🏛️ The Vault · № 7 · Wing 2 — slicing & stacking · watch, then wonder
Archimedes discovered levers, buoyancy, and machines that terrified an empire. But when it came to his own grave, he asked for none of that — just a carving of a ball sitting snugly inside a can. Why? Because of one result he loved above all the others: the ball fills an exact, perfectly clean fraction of the can — no rounding, no "roughly". Which fraction? The video slices its way to it. The slicing game has a name now — Cavalieri's principle, played by Archimedes 1,800 years before Cavalieri.
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One ball, one can, one upside-down cone — sliced floor by floor like stacks of coins. Watch it once all the way, then scroll down for the story of the grave, the lost notebook, and the slicing you can do with your own hands.
If you visited the Vault's first treasure — why a circle's area is πr² — you already met the promise at the end of Archimedes' story: a tombstone with a sphere and a cylinder carved on it. This lesson is that carving, paid in full. Of everything the greatest mathematician of the ancient world ever discovered, this is the one he chose to be remembered by.
Archimedes asked his friends for one thing on his grave: a sphere inside a cylinder, with the ratio 2 : 3 — his favorite of all his results. He got his wish… and then Syracuse forgot where he was buried. 137 years later, in 75 BC, the Roman writer Cicero went hunting for the lost grave and found it buried in brambles — by searching for that exact carving.
A fraction good enough to be found by. 🥷
Two stacks of coins — one straight, one leaning — hold the same money, because they match floor by floor. That's the whole rule: if two shapes have the same slice area at every height, they hold the same amount. It's named after Bonaventura Cavalieri, an Italian monk-mathematician of the 1600s who turned slicing into a proper method. Archimedes was playing the same game 1,800 years earlier.
Sometimes the trophy gets named after the second player.
True story. Archimedes wrote a secret notebook, The Method, explaining how he found results like this one — by weighing shapes against each other on an imaginary lever! Then the book vanished for centuries: monks scraped the parchment clean and wrote a prayer book over it. In 1998 the battered volume surfaced at auction and sold for $2 million — and modern imaging pulled the hidden math back out from under the prayers. It's called the Archimedes Palimpsest.
A 2,000-year-old secret, rescued by X-rays. 📖✨
A tube of tennis balls is the tombstone shape, stacked three high. Each ball sits in a slot exactly its own height and width — and fills exactly ⅔ of it. Which means ⅓ of every tin of tennis balls is air. Not because the factory is lazy. Because of a theorem carved on a grave in Sicily.
Next time you pop a tube open: that hiss is ⅓ theorem.
A fox scholar in little round glasses runs a stone workshop where shapes get sliced floor by floor. It lays sphere-slices next to cylinder-slices and, instead of trusting any formula, keeps asking one stubborn question: are these floors really the same size? That's how a tombstone theorem gets checked.
Here's the move from the video. Take a half-ball (a dome of radius r) and a can of the same radius and height r with an upside-down cone drilled out of it — point at the bottom, opening at the top. Now slice both at any height h. The dome's floor is a solid disc; the drilled can's floor is a ring.
Disc area: π(r² − h²). Ring area: πr² − πh². The same number at every height — so by the coin-stack rule the two shapes hold exactly the same amount. And the drilled can is easy: a cone is ⅓ of its can, so what's left is ⅔. Half-ball = ⅔ of the half-can… so the whole ball = ⅔ of the whole can. That's the tombstone.
⚠️ Compare floor areas, never widths! Near the top the ring still stretches the full width of the can while the disc has shrunk — but the ring is mostly hole. The areas match; the widths don't even try.
Time to check the theorem the wet way. A ball sits snugly in its can — same height, same width. Pour water into the gaps until the can is full to the brim, and watch the meter: how much water did the can actually swallow?
water poured (in cans): the dashed line is ⅓ of a can
poured so far: 0 of a can
Slicing a solid into floors and adding them up has a modern name too: integration — the other half of calculus. And the tombstone ratio hides the famous formula: the can holds 2πr³, so the ball holds ⅔ of that — 4/3 × πr³, the volume of a sphere. You just proved it with coins and a cone.
Archimedes squeezed his circle with the method of exhaustion, tamed infinity like the infinite snack, and sliced his sphere like a stack of coins. Three Vault treasures, one restless brain.
Next time someone shows off a gold trophy, remember: the greatest mathematician of antiquity chose a fraction. 🥷