Tessellations and Patterns

Origami and Paper Folding 7 joined many sheets into one model. This chapter goes the other way: it makes a single sheet carry a repeating pattern. An origami tessellation is a flat sheet covered in a regular grid of creases, folded so that the same shape repeats across the whole surface. The finished piece looks like a woven or tiled texture, and it is often lit from behind so the layers show.

The Crease Pattern as a Map

Until now you have followed steps. A crease pattern (CP) is different: it is a map of every fold in the finished model, drawn on the unfolded sheet. It shows where the creases are and whether each is a mountain or a valley, but not the order in which to make them. Reading a CP is the key skill for this chapter and for Origami and Paper Folding 9 and 10.

• Valley lines are drawn dashed, and mountain lines are drawn dot-dash, using the symbols from Origami and Paper Folding 2.
• Every crease is folded at the end. Tessellations are usually pre-creased first, then collapsed all at once.
• Repeat means copy. If you can read one tile of the pattern, you can read the whole sheet.

A small corner of a square-twist pattern looks like this. The letters show the crease type, with V for valley and M for mountain:

Pre-Creasing a Grid

Nearly every tessellation starts from an even grid, so the first job is accurate division of the square.

1. Fold in half both ways and unfold, giving a cross of four equal squares.
2. Fold each edge to the centre line and unfold, giving quarters.
3. Repeat once more to reach eighths. An 8 by 8 grid suits a first tessellation.
4. Crease the whole length of each line firmly, so that all creases are the same sharpness.

The Twist Fold

A twist fold is the building block of most tessellations. A square of the grid is collapsed so that its middle rotates, leaving a small turned square surrounded by triangles. The basic square twist works like this:

1. Pre-crease a 4 by 4 grid and add the diagonal creases inside the centre four squares.
2. Pinch the centre so the creases that should be valleys fold down and the mountains fold up.
3. Rotate the middle square clockwise (or anticlockwise, the direction is a choice) while flattening the sheet.
4. Press flat. A rotated square sits in the centre, with four triangles around it.

From One Twist to a Tessellation

A tessellation repeats the twist across the whole grid. Neighbouring twists share edges, so the rotation directions must alternate in a chessboard pattern. If two neighbours turn the same way the paper locks and will not lie flat.

Counting Tiles

The number of twists grows quickly with the grid. This small script counts the twists and crease lines for a square grid of a given size:

Output when run:

Paper and Finishing

Tessellations pack many layers into a small area, so paper choice matters more here than in earlier chapters.

• Thin, strong paper. Tissue-foil, kozo or thin origami paper folds in many layers without bulk. Standard copy paper is too thick for large grids.
• Start small. A 4 by 4 grid on 15 cm paper is a good first tessellation, and 8 by 8 follows once that feels easy.
• Back-light to check. Hold the finished piece up to a window to see the layers and spot any uneven twist.
• Frame it. Press the piece flat under a book overnight, then mount it between glass or on card.

Challenges

Each challenge has a worked solution you can open after you have tried it.

Next, in Origami and Paper Folding 9, we look at the mathematics of folding: flat-foldability rules, folding angles and thirds, and origami in engineering.