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The eight-year-old Roger Penrose was not a promising mathematician. He was so slow at mental arithmetic that his school held him back a year. Only after a sympathetic teacher gave him extra time to complete his mathematics tests did Penrose catch up. In those days, he was outshone by his precocious elder brother Oliver, who was a powerful influence on him and later became an accomplished theoretical physicist. Their two younger siblings were also outstandingly talented: their brother Jonathan became a chess grandmaster, their sister Shirley, a professor of cancer genetics. In later life, Roger remembered receiving generous emotional support from his mother and a good deal of intellectual stimulation from his father Lionel, a professor of genetics with a keen interest in mathematics and astronomy.

Roger’s mathematical talent began to flower at senior school. He thought about the subject in a distinctive way, not much interested in manipulating symbols, but given to thinking about mathematics geometrically, using his powerful visual imagination. He loved to draw mathematical shapes and make cardboard models of the symmetric geometric shapes known as polyhedra, developing skills he would use later in his career. In the early 1950s, he studied mathematics at University College London, while also cultivating an interest in the cosmos, his interest having been piqued by the popular lectures on BBC radio delivered by Fred Hoyle, a splendidly imaginative astrophysicist and a gifted popularizer. In the spring of 1952, Roger spent a weekend in Cambridge with his brother Oliver, a PhD student, and a friend of one of his roommates, the astrophysicist Dennis Sciama. During a restaurant meal, Oliver introduced his brother to Sciama, who happened to be dining there and sitting nearby. This was to be perhaps the most consequential meeting of Penrose’s life.

Dennis Sciama was shaping up to be one of the talented physicists who chose to focus not on this or that phenomenon but on the entire universe – everything that exists. No less important for the scientific community, he had a prodigious generosity of spirit, always seeking to support his colleagues – and, later, his many students – by highlighting research areas that would benefit most from their talents. Sciama saw that Penrose was a thinker of unusual talent and originality, a rare example of a first-class mathematician with the ability to identify key problems in physics and the determination to solve them. Like all mathematicians in the past few centuries, he had a keen eye for symmetry, a quality that had become no less important for theoretical physicists.

The main theme of those early conversations between Penrose and Sciama was the nascent discipline of cosmology, the study of the structure and development of the universe, apparently shaped by gravitational forces. Einstein had pioneered modern cosmology using his theory of gravity, which he had developed by generalizing his 1905 special theory of relativity, which modified Newton’s view that space and time are independent concepts. Two years later, the German mathematician Hermann Minkowski introduced the unified concept of space-time, which Einstein used to forge his theory of gravity, which he built from geometric concepts. According to Einstein, space-time can be represented by curved surfaces on which the angles of a triangle do not usually add up to 180 degrees, as they always do on a flat piece of paper. The force of gravity of common experience should be re-interpreted as the curvature of space-time, Einstein argued – not an easy transition for most of us to make. For the geometrically minded Penrose, however, it was a piece of cake. Yet he knew that Einstein’s theory was extremely difficult to apply to the cosmos and therefore to check, mainly because its mathematical structure was so difficult to understand.

In the autumn of 1952, Penrose began his PhD at Cambridge, supervised by William Hodge, a top-quality pure mathematician. It was not the happiest of academic relationships for Penrose, because the mathematics Hodge suggested he study was too algebraic, insufficiently geometric. Thrown out by Hodge after only a year, Penrose spent his remaining postgraduate years advised by the more sympathetic John Todd, and finally took his PhD in 1957. During those postgraduate years, Sciama had taken Penrose under his wing, repeatedly urging him, “What are you doing with this pure mathematics nonsense? Come and work on physics and cosmology”. Penrose was not quite ready to take that advice – he was determined to complete his PhD – but chose to moonlight in the world of physics, attending lectures on quantum mechanics by the theoretical physicist Paul Dirac and energetic talks by the cosmologist Hermann Bondi. Inspired by these presentations and won over by the charm and persuasiveness of Sciama, Penrose switched his focus to physics. It was a wise move for him and a hugely productive one for physics.

In the early 1960s, the renaissance in gravity theory was well underway, though still a backwater in physics research. Most of its advances were being made by only a few dozen experts. Penrose, then in his early thirties, was rapidly coming to the fore as one of the subject’s most powerful thinkers, well known in the community for his originality and mathematical skill. Urbane, quietly-spoken and unassuming, he was a popular member of the research community, wary of bandwagons and happiest when working alone. He later recalled: “It was important for me always … to think I had a different angle on it from other people. Because I wasn’t good at following where everybody else went”.

His methods were delivering such good results that by the early 1960s he had secured a senior faculty appointment at Birkbeck College, London. In the autumn of 1964, he made his most startling and impressive advance, later to win him a Nobel Prize. The story of the discovery began when Sciama suggested that Penrose attend a lecture by the distinguished American theorist David Finkelstein about a controversial aspect of the equations at the heart of Einstein’s theory of gravity. It seemed to some experts that the equations often generated “singularities” tiny regions where the mathematics of the theory goes haywire. The curvature of space-time in these regions becomes not just extremely large but infinite, impossible to deal with using conventional techniques. Other experts believed that the existence of such points in the theory was illusory. The idea that turned out to be crucial to his new way of thinking about these singularities occurred to him when he was crossing a London street, Penrose later recalled. From this epiphany, he developed an innovative way of applying Einstein’s theory to a star imploding under its own gravity to form an object now known as black hole (the term became popular only in 1967, after it was promoted by the physicist John Wheeler, three years after the American journalist Ann Ewing first used it in print). Penrose crystallized his reasoning in the form of a rigorous mathematical theorem that told physicists something new and revealing about the cosmos and our understanding of it: if they used Einstein’s theory to describe any imploding star whatsoever, then the presence of a singularity in the region was not only likely; it was inevitable. In modern language, he had proved that a black hole must contain a singularity.