Tessellation Repeat Calculator
Grid size and repeat count from paper size and molecule.
About this calculator
Origami tessellations are built from a single repeating "molecule" — a twist fold, a pleated hexagon, whatever the base unit is — stamped out in a grid across the sheet. This calculator starts by subtracting your border margin from both edges of the square paper to get the usable working area, then figures out how much room each molecule actually needs once its pleats are accounted for: effective spacing equals the molecule's flat size times (1 + pleat depth ratio), since the paper folded underneath a pleat still has to come from somewhere. Dividing usable size by that effective spacing and flooring the result gives repeats per axis, which is squared for the total molecule count in the grid.
Because the grid rarely divides the paper perfectly, the tool also reports the actual leftover border once the grid is packed in, plus a coverage percentage comparing the visible tessellation area to the full sheet — the difference is paper that either sits in the border or gets consumed by pleats folded out of sight. The biggest source of error here is pleat depth ratio, which is a rough multiplier (shallow ~0.3, deep 1.0+) rather than a measured constant — it varies by molecule design, so test-folding one repeat first and back-calculating your own ratio will beat guessing. Likewise, molecule size should be measured from an actual practice fold, not estimated, since small errors compound across dozens of repeats.
Inputs
Results
Repeats per axis
5
Total molecules
25
How to Use This Calculator
- Enter the side length of your square paper in centimeters.
- Set the size of one tessellation repeat unit (molecule) in centimeters — measure a practice molecule if possible.
- Enter the pleat depth ratio: shallow pleats are ~0.3, medium ~0.5, deep or complex molecules ~1.0 or more.
- Set the border margin in centimeters to leave unfolded space around the edges for finishing.
- Read the number of repeats per axis and total molecules to know how many pattern units will fit.
- Use the minimum paper size output to check whether your paper is large enough for the desired grid.
How the result changes with Paper size (cm)
| Paper size (cm) | Repeats per axis | Total molecules |
|---|---|---|
| 13 | 2 | 4 |
| 19 | 3 | 9 |
| 38 | 8 | 64 |
| 63 | 13 | 169 |
What each input means
- Paper size (cm)
- Side length of the square paper in centimeters.
- Molecule size (cm)
- Size of one tessellation repeat unit (molecule) in centimeters.
- Pleat depth ratio
- How much extra paper each pleat consumes as a ratio of molecule size. Shallow pleats ≈ 0.3, deep ≈ 1.0+.
- Border margin (cm)
- Minimum untiled margin around the edges for finishing/anchoring.
What each result means
- Repeats per axis
- Number of molecule repeats fitting along one side of the paper.
- Total molecules
- Total tessellation units in the grid (rows × columns).
- Effective spacing (cm)
- Actual center-to-center spacing including pleat consumption.
- Actual border (cm)
- Resulting border margin on each side after fitting the grid.
- Visible area (cm²)
- Surface area of the visible tessellation pattern.
- Coverage (%)
- Percentage of total paper area covered by the visible pattern.
- Paper in pleats (cm²)
- Estimated paper area consumed by pleats underneath the surface.
- Min paper for this grid (cm)
- Minimum paper side length needed for the calculated grid size.
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersPaper size (cm) = 25, Molecule size (cm) = 3, Pleat depth ratio = 0.5, Border margin (cm) = 1 = 4 input(s) provided
- Calculate Repeats per axisRepeats per axis5 = 5
- Calculate Total moleculesTotal molecules25 = 25
- Calculate Effective spacingEffective spacing = moleculeSize * (1 + pleatDepth)4.5 = 4.5
- Calculate Actual borderActual border = (paperSize - gridCoverage) / 21.25 = 1.25
Engine last updated . Checked against 2 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why is my actual border bigger than the border margin I entered?
Border margin is only the minimum you're willing to accept — the calculator subtracts it once to find usable space, then fits as many whole molecule repeats as possible into that space. Because repeatsPerAxis is floored to a whole number, the grid almost never uses every last millimeter of usable space, so the leftover gets pushed back into the border on each side. The 'actual border' output tells you what you'll really see once the grid is packed in, which is always equal to or larger than what you entered.
What's the difference between coverage percentage and effective spacing?
Effective spacing is how much room each molecule takes up including the paper folded underneath its pleats — it's what determines how many repeats fit. Coverage percentage compares only the visible tessellation area (molecule size squared, without the pleat allowance) to the total sheet area, so it answers a different question: how much of the paper you'll actually see patterned versus left blank or hidden in pleats. A design can have tight effective spacing but still show a low coverage percentage if pleat depth is high.
How do I find my own pleat depth ratio instead of guessing?
Fold one full repeat of your molecule on scrap paper, then measure the actual center-to-center distance between where two repeats would sit versus the molecule's flat, unfolded size. Divide the difference by the molecule size to get your ratio — for example, if a 3cm molecule ends up needing 4.5cm of spacing once pleated, that's a ratio of 0.5. This beats guessing because pleat consumption depends heavily on your specific fold sequence, not just a generic 'shallow' or 'deep' label.
Why does 'paper in pleats' show up separately from the visible tessellation area?
The pleats are real paper — folded under the surface to create the 3D texture — but they don't add to the flat pattern you see from above, so the calculator tracks them as a separate area (totalMolecules × moleculeSize² × pleatDepth) rather than lumping them into visible coverage. This split matters if you're trying to estimate how much of your sheet is 'used' for the design versus how much is functionally load-bearing structure hidden from view.
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