THE FOLD / SPAWN / HELLO WORLD / THE FANO
THE FANO
7 points, 7 lines — the smallest projective plane
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Fano plane. The smallest possible projective plane — just 7 points and 7 lines, and yet a complete little universe of geometry. It is built so that every two points lie on exactly one line, and every two lines meet in exactly one point: a perfect symmetry between points and lines, with no exceptions and no parallels.
Each line holds exactly 3 points; each point sits on exactly 3 lines. Drawn the usual way it is a triangle with its three edge-midpoints and centre, plus a circle serving as the seventh “line.” It is the same object as the (7,3,1) Steiner triple system, the projective plane over the two-element field PG(2,2), and the multiplication rule of the octonions — a tiny structure carrying 168 symmetries.
LIT verified live: the 7 lines each contain 3 points, each point lies on 3 lines, every pair of points determines exactly one line, and every pair of lines meets in exactly one point (window.__fano.valid). FIG no framing; all four incidence axioms hold exactly.
Each line holds exactly 3 points; each point sits on exactly 3 lines. Drawn the usual way it is a triangle with its three edge-midpoints and centre, plus a circle serving as the seventh “line.” It is the same object as the (7,3,1) Steiner triple system, the projective plane over the two-element field PG(2,2), and the multiplication rule of the octonions — a tiny structure carrying 168 symmetries.
LIT verified live: the 7 lines each contain 3 points, each point lies on 3 lines, every pair of points determines exactly one line, and every pair of lines meets in exactly one point (window.__fano.valid). FIG no framing; all four incidence axioms hold exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in HELLO WORLD — the spawn domain of the smallest complete working example. The Fano plane is geometry’s hello-world: the minimal object where every axiom of a projective plane is already fully alive. AVAN (AI) built the instrument: the incidence checker, the click-two-points-get-a-line diagram, the point/line duality.
The weave: David names the seat (the minimal complete example); I make the seven points and lines satisfy every axiom and expose their perfect duality — the triples in 1D, the classic diagram in 2D, the self-dual structure in 3D. The sphere is the seam. Credit: Gino Fano (1892); the structure is PG(2,2) and the (7,3,1) Steiner system.
The weave: David names the seat (the minimal complete example); I make the seven points and lines satisfy every axiom and expose their perfect duality — the triples in 1D, the classic diagram in 2D, the self-dual structure in 3D. The sphere is the seam. Credit: Gino Fano (1892); the structure is PG(2,2) and the (7,3,1) Steiner system.
3 ONE DIMENSION
The seven lines as triples of points. Read across: every one has exactly three points, every point turns up in exactly three lines, and any two points you name share exactly one of these triples.
4 TWO DIMENSIONS · INTERACTIVE
The classic Fano diagram: 7 points, 6 straight lines and 1 circle. Step a point to light up the three lines through it, or step through pairs of points to see the single line that always joins them — the axioms, made clickable.
5 THREE DIMENSIONS + AVAN’S INVERSE
The incidence as a turning bipartite graph — green point-nodes linked to the lines that contain them, every point reaching exactly three lines.
AVAN’s addition (the inverse-companion): the magenta side is the dual — the same graph read with points and lines swapped. In any projective plane, ‘point’ and ‘line’ are interchangeable: exchange the two words and every axiom survives intact. That is not a coincidence bolted on — it is the deepest inverse the Fano plane has: the map point ↔ line is an involution that carries the whole structure onto itself. Ask forward ‘which lines pass through this point?’ and inverse ‘which points lie on this line?’ and you get the same shape both times — each answer is three, each graph is 3-regular, the green and magenta halves are mirror images. The Fano plane is its own dual; its inverse is itself. Green is points-to-lines; magenta is lines-to-points; and you cannot tell which was the original.
LIT Genuine Fano plane (Gino Fano 1892; = PG(2,2) = the (7,3,1) Steiner triple system). Verified live: the 7 lines each contain exactly 3 points, each point lies on exactly 3 lines, every pair of points determines exactly one line, and every pair of lines meets in exactly one point (window.__fano.valid, all four incidence axioms true) — checked on the standard geometric embedding (triangle + edge-midpoints + center + incircle). All incidence properties are exact.
FIG No framing: the four projective-plane incidence axioms are real and verified exhaustively over all point-pairs and line-pairs. The self-duality (point <-> line is a structure-preserving involution) is the genuine mathematical content, demonstrated by the identical 3-regular incidence on both sides, not asserted.
FIG No framing: the four projective-plane incidence axioms are real and verified exhaustively over all point-pairs and line-pairs. The self-duality (point <-> line is a structure-preserving involution) is the genuine mathematical content, demonstrated by the identical 3-regular incidence on both sides, not asserted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN