A flat sheet whose corners are special points of the map, flying to where the map sends them. Every corner lands on the same red dot — exactly, not approximately — and the map never squishes space anywhere: its stretch-number (the Jacobian determinant) is the same constant at every point of the flight's start and end. Drag to orbit · scroll to zoom · slider is the morph. At t = 1 every corner sits on the red point. Degree 3: three corners → (−¼, 0, 0). Degree 5: five corners → (0, −3/2, 1). Nothing crushed — the Jacobian determinant is constant — and nothing torn. After Bryce Drennan.
Degree 3 — L. Alpöge, with Claude Fable 5 (announced 2026-07-19). With \(u=1+xy\):
\(F(x,y,z)=\big(u^3z+y^2u(4+3xy),\;\; y+3xu^2z+3xy^2(4+3xy),\;\; 2x-3x^2y-x^3z\big)\)
\(\det JF\equiv-2\), and his certificate: \(F(0,0,-\tfrac14)=F(1,-\tfrac32,\tfrac{13}2)=F(-1,\tfrac32,\tfrac{13}2)=(-\tfrac14,0,0)\) — the three corners of the first flight above.
Independent certificate (this project, 2026-07-20, derived from the map alone): \(F(1,-2,9)=F(-\tfrac13,4,27)=F(-\tfrac23,-\tfrac12,-\tfrac98)=(-1,1,-1)\).
Degree 5 — from A. Gallagher's weighted-lift construction (with Claude; jacobianfun.org), transcribed and re-verified here exactly: \(\det JF_5\equiv 1\), five rational points onto \((0,-\tfrac32,1)\) — the five corners of the second flight. Full coefficients: page source, or the ledger.
The generator. Every member comes from a one-variable seed \(p\) with \(p(0)=0,\; p(1)=-c,\; \int_0^1 p=0\): fiber degree \(=\deg p+1\) (Gallagher). Alpöge's map is the unique degree-2 seed \(p(w)=2w-3w^2\). Generalized weights — \(A=1+xy^m\), degree \(r(m+1)+1\) — give the Zyskind–Sol factory. Build seeds yourself: the forge, on the atlas.
Each color is a corner's region. Every vertex moves along (1−t)·q + t·F(q), with F evaluated exactly. The log lens (default at degree 5) radially compresses distance from the destination — the map's values span seven orders of magnitude, beyond any linear camera; the lens is a display coordinate, never a change to F. Back to the atlas.