Origami Tessellation Grid Calculator

What are origami tessellations?

Tessellations are repeating patterns folded from a single uncut sheet of paper. They require a precise grid of pre-creases (mountain and valley folds) that serve as the foundation for collapsing the pattern. The grid type (square, triangular, or hexagonal) determines which tessellation patterns are possible.

Grid types and their uses:

Pre-crease count formula:

Everything below counts crease lines, meaning one fold you make edge to edge across the sheet. That is the number that matters when you are estimating how long pre-creasing will take, because each line is one pass with the bone folder. It is not the same as counting unit segments between intersections, which is the convention the crease count calculator uses for box-pleat grids. Same paper, two legitimate numbers, roughly sixteen times apart on a 16 grid. Check which one a pattern designer means before you trust a figure.

For a square grid of n × n divisions: Horizontal lines = n - 1 Vertical lines = n - 1 Total crease lines = 2 × (n - 1) Crease intersections = (n - 1)²

For a triangular grid of n divisions: Horizontal lines = n - 1 Diagonal lines (left) = n - 1 Diagonal lines (right) = n - 1 Total crease lines = 3 × (n - 1)

Paper size formula:

The final tessellation shrinks significantly from the original paper size:

Shrink ratio = Final size / Original size

Required paper size = Desired finished size / Shrink ratio

Example calculation (16 × 16 square grid):

• Grid: 16 × 16 divisions
• Horizontal lines: 15
• Vertical lines: 15
• Total crease lines: 30
• Crease intersections: 225
• Paper: 30 × 30 cm (for a finished piece around 14 cm)
• Time: 30 to 45 minutes of pre-creasing

Thirty lines in forty minutes sounds slow until you try it. You do not fold them in order; you fold halves, then quarters, then eighths, and the last round of sixteenths is fifteen separate folds each of which has to land exactly on a mark you made three rounds ago. Reckon on a minute and a half per line and you will be about right.

Paper weight recommendation:

Thinner paper is essential for high-division grids because each fold adds thickness. A 64-grid on thick paper becomes impossibly bulky at the center where creases overlap.

Diagonal creases (for twist folds):

Many tessellations add diagonals inside the grid cells, and this is where the crease count stops being polite.

Counting lines again: the diagonals of a square grid run corner to corner across the sheet, and there are 2n − 1 of them in each direction, so Diagonal lines = 2 × (2n − 1). A 16 grid gains 62 diagonal lines on top of its 30 orthogonal ones, giving 92 lines.

Counting segments tells a different story. Two diagonals in every one of the n² cells is 2n² short creases, which for a 16 grid is 512. Both numbers describe the same sheet. The first is how many folds you make; the second is how many little creases end up in the paper.

The important consequence either way: a diagonal grid roughly triples the work of a plain one, and it triples it in the fiddliest possible way, because a diagonal fold has no edge of the paper to align against. Most people pre-crease the orthogonal grid first, then use the intersections as landmarks for the diagonals.