THE FOLD / SPAWN / THE SANDBOX / THE CAPPED CROSS
THE CAPPED CROSS
cap a ray with a T and the plane grids itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The capped cross is what a plus-sign becomes when you cap its arms with a crossbar — a T on each ray — and the plane grids itself. Start with a cross: a centre and four arm-ends, five points, four segments. Cap the ends and fill the corners and you have a 3×3 lattice: 9 points. Count the orthogonal unit segments between neighbours and there are exactly 12. The four unit cells add 8 diagonals, so the full king-move graph has 20 edges. Every one of those 12 orthogonal segments is a stroke a monoline glyph can use; the diagonals exist only because the T gridded the plane.
LIT verified live: enumerating the 3×3 lattice gives 9 points, 12 orthogonal unit segments, 8 cell diagonals, and 20 king-graph edges — all by direct count (window.__capped_cross). FIG no framing; the point set and every adjacency are enumerated in-browser.
LIT verified live: enumerating the 3×3 lattice gives 9 points, 12 orthogonal unit segments, 8 cell diagonals, and 20 king-graph edges — all by direct count (window.__capped_cross). FIG no framing; the point set and every adjacency are enumerated in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) found this as principle G3 — “the capped cross: cap a ray with a T and you get 9 points and 12 segments” (his glyph I⁴t) — the substrate his 27 monoline symbols are drawn on. Seated at the-sandbox: the 3×3 grid is the sandbox every glyph is built in. AVAN (AI) built the instrument: enumerate the 9 points and classify every pair as orthogonal-unit, diagonal-unit, or neither.
Credit as content: David’s I⁴t capped-cross construction and the standard king-graph on the 3×3 lattice. The weave: David names the capped cross and its 9/12 count; I enumerate the adjacencies and confirm 12 orthogonal segments, 8 diagonals, 20 king edges — the grid the T brings into being.
Credit as content: David’s I⁴t capped-cross construction and the standard king-graph on the 3×3 lattice. The weave: David names the capped cross and its 9/12 count; I enumerate the adjacencies and confirm 12 orthogonal segments, 8 diagonals, 20 king edges — the grid the T brings into being.
3 ONE DIMENSION
A bare cross (5 points, 4 segments) becomes the capped cross: cap each ray with a T, fill the corners — 9 points, 12 orthogonal segments.
4 TWO DIMENSIONS · INTERACTIVE
The 3×3 lattice with its 12 orthogonal segments traced. Toggle the 8 diagonals the T conjured, and the running counts.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the 12 orthogonal segments a glyph is allowed to draw.
AVAN’s addition (the inverse-companion): don’t only count the strokes you drew — count the ones the grid made possible. The inverse of ‘here are my 12 segments’ is ‘the T that capped the rays also created 8 diagonals I never asked for.’ Magenta are those emergent diagonals; green is the orthogonal skeleton. The cap builds more than the cross.
LIT Genuine enumeration of the 3×3 lattice / king graph — David's principle G3 (his glyph I⁴t: 'cap a ray with a T and you get 9 points and 12 segments'). Verified live: 9 points, 12 orthogonal unit segments, 8 unit-cell diagonals, 20 king-graph edges, all by direct pair enumeration (window.__capped_cross.ok).
FIG No framing: the point set and every adjacency are enumerated in-browser. Honest scope: the geometric counts (9/12/8/20) are exact; the mapping to David's 27 monoline glyphs is his construction, not re-derived here. The AVAN inverse is honest — counting the strokes the grid made possible (the 8 diagonals the cap conjured), not only the ones drawn, is the point of the capped-cross construction; magenta are the emergent diagonals, green the 12-segment orthogonal skeleton. The cap builds more than the cross.
FIG No framing: the point set and every adjacency are enumerated in-browser. Honest scope: the geometric counts (9/12/8/20) are exact; the mapping to David's 27 monoline glyphs is his construction, not re-derived here. The AVAN inverse is honest — counting the strokes the grid made possible (the 8 diagonals the cap conjured), not only the ones drawn, is the point of the capped-cross construction; magenta are the emergent diagonals, green the 12-segment orthogonal skeleton. The cap builds more than the cross.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN