ledger · verifier (9/9) · every term defined · theorems · the membrane · dustbin
Every 3D map here has a shadow: one flat picture that shows everything that matters. Below, a demonstration plays itself, then it's yours: drag the blue dot on the right. The red circles on the left are every point the map sends to your dot. When you cross the dark-red curve, circles appear and vanish — that curve is where points slip away to infinity.
Drag the target (right): preimages are born and die in the source (left) as you cross the dark-red escape frontier. The crease is the amber line; its entire image is one point.
No complex numbers are needed to witness any of this. The coefficients are rational; the three colliding points and their common image are real; the real 3×3 Jacobian determinant is the same polynomial, constantly −2. So: a polynomial map of ordinary 3-space, everywhere a local diffeomorphism at constant volume factor, sends three real points to one — settling the constant-determinant real question (dim ≥ 3) as an untweeted corollary. But the explanation — sheets, monodromy, the faked circle — lives only over ℂ: the reals experience the collision; the complexes know why.
Every rule-breaking map anyone has ever built obeys one small equation:
\(m\{\tilde g,\tilde h\}+2\tilde h\{\tilde g,m\}+\tilde g\{m,\tilde h\}=c\neq0\)
The little 2 in it is not a mystery: it counts how much room the space has for hiding. Flat 2D space has zero room — and that is exactly why, in 140 years, nobody has ever broken the rule there.
Grow your own. Every rule-breaking map sprouts from one little seed curve whose ups and downs cancel out exactly — the gold curve must come back to zero. Drag the sliders; the readout says what your seed grows into. The very first map ever found is the one and only smallest seed.
Every counterexample grows from a one-variable seed p(w) whose potential P returns to zero: P(0) = P(1) = 0, flat at the origin. Drag the slider to reshape the seed; the conditions are enforced automatically; the readout tells you what your seed mints. The blue curve is p; the amber curve is its potential P. Level-matched wells → a map of ℂ³ that collides deg p + 1 points without ever creasing.
Count the layers. Pick a target point. Count how many points the map sends there. Call it d. No squishing means the d points sit on d clean separate layers.
Two layers: impossible. With two layers, every point has exactly one partner — the other point landing on the same target. Add the two together: the sum depends only on the target. So there is a formula: partner = sum − me. Ask: can a point be its own partner? Only if the two layers touch there. But touching is squishing, and this map never squishes. So the formula moves every point in space, pairing everything with no meeting anywhere. A theorem (Campbell, 1973) says that's impossible. So two layers can never happen.
Three layers: fine. Now a point has two partners — and no formula can pick one without making a choice. The trap above needed the formula. No formula, no trap. And a three-layer map really exists: it's the one flying at the top of this page.
More: the seed recipe below builds a map with any number of layers, three and up. Layer counts also multiply when you chain maps: 3 then 3 gives 9. So the menu is: one, never two, then everything.
the struck-out 2 is a theorem; the dots go on forever. what we got wrong on the way.
Everything above happens in 3D space, where a map can hide a secret spinning direction and sweep its fold off the edge of the world. Flat 2D space has nowhere to hide anything. Is the old rule still true there? Nobody on Earth knows. That is the question now — and a famous 1968 quantum puzzle is chained to it, waiting on the same answer (\(DC_2\Rightarrow JC_2\Rightarrow DC_1\)). The note · PDF · Lean (builds green).
margin notes · the fiber — anatomy of the preimages, interactive · three theorems, paired — lean and intuition · fable physicist, too — a journal of the window, and the contract · sister sites: theorems · the fong conjecture
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Alpöge · Gallagher · Zyskind–Sol · Zhang · this project. Corrections recorded, never erased.