In triangle geometry, he invented a formulation that resulted in simpler formulas (Conway triangle notation). The Conway circle is also named after him: If the sides of a triangle are each extended by the length of the opposite side of the triangle, then the end points of these segments lie on a circle with radius (R = sqrtr^2+s^2 ), where (s= frac12 cdot (a+b+c) ) half the perimeter of the triangle and right are the incircle radius. The center of the Conway circle is also the incircle center of the triangle.
He also brought new order to knot theory by introducing a new notation for knots and correcting errors in old tables. He also found the Conway knot named after him with eleven crossings.
Through his diverse contributions, Conway coined a number of terms that have been widely adopted: When his friend, the English physicist and mathematician Roger Penrose, examined tessellations with “golden” triangles in the 1970s, Conway gave the two basic shapes the designations “darts” (arrows ) or »kites« (kites). Conway suggested adding additional arcs to these basic elements. The radius of these curves is chosen so that the “kites” and “darts” sides are each divided in the ratio of the golden ratio. In addition, matching rules should apply: Only those »kites« and »darts« may be placed next to each other in which the arcs of the same color merge into one another.
© Whiteway / Getty Images / iStock; Processing: spectrum of science (excerpt)
Penrose Paving | This Penrose paving of the plain can be continued ad infinitum. It is five-fold symmetrical about its center; and although the same motifs appear again and again, it never becomes periodic.
According to this rule, wonderfully symmetrical figures can be gradually inflated (here Conway coined the term “inflation”, as well as “star pattern” and “sun pattern”). Conversely, the golden triangles can also be broken down (»deflation«).
Conway also had the original idea of using footprints to characterize the seven possible types of frieze ornament: F1 (hop), F2 (jump), F3 (sidle), F4 (spinning hop), F5 (step), F6 ( spinning side), F7 (spinning jump).
© Heinz Klaus Strick (detail)
In order to analyze games, Conway invented the so-called surreal numbers in 1974 – a class of numbers that includes real numbers as well as infinitesimally small or infinitely large ones. He considered it the most important discovery of his life.
In 1986 Conway left Cambridge and accepted a call to the renowned John von Neumann Chair of Applied and Computational Mathematics in Princeton. He has received numerous honors and awards over the decades, including being the first laureate of the London Mathematical Society’s Pólya Prize.
Throughout his academic life, Conway was restless and constantly coming up with new ideas. His study was overflowing with manuscripts and colorful home-made models; yet he almost always found what he had jotted down on some piece of paper. He usually wore sandals year in and year out, in summer he also liked to go around barefoot; he did not change this habit from his student days even in old age. He happily accepted invitations to give popular science lectures or summer camps for young people, and initially had the audience hanging on his every word vote on the topic of his lecture (from a spontaneously compiled list of ten suggested topics); however, it also happened that he forgot such an appointment. With his unconventional way of lecturing, he not only fascinated laypeople, but also experts, such as the 3,000 listeners at the International Mathematics Congress in Zurich in 1994.
He gave his students two thoughts to take with them:
- Take it as axiomatic that you are stupid. If you think you have proved something, think again. Find the holes in your own proofs.
- If you have indeed discovered something, but then discover that someone else discovered it before you, consider yourself in good company, and mark your progress. If you find something already discovered 2000 years ago, then 200, then 20, at least you are improving. And then, if you’re lucky, next maybe you’ll discover something new.
Two of his marriages failed, not only because of numerous affairs – a total of seven children emerged from his three marriages. After the failure of the second marriage, he attempted suicide and suffered from depression. His unhealthy lifestyle led to two heart attacks, among other things, but he always recovered – until a severe stroke in 2018: he had to go to a nursing home, where he still received regular visitors until the beginning of 2020 because of the Covid 19 pandemic visits were no longer possible
Siobhan Roberts writes about him in her biography: »(Conway is) a singular mathematician with a lovely loopy brain. He is Archimedes, Mick Jagger, Salvador Dali, and Richard Feynman all rolled into one—a singular mathematician, with a rock star’s charisma, a sly sense of humor, a polymath’s promiscuous curiosity, and a burning desire to explain everything about the world to everyone in it.«