In the 1960s, the Soviet climatologist and mathematician Mikhail Budyko set out to investigate the potential future of a planet on the brink of nuclear Armageddon. He started by looking some 600 million years into the past. Back then, some scientists claimed, the ancient planet was an iced-over snowball. Most researchers considered that a crackpot theory. Ice over the equator? Please. But Budkyo developed a mathematical model to back it up. If sea ice had been able to expand past a critical latitude, he suggested, then its reflective surface would have returned more sunlight to space. This would have kicked off an out-of-control feedback loop: The planet would cool further and ice would build up until it spread everywhere. The Earth would, in other words, tip from one equilibrium into a different one, reaching a new stable — and frozen — state. Budyko’s investigation was motivated by a pressing question: If the global climate had tipped dramatically and catastrophically in the past, could humans tip it in the present? He and others feared what would happen if the United States and the Soviet Union launched their nuclear arsenals. “They realized, look, if we block the sun for sufficiently long, we’re going to just destroy life on the planet,” said Valerio Lucarini, a statistical physicist studying the Earth’s climate at the University of Leicester. “Not even the cockroaches will survive.” The missiles didn’t launch. But it turned out that nukes weren’t necessary for humans to tip the climate. By the time Budyko was building his Snowball Earth models, it was clear that atmospheric carbon dioxide was rising, and with it global temperatures. Since then, mathematicians have uncovered the potential for abrupt and radical shifts in Earth’s climate — known popularly as tipping points. The loss of sea ice could cause the oceans to absorb more of the sun’s heat, crossing a threshold that kicks off runaway ice melt and rising seas. The Amazon rainforest could wither into a savanna; coral reefs could bleach ghost-white; a major current in the Atlantic Ocean might go slack and fail to deliver warmth to Europe, turning Scotland into Siberia. Tipping points often capture the worst-case scenarios of climate models: the reorganization of the world we know, and the human civilization we’ve built within it, into a new equilibrium state — an unimaginable, frightening unknown. Yet the math of tipping points is fraught with uncertainty. The Earth is certainly warming, and the effects of that warming, if left unchecked, will be dire. But tipping points are subtler phenomena. Slight changes in the assumptions a mathematical model is built on can cause tipping points to unfold very differently or even slip away entirely. And in most cases, scientists are armed with relatively little data, making it challenging to understand the chaotic nature of tippable climate systems, much less predict where they’re headed.