Stacking and Conservation

A fold moves stacks, it never destroys them

Picture the flat sheet as sixteen numbered squares, each one layer thick. A fold picks up everything on one side of the crease, reflects it across, and sets it down on top of whatever is already lying there. Nothing is cut and nothing vanishes. Paper only changes address.

Fold the right half onto the left half and the sheet still owns all sixteen squares — they are simply stacked two deep across the half that stayed put.

Flat sheet

1 layer

→

After the half fold

2 layers

The solid line is the paper's real edge. The dashed square is the sheet's original footprint, redrawn in every panel so you can see how much of it the paper still covers; after one half fold, exactly half.

Read the badge under each panel as the deepest stack anywhere on the sheet, not as a uniform thickness. For a half fold the two happen to agree. For every other fold they do not — which is why the next figure counts positions one at a time instead of trusting a single number.

The layer grid

Here is that same folded state drawn a second way: every covered position carries the number of original squares stacked at it.

Two columns of 2s and two empty columns. Add across a row: 2 + 2 + 0 + 0 = 4. Every row of the flat sheet held four squares and it still holds four — they are just standing on each other now.

Add every cell of any layer grid and the total is always 16. That is the conservation law, and it is the cheapest sanity check in this section: a layer grid that does not total 16 has lost paper, and folding cannot lose paper.

Folding empties one region and thickens another

A quarter fold creases one column in from the edge and turns that narrow strip inward. The strip lands one column over.

The first column is empty — the paper left. The second column reads 2 — the paper arrived. The last two columns never moved and are still one layer. The row still totals 4.

A quarter fold thickens only the strip it lands on, so one sheet can be two layers thick in one place and one layer thick immediately beside it. Half folds are the exception, not the rule: they are the only folds that leave the covered paper at an even thickness. That unevenness is what makes this section hard, and it is what the next two topics are built on.

The crease band

Both folds so far ran along a grid line, so every square stayed whole. Two fold types do not. A center fold creases through the middle of a column or row of cells; a 45-degree fold creases through a diagonal run of them. Those cells are split down the middle and fold onto themselves.

Flat sheet

1 layer

→

After the center fold

2 layers

The crease runs down the middle of the second column, so the paper's edge stops halfway across it rather than at a grid line. Now count the same state:

The first column has emptied onto the third, which reads 2. The crease column still reads 1. Its original square never went anywhere: half of it folded onto its other half, so exactly one original square is still present there — doubled over, covering half the width it used to. The row totals 4, as always.

A corner-to-corner diagonal does the same along a diagonal run.

The upper triangle has folded down across the main diagonal. Six positions are now empty, the four creased positions on the diagonal read 1, and the six positions below them read 2. Nothing is lost: 4 + 12 = 16.

The takeaway

A fold reflects the moving side across the crease and lays it on top of whatever is already there, so every position on the sheet carries a stack of original squares. Folding empties the region the paper left and thickens the region it landed on, and outside a half fold that thickening is never uniform. Cells split by a center or diagonal crease fold onto themselves and keep the one square they started with. Whatever the fold list, all sixteen original squares are still somewhere: every layer grid totals 16.