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Beloch

Your first fold

The @map statement folds the sheet by mapping one point (or edge) onto another. Beloch finds the crease that makes the mapping exact, then folds.

Folding in half

Section titled “Folding in half”
paper square
@map .b onto .a moving .b

@map .b onto .a moving .b says: bring point .b (bottom-right corner) onto .a (bottom-left corner). The .b at the end names which side to fold — the half of the sheet that contains .b lifts and lands on top of the other half. Beloch computes the perpendicular bisector of the segment .a–.b as the crease (axiom 2).

; status: works — fold the right half onto the left (valley), one crease
paper square
@map .b onto .a moving .b
.a .d .b .c axiom 2 · map onto
.a .d .a .d axiom 2 · map onto
fold-half.bel

The left diagram is the crease pattern; the right is the folded state.

What @map does

Section titled “What @map does”

@map P onto Q moving P is the most direct form: find the unique fold that takes point P to point Q. Beloch validates that such a fold exists (that P and Q are equidistant from the crease) and then applies it.

The moving clause controls which side of the crease is folded. Any named point on that side will move; points on the other side stay.

Next steps

Section titled “Next steps”