Nine inflection points on twelve lines — a configuration so symmetric it cannot be drawn with real straight lines
Draw a smooth cubic curve. Somewhere on it are its inflection points — the places where the curve stops bending one way and starts bending the other. Over the complex numbers there are always exactly nine of them, on every smooth cubic, no exceptions.
Those nine points are arranged with an almost unreasonable amount of symmetry. Take any two of them and draw the line through them: that line always hits a third inflection point. Never zero, never two — always exactly one more. Counting up, the nine points lie on twelve such lines, with three points on each line and four lines through each point. This is the Hesse configuration, written (94, 123).
And here is the part that stops people short: you cannot draw it. Not because it is hard, but because it is impossible. No nine real points and twelve real straight lines in the plane have this incidence pattern. The configuration exists in full only over the complex numbers — which is why a real cubic will only ever show you three of its nine inflection points.
A real cubic curve with its real inflection points marked. However you push the coefficients around, you will find exactly three — and they are always collinear, a fact first noticed by Newton. The other six inflection points are complex, and no real picture can show them.
An inflection point of a plane curve F = 0 is a point where the curve meets its own tangent line three times over. The condition is that the Hessian — the determinant of the 3×3 matrix of second partial derivatives of F — also vanishes there. For a cubic, that Hessian is itself a cubic curve, so we are intersecting two cubics. By Bézout’s theorem they meet in 3 × 3 = 9 points. That is where the nine comes from, and it is why the count never varies.
The nine inflection points also form a group. Fix one of them as the identity and the other eight become the 3-torsion points of the elliptic curve: the points P with P + P + P = 0. That group is Z/3 × Z/3 — nine elements — and three inflection points are collinear precisely when they add up to zero in it. The collinearity rule is not geometric happenstance; it is addition.
Complex conjugation permutes the nine points, and a real cubic’s real inflections are the ones fixed by it. Since the group is Z/3 × Z/3 and conjugation is an involution acting on it, the fixed subgroup has order 3 — so exactly three inflection points are real, and being a subgroup summing to zero, they are collinear.
Sweep through the family of cubics x³ + y³ + z³ + t·xyz = 0. The curve changes shape wildly, but the nine base points never move — every member of the family passes through all of them. Snap to the degenerate members and the cubic collapses into a triangle of three straight lines; those triangles supply the twelve lines of the configuration.
The fuchsia dots never move: they are base points of the pencil — every single member threads through them, no matter how wildly the curve morphs. All 9 base points are shared; the other 7 are complex or at infinity.
The Hesse pencil is the one-parameter family of cubics λ(x³ + y³ + z³) + μ·xyz = 0. Every member passes through the same nine points — the base points of the pencil — and those nine points are exactly the inflection points of every smooth member. One family, one set of nine points, shared by all of its curves.
Almost every member is a smooth elliptic curve, but exactly four members degenerate into a triangle of three lines: the member xyz = 0 (the three coordinate axes), and the three members at t³ = −27, of which only t = −3 is real. Four triangles, three lines each, gives 4 × 3 = 12 lines — precisely the twelve lines of the Hesse configuration. The configuration is not imposed from outside; it falls out of the pencil.
At t = −3 the cubic factors as (x + y + z)(x + ωy + ω²z)(x + ω²y + ωz), where ω is a primitive cube root of unity. Only the first factor is a real line; the other two are complex conjugates, and they meet the real plane in a single isolated point. Watch for it in the demo — a lone dot with no curve attached, which is exactly what a pair of complex lines looks like from inside the real world.
Nine points, twelve lines, four lines through every point and three points on every line. Hover to trace the incidences. Then tick “try to straighten” and watch four of the lines fail — not for want of trying, but because Sylvester–Gallai forbids it.
Hover a point to see its four lines, or a line to see its three points. Every line through two inflection points automatically contains a third — that is the whole miracle in one sentence.
Label the nine points by pairs of numbers mod 3 and the configuration becomes AG(2, 3): the points are F₃², and the 12 lines are the 12 affine lines of that tiny plane — 3 vertical, 3 horizontal, and 3 for each of the two remaining slopes. The 9 inflection points of a cubic form a group isomorphic to Z/3 × Z/3, and three of them are collinear exactly when they sum to zero.