There’s a running joke among evolutionary biologists that given enough time, everything wants to become a crab. This isn’t entirely hyperbole, at least five separate groups of crustaceans have independently evolved into crab-like forms over millions of years. This phenomenon, called carcinization, has become something of an internet sensation, spawning memes about crabs being the “ultimate life form” and jokes about humanity’s eventual crabby destiny.<\/p>\n\n\n\n
But beneath the humor lies a profound truth about how complex systems evolve. Whether we’re talking about biological organisms, mathematical discoveries, or even social media platforms, there seems to be an inexorable pull toward certain optimal forms. These “attractors” in the space of possibilities reveal something fundamental about the nature of innovation, discovery, and adaptation.<\/p>\n\n\n\n
Carcinization describes how at least five groups of decapod crustaceans, including king crabs, porcelain crabs, and various hermit crab relatives, independently evolved a crab-like body plan with a wide, flattened shape and a bent, reduced abdomen. The term was coined in 1916 by biologist Lancelot Alexander Borradaile, who described it as “the many attempts of Nature to evolve a crab.”<\/p>\n\n\n\n
But why crabs? The answer lies in the concept of convergent evolution, when unrelated species independently evolve similar traits in response to similar environmental pressures. The crab body plan offers specific advantages: the ability to walk sideways means crabs can make speedy exits in either direction without losing sight of predators, and their flattened bodies allow them to hide in crevices and under rocks. In the ecological niche of the seafloor, the crab form is simply one of the best solutions available.<\/p>\n\n\n\n
This isn’t unique to crabs. Camera eyes evolved independently in humans and octopuses. Echolocation developed separately in both whales and bats. Flight evolved independently in birds, bats, pterosaurs, and insects. When faced with similar challenges, life repeatedly discovers similar solutions.<\/p>\n\n\n\n
Perhaps nowhere is convergent evolution more striking than in mathematics, where the constraints aren’t physical but logical. The history of mathematics is littered with examples of independent, simultaneous discoveries that would seem almost impossibly coincidental if they weren’t so common.<\/p>\n\n\n\n
The most famous example is the independent development of calculus by Newton and Leibniz. Newton developed his “Method of Fluxions and Infinite Series” around 1666, while Leibniz began working on his differential calculus in 1674, publishing it in 1684. Despite their different approaches, Newton relied more on geometric intuition with concepts rooted in kinematic problems, while Leibniz approached calculus from an algebraic mindset, they arrived at essentially the same mathematical framework.<\/p>\n\n\n\n
This wasn’t a fluke. As Newton himself acknowledged: “If I have seen further than other men, it is because I stood on the shoulders of giants.” Both men were building on centuries of accumulated mathematical knowledge, and the time was simply ripe for calculus to emerge.<\/p>\n\n\n\n
The phenomenon continues today, often at an accelerated pace. In May 1953, Robinson and Frame proved the hook length formula on a Thursday. When they presented it that Saturday at the University of Michigan, Thrall in the audience had independently proved the same result on the same day.<\/p>\n\n\n\n
Even more remarkably, in 2022, Terence Tao observed “Maths at internet speed” when Justin Gilmer made a breakthrough on the union-closed sets conjecture, and within a day, other mathematicians were already building on and extending his work. The internet has transformed mathematical convergent evolution from a years-long process to one that can happen in days or even hours.<\/p>\n\n\n\n
Sometimes convergent evolution in mathematics takes an unexpected form, when someone outside the established community independently rediscovers or disproves long-standing beliefs. Seventeen-year-old Hannah Cairo stunned the mathematics community by disproving the Mizohata-Takeuchi conjecture, a 40-year-old problem in harmonic analysis. After months of failed attempts to prove it, she shifted gears and constructed a counterexample using fractals.<\/p>\n\n\n\n
Cairo’s breakthrough illustrates another aspect of mathematical convergent evolution: sometimes fresh perspectives, unburdened by conventional thinking, can see solutions that elude experts who have been immersed in a problem for decades.<\/p>\n\n\n\n