Origami & Folding
what one sheet can do
Concept

The folded period — where it appears

The smallest translation under which a pattern's folded state repeats. It is a multiple of the drawn period and need not equal it: folding the equilateral Yoshimura turns its lattice through 240° per column, so its folded period is three drawn ones.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

flat-folding · Two-colourability the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The turn a column costs

The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

tessellation · Shrinkage the folded period of the Yoshimuradrawn periods across the top1234567the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyes240°yesnoa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The period nobody measured

Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.

tessellation · Found patterns the impossible lettering, on ordinary patchessquare ×140 creases16 vertices · every condition holds · no forced loopsquare ×2144 creases64 vertices · every condition holds · no forced loopsquare ×3312 creases144 vertices · every condition holds · no forced looptriangular ×1116 creases48 vertices · every condition holds · no forced looptriangular ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×1116 creases48 vertices · every condition holds · no forced loophexagonal ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×3924 creases432 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said

The symmetry a gluing adds

A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.

design · Symmetry

Named alongside it

The objects these essays reach for when they reach for this one.

GluingPeriodicitySymmetryUnit cellFolded statePanelShrinkageTessellationTorusBoundaryCorrugationCrease assignment

All concepts