Geometry's wild decades

In 1963 a meteorologist named Edward Lorenz went to get a coffee, came back, and accidentally discovered that the future is unknowable. Not because of randomness. Because of rounding.

Lorenz was running a toy weather model on a vacuum-tube computer. To re-examine a run, instead of starting over he typed in numbers from a mid-run printout and let it go. When he came back the new weather had nothing to do with the old — the two runs had drifted from "basically identical" to "completely different" in simulated weeks.

The bug wasn't a bug. The printout rounded the computer's internal 0.506127 to 0.506 — one part in a thousand. That difference didn't stay tiny; it doubled, and doubled, and doubled until it swamped the whole forecast. The system was perfectly deterministic: same input, same output, every time. But "same input" turned out to be a fantasy, because no measurement is infinitely precise and any error you start with grows exponentially.

This is the thing to hold onto: deterministic does not mean predictable. The equations have no dice in them. Run them twice from the exact same number and you get the exact same answer forever. But nudge the start by a hair and the trajectories fly apart — at an exponential rate. That sensitive dependence on initial conditions is what we now call chaos, and the popular name for it — the butterfly effect — comes straight from Lorenz, who asked in a talk title whether a butterfly's wings in Brazil could set off a tornado in Texas.

You don't need a weather model to see it. You need one line of arithmetic. The logistic map — x ← r·x·(1−x) — was studied by biologist Robert May in 1976 as a model of how an animal population grows and gets capped by its food supply. Crank the growth rate r and a population that used to settle on one steady value starts oscillating between two, then four, then descends into full chaos. One multiplication and one subtraction. Drag r below and watch it happen.

Back in Chapter 4 we met Weierstrass's "monster" — a curve that wiggles at every scale and has a tangent nowhere. The 19th century filed it under pathology: a counterexample built to embarrass the calculus, the kind of thing real functions would never do. Nature, it turns out, does almost nothing else.

In 1982 Benoit Mandelbrot published The Fractal Geometry of Nature and made the case bluntly: coastlines, clouds, mountains, blood vessels, broccoli, and stock-market charts are all rough at every zoom. Magnify a coastline and you don't get a smooth line — you get more coastline, with the same jagged character, all the way down. The monster wasn't a bug in mathematics. It was the rule that physical shapes actually follow. Mandelbrot coined the word fractal for them, from the Latin for "broken".

The sharp idea inside this is fractal dimension. A smooth line is 1-dimensional; a filled plane is 2-dimensional. A coastline is neither: it's so wrinkled that it fills more space than a line but never enough to be a sheet — so its dimension is a fraction, say 1.25. The more wrinkled the curve, the higher the number climbs toward 2. Dimension stops being a count of axes and becomes a measure of roughness. That's the whole trick, in one breath: a coastline is more than a line, less than a plane.

Here's a question that sounds like it should have a one-line answer and instead ate fifty years: is there a single shape that tiles the whole plane — covers it with no gaps and no overlaps — but can never do so in a repeating pattern?

Every tiling you've ever seen on a bathroom floor is periodic: find the repeating block, slide a copy, and it lands perfectly on itself. The challenge was to forbid that forever. In the 1970s Roger Penrose found tilings that never repeat — but his famous set needed two different tiles. The dream was one. Mathematicians even gave the imaginary shape a name: an einstein, German for "one stone" (the physicist pun is free). For decades nobody knew if it existed.

Then, in November 2022, a retired print technician in Yorkshire named David Smith — a hobbyist who relaxes by cutting shapes out of cardstock and shoving them together — found a 13-sided tile he couldn't get to repeat no matter how he tried. He called it "the hat". Suspecting he'd found something real but unable to prove it, he emailed a mathematician, Craig Kaplan. Kaplan, Smith, software engineer Joseph Myers, and mathematician Chaim Goodman-Strauss spent months turning the hunch into a theorem. In March 2023 they announced it: the hat is an einstein. The fifty-year hunt was over, and it was cracked open by a man with paper cutouts and a hunch.

There was one asterisk: the hat needed a few of its copies to be mirror-flipped. Purists grumbled. So Smith and the team went back and, months later, produced "the Spectre" — a tile that needs no reflection at all. The cleanest possible answer to a question half a century old. Below: a sandbox. The hat is built from kites of a hexagonal grid; drop tiles, rotate, flip, snap them to the grid, and then run the ghost test to feel why no slide ever maps the pattern onto itself.

1917, Japan. Sōichi Kakeya asks a question a child could pose: you have a needle of length 1 lying on a table. You want to rotate it a full 180° — point it in every direction — sliding and turning however you like. What's the smallest area you can do this in?

The obvious move is to spin it about its center, sweeping a disc of radius ½. Area π/4 ≈ 0.785. You can do better: slide-and-turn inside a three-cusped curve called a deltoid and the area drops to about π/8, roughly half. A reasonable person stops here and assumes there's a sensible minimum.

There isn't. In 1928 Abram Besicovitch proved the answer is zero — you can turn the needle through every direction inside a region of arbitrarily small area. The construction (the "Perron tree") chops a triangle into thin slivers and slides them so they overlap almost perfectly, like a hand of cards fanned and then squeezed shut. The slivers still cover every direction the needle needs, but the total ink approaches nothing. It's deeply counterintuitive and completely correct.

That would be a fun curiosity if it stopped there. It didn't. The 3D version — how thin can a set be that contains a unit line segment in every direction? — became load-bearing for harmonic analysis, the math under signal processing and PDEs. The conjecture that such "Kakeya sets" can't be too thin (they must have full dimension) resisted for decades. In February 2025, Hong Wang and Joshua Zahl finally settled the three-dimensional case — a result other mathematicians called "once in a century". Walk through the three stages below.

In 1611 Johannes Kepler looked at how grocers stack oranges — that staggered pyramid on every fruit stand — and conjectured it's the densest possible packing of equal spheres, filling about 74% of space. The grocers were right. It took 387 years to prove it: Thomas Hales finally did in 1998, with a proof so large it leaned on a computer for thousands of cases — and referees who admitted they were only "99% certain". (The 2D version — hexagonal circles are densest — wasn't pinned down until 1940, by László Fejes Tóth.) Proving the obvious is brutal.

Then dimensions 8 and 24 fell, and they fell cleanly. In 2016 Maryna Viazovska, a Ukrainian mathematician, solved sphere packing in dimension 8 exactly — not a computer slog but a few pages built around a single hand-crafted "magic function" that nails the optimal density on the nose. The one-woman proof won her the Fields Medal in 2022, the first ever for sphere packing; within a week of the 8D paper, she and collaborators knocked over dimension 24 too.

Why those dimensions? Because 8 and 24 host two freakishly symmetric lattices — the E8 lattice and the Leech lattice — where the spheres lock together with no slack. And here's the callback to Chapter 8: those exact lattices are the geometry secretly underneath good error-correcting codes. Packing spheres tightly and spacing codewords far apart so noise can't confuse them are the same problem. Your phone's radio is, in a real sense, doing geometry in dimension 8.

The lab below has two faces. Panel A is the 2D truth you can see. Panel B is where intuition dies: in high dimensions, space is almost entirely corners — and that's why packing there is so strange.

One more problem you can hand to a child, that nobody can finish. Color every single point of the infinite plane so that any two points exactly distance 1 apart get different colors. How many colors do you need? This is the Hadwiger–Nelson problem, posed around 1950, and its answer is the chromatic number of the plane.

Here's everything that was known for nearly seventy years: the answer is somewhere between 4 and 7. Four, because you can draw a little graph of points (we'll build it below) that forces it. Seven, because there's a clever way to tile the plane with seven colors of hexagon that never puts the same color a unit apart. Between 4 and 7. For decades, dead still.

Then in April 2018 the lower bound moved — and the person who moved it has no math job. Aubrey de Grey is a biologist, famous for arguing that aging is an engineering problem we might one day cure. As a hobby, he chips at hard combinatorics. He constructed an explicit graph with 1581 vertices that simply cannot be colored with four colors under the unit-distance rule — proving the plane needs at least 5. A whole community then raced to shrink his monster graph. The gap (5 to 7) is still open. The entry fee to work on it is a pencil.

The little graph that forces 4 is the Moser spindle: seven points, eleven unit-length edges, arranged so that no four colors can satisfy all of them at once — wait, the other way: four colors can do it, but three provably cannot. Try to color it below.

This chapter is the frontier doing what frontiers do: refusing to behave. A rounding error told us the future is deterministic and unpredictable. The "monster" curve turned out to be how nature actually builds coastlines. A hobbyist with cardstock ended a fifty-year tile hunt; a needle turned full circle inside almost no area; sphere packing fell in dimension 8 to a single magic function; and an anti-aging biologist nudged a seventy-year-old coloring bound. The recurring theme: in geometry, the open problems are stated in plain language, and the entry fee is sometimes just a pencil.

Notice that these shapes misbehaved — they fought intuition at every step. The next chapter flips it. We hand the wheel to randomness, the thing that should produce pure mess, and watch it snap into eerily precise order instead.