THE SHOELACE lace the coordinates and halve — a polygon’s area from its corners
The shoelace formula gives the area of any simple polygon from its vertex coordinates alone: sum the cross products of consecutive corners and halve. Written out, the terms cross like laces — ½ |∑ (xiyi+1 − xi+1yi)| — hence the name. It is a chain of the cross products from dart 486, and its sign even tells you the polygon’s winding. For the triangle (0,0),(4,0),(0,3) the laced sum is 12, so the area is 6. Gauss and Meister knew it in the 18th century; it underlies every polygon-area routine since.
THE TECHNIQUE ½ |∑ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| — area from the corners
The demo laces the triangle (0,0),(4,0),(0,3) — the summed cross products give twice the area, halved to 6: live demo
HISTORY & CREDIT Meister 1769 · Gauss (shoelace)
“Area needs a base and a height.” — the shoelace formula gets it straight from the coordinates, for any polygon. cited
the lace · ∑ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) — consecutive cross products, summed. the halve · area = half the absolute sum; the sign gives the winding. 18th c. · Meister (1769) and Gauss — area of a polygon from its vertices.
The corners laced into a sum and halved — a polygon’s area with no base, no height, just the cross product chained. theorem
RECOMMEND FOR I-13 the laced area, on the compiler
On the canonical compiler, the shoelace sum of the triangle is 12, so its area is 6:
$ i13 run cg_shoelace.i13 # sum of consecutive cross products
RUN OK · 38 step(s) · peak stack 4 · call depth 0
area2 = 12 -- the laced sum (twice the area)
area = 6
is6 = 1
Recommend as a NULL — a theorem, chained from the cross product. The shoelace formula is a closed-form identity for polygon area (B39): every correct implementation returns 6. It is dart 486 summed around the boundary — beautiful, but a witnessed theorem, not an enacted invariant. NULL.