Layers & Ordering

Every tutorial so far has folded at most once. This one folds twice, and shows what happens to the paper in between: it stacks. Once there is a stack, a new fold has to decide how far down it reaches — and by default it reaches as little as it can.

One fold, two layers

Start with a single fold.

; one-fold.bel
paper square
fold map .d onto .a

fold map .d onto .a folds .d (top-left) onto .a (bottom-left) — the same kind of reflection from earlier tutorials, but this time it's a fold, not a mark, so the paper actually moves. Toggle to the folded view: the top half of the sheet is now lying on top of the bottom half. The paper is no longer one flat face — it is two stacked layers.

A second fold takes only the top flap

Now there is a stack, and a second fold has to choose which layers it lifts. By default it takes the smallest bite it can: the outside run of layers down to the fold's anchor — here, just the top flap. The layer underneath stays put.

Here's a second crease that folds cleanly by default. Fold the top down as before, find the point where that hinge meets the left edge, and fold .d back up onto it.

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6; top-flap.bel
paper square
mark --left = through .a .d
fold --hinge = map .d onto .a
.h = --left * --hinge
fold map .d onto .h

--left * --hinge is a meet: the point where two lines cross. .h sits exactly on the first fold's crease, so folding .d up onto it makes a new crease parallel to that hinge. Because it runs parallel, lifting the top flap doesn't drag the bottom layer along. Toggle to the folded view and count faces: 3, not 4 — the bottom layer was left uncreased.

That is the default now: a fold moves the outside flap(s) down to its anchor and nothing deeper. You don't have to ask for "just this flap" — you already have it.

Folding through the whole stack

To crease every layer, you have to say so. Naming a point on the deepest flap you want to move — moving .a — extends the fold down through the stack.

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4; whole-stack.bel
paper square
fold map .d onto .a
fold map .c onto .d moving .a

fold map .c onto .d creases a line straight down the middle, and moving .a tells it to carry the bottom flap (the one .a sits on) along with the top. Both layers fold → 4 faces.

Leave off moving .a and this fold is an error, not a silent whole-stack fold: the crease down the middle crosses the hinge joining the two layers, so lifting only the top flap would tear the paper along that hinge. Beloch stops you and tells you to move the layers underneath too — which is exactly what moving .a does.

It helps to see where that middle crease falls before the paper moves. Here is the same crease — named --second so we can point at it — ghosted onto the one-fold stack:

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4; whole-stack.bel
paper square
fold map .d onto .a
fold --second = map .c onto .d moving .a

The dotted line is --second projected onto the stack from the first fold. It runs straight down the middle, crossing both layers at once — which is why folding along it (with moving .a) creases all of them into the 4 faces above. (This is a static "ghost" view: the geometry is the one-fold stack, the decoration is a crease from a later step.)

Reaching past the anchor with up to

The default stops at the fold's anchor flap. up to <point> is the interim way to reach past it — to a flap deeper in the stack than the default would take, without moving the entire stack the way moving does. On the simple two-layer stacks here the default already lands where you want, so you rarely need it yet; it earns its keep once a model has several layers and you want to fold down to a specific one.

What you learned

• Folding stacks the paper: after one fold you have layers, not a single flat face.
• By default, a second fold takes only the outside flap(s) down to its anchor — the layers underneath stay uncreased.
• moving <flap> drives a fold through the whole stack; without it, a fold that would tear the paper is rejected rather than forced.
• up to <point> reaches past the default anchor to a deeper flap.

Try adjusting the anchor in the Playground to see which flaps fold and which stay put.