How many conics are tangent to five given conics? Steiner guessed 7776. The truth is stranger — and all of them can be real
Some numbers in mathematics are famous for being large, some for being strange. The number 3264 is famous for being exactly right after a very distinguished mathematician got it very publicly wrong.
The question is easy to state. Draw five conics — ellipses, say — in the plane. How many other conics are tangent to all five at once? In 1848 Jakob Steiner, one of the great geometers of the century, worked it out as 7776. The reasoning looked impeccable. It was not: the true answer, found by Chasles in 1864, is 3264, less than half of Steiner’s.
What went wrong is genuinely interesting, and the fix launched a whole field. Enumerative geometry is full of questions like this — how many curves of some kind satisfy some conditions — and the surprises are almost always about objects sneaking into the count that had no business being there.
Before the hard question, an easy one with the same flavour. Four lines float in space, no two of them meeting. How many straight lines cross all four? Have a guess, rotate the scene, then reveal the answer — and note that it never changes.
How many straight lines meet all four of these at once? Not approximately — exactly. Rotate the scene and try to guess before revealing the answer.
This is the first non-trivial result of Schubert calculus, the machinery for counting geometric figures satisfying constraints. Hermann Schubert developed it in the 1870s with an intuitive “principle of conservation of number” that produced correct answers by methods nobody could justify. Making that rigorous became Hilbert’s fifteenth problem, and it took most of the twentieth century and the invention of modern intersection theory to finish the job.
The trick that makes enumerative geometry work is to stop thinking about one curve at a time and think about the space of all of them. A conic is a x² + b xy + c y² + d x + e y + f = 0, six coefficients, and scaling all six gives the same curve — so the conics form a five-dimensional projective space P⁵. One conic, one point.
Now geometric conditions become subvarieties. “Passes through a fixed point” is one linear equation in the coefficients, so it is a hyperplane. “Tangent to a fixed line” is a quadratic condition. “Tangent to a fixed conic” is a degree-6 condition. Counting solutions becomes intersecting subvarieties, and Bézout’s theorem says to multiply the degrees.
Five dimensions means five conditions give a finite answer. Mixing point and tangency conditions gives the six characteristic numbers of conics — and they read 1, 2, 4, 4, 2, 1, a palindrome. That symmetry is not a coincidence: projective duality swaps points with lines and takes conics to conics, so it reflects the list onto itself.
Choose how many point conditions and how many tangency conditions to impose, then drag the constraints around. Every case is solved live — the line-heavy ones by computing in the dual plane and dualising back — and the panel reports how many of the complex solutions happen to be real for your arrangement.
Four points leave a pencil of conics; tangency to a line is one quadratic condition on the pencil, giving 2 solutions.
1, 2, 4, 4, 2, 1 — a palindrome, because duality reflects the list onto itself.
Walk through the whole story: the space of conics, Steiner’s plausible count, the degenerate curves that poisoned it, Chasles’ repair, and the modern discovery that all 3264 solutions can be made real at once.
A conic is the zero set of a x² + b xy + c y² + d x + e y + f. That is six coefficients, and multiplying all of them by the same non-zero number gives the same curve — so conics form a five-dimensional projective space, P⁵. Each conic is a single point of it. Push the sliders and travel around that space.
Five dimensions, five conditions — which is exactly why five points pin down a unique conic, and why asking for tangency to five conics is a sensible question with a finite answer.
Steiner’s error was not careless arithmetic; it was a subtle failure of Bézout’s theorem. Bézout counts intersection points correctly only when the intersection is genuinely a finite set of points. Here the five hypersurfaces share an entire surface of double lines — the Veronese surface — and that shared component absorbs a great deal of the count. The number 7776 is the honest total of real solutions plus the excess contributed by that surface.
Extracting the true count means separating the genuine solutions from the parasitic ones, and the modern tool for that is excess intersection theory: blow up along the offending surface and compute a correction term. Fulton and MacPherson put this on a rigorous footing in the 1970s, and the 3264 conics are its showcase example — the title of Eisenbud and Harris’s standard textbook on the subject is simply 3264 and All That.
The same pattern recurs throughout enumerative geometry, and it is why the subject needed a century of foundations before its answers could be trusted. Hermann Schubert computed dozens of such numbers in the 1870s by a “principle of conservation of number” that worked but could not be justified; justifying it became Hilbert’s fifteenth problem. The counting was right. The reasoning took a hundred years to catch up.