eventually becoming a Professor Emeritus. His career spanned the transition of mathematics from a purely pen-and-paper pursuit to one deeply integrated with computer science—a transition he helped facilitate.
2. Major Contributions: Points, Permutations, and Pancakes
Goodman’s work bridged the gap between combinatorics (the study of counting and arrangement) and geometry (the study of shape and space).
Allowable Sequences of Permutations
In collaboration with his long-time research partner Richard Pollack, Goodman introduced the concept of "allowable sequences." This was a breakthrough in understanding the combinatorial properties of finite sets of points in a plane. By translating geometric configurations into sequences of permutations, Goodman and Pollack provided a way to study the "topology" of point sets without needing exact coordinates. This work led to the discovery of "non-realizable" configurations—sets of sequences that look like they should come from points in a plane but mathematically cannot.
Pancake Sorting (The "Harry Dweighter" Problem)
In 1975, Goodman posed a seemingly whimsical problem in the American Mathematical Monthly under the punning pseudonym Harry Dweighter ("Harried Waiter").
What is the minimum number of flips required to sort a stack of different-sized pancakes using a spatula?
While it sounds trivial, "Pancake Sorting" became a classic problem in theoretical computer science. It notably attracted the attention of a young Bill Gates, who co-authored a 1979 paper providing an improved upper bound for the problem. It remains the only academic math paper Gates ever published.
Defining a Field
Perhaps his greatest contribution was not a single theorem, but the synthesis of a field. In the mid-1980s, Goodman recognized that researchers in computer science and pure mathematics were working on similar geometric problems but not talking to one another. He acted as the "connective tissue" that bound these groups together.
3. Notable Publications
- On the Combinatorial Type of Sets of Points in the Plane (1980): Co-authored with Richard Pollack, this paper laid the groundwork for the theory of allowable sequences.
- Pancake Sorting (1975): Published as a problem in American Mathematical Monthly, launching a sub-genre of combinatorial research.
- The Handbook of Discrete and Computational Geometry (1997, 2004, 2017): Co-edited with Joseph O'Rourke. This massive volume is universally regarded as the "Bible" of the field. It consolidated thousands of papers into a structured reference that defined the state of the art.
- Discrete & Computational Geometry (Journal): In 1986, Goodman and Pollack founded this journal (published by Springer). It became the premier venue for research in the field and remains highly influential today.
4. Awards and Recognition
Goodman’s honors reflect his status as a foundational pillar of the mathematical community:
- Fellow of the American Mathematical Society (2012): He was selected for the inaugural class of AMS Fellows, recognized for his contributions to discrete and computational geometry.
- The Goodman-Pollack Theory: The theoretical framework he developed with Richard Pollack is named in their honor and remains a staple of graduate-level geometric combinatorics.
- Founding Editor Emeritus: His leadership of the journal Discrete & Computational Geometry for over 30 years earned him a status of "elder statesman" in the math world.
5. Impact and Legacy
Jacob Goodman’s legacy is visible in every modern application of computational geometry. When a robotic arm calculates a path to avoid obstacles, or when a geographic information system (GIS) processes map data, they use algorithms rooted in the discrete geometry Goodman helped formalize.
He transformed a collection of "geometric curiosities" into a rigorous mathematical discipline. By creating the Handbook and the DCG Journal, he provided the infrastructure that allowed subsequent generations of mathematicians—including those working in data science and AI—to thrive.
6. Collaborations
Goodman was a famously social and collaborative mathematician.
- Richard Pollack: Their partnership was one of the most productive in 20th-century geometry, spanning decades and dozens of papers.
- Joseph O'Rourke: A computer scientist with whom Goodman edited the Handbook, representing the bridge between the "math" and "CS" sides of the field.
- The CUNY Geometry Seminar: Goodman was a fixture of the New York math scene, mentoring dozens of PhD students at the CUNY Graduate Center and fostering a vibrant research community in Manhattan.
7. Lesser-Known Facts
- The Musical Mathematician: Goodman was a deeply talented composer and musician. He didn't just "enjoy" music; he wrote it with professional rigor. He composed several operas and musical works, often finding parallels between the structure of a mathematical proof and the structure of a musical score.
- The Gates Connection: For decades, the "Gates-Papadimitriou" bound for pancake sorting (inspired by Goodman's problem) was the gold standard, until it was finally improved upon in 2008 by a team at the University of California, Berkeley.
- Pseudonyms: He enjoyed the playful side of mathematics. Using the name "Harry Dweighter" allowed him to pose "recreational" problems that actually had deep, unsolved complexities beneath the surface.