Crease Pattern Analyzer Calculator
Flat-foldability check from crease pattern.
About this calculator
Flat-foldability in origami is governed by two classical results: Maekawa's theorem, which says that at any interior vertex the number of mountain folds minus valley folds must equal plus-or-minus 2, and Kawasaki's theorem, which requires the angles around a vertex, taken in alternating order, to sum to 180 degrees on each side. This calculator checks the Maekawa condition directly and exactly from the mountain and valley fold counts you enter for a single vertex. It cannot check the full Kawasaki condition, since that requires the actual angle measurements around the vertex, which aren't collected as inputs — instead it checks only a necessary-but-not-sufficient proxy, that the total number of creases meeting at the vertex is even, since Kawasaki's alternating-angle condition can only hold with an even crease count.
A pattern can pass this even-count check and still fail the true Kawasaki condition, so a pass here should be read as "not yet disqualified" rather than "proven flat-foldable." Thickness estimates are separate and heuristic: the simple accordion model doubles thickness with every crease (2^n), which quickly becomes absurd for real models, so the calculator instead reports a realistic estimate that grows more slowly (roughly creases^1.3, capped at 1000x paper thickness) to approximate how actual origami layers overlap rather than stack uniformly. Use this as a quick single-vertex sanity check on a design, not a full crease-pattern validator — genuine flat-foldability analysis of a complete pattern requires checking every interior vertex and the full angle set.
Inputs
Results
Maekawa's theorem (1=pass)
1
Kawasaki's check (1=pass)
1
Figures current as of 1979. Sources: General case first proved by S. Murata (1966); independently rediscovered by Jacques Justin (1986) and popularly named for origami artist Jun Maekawa, Independently discovered in the late 1970s/early 1980s by Toshikazu Kawasaki, Jacques Justin, and Kôdi Husimi, whose special-case result appeared in his 1979 book The Geometry of Origami
How to Use This Calculator
- Count the mountain folds (convex ridges) meeting at the vertex you are analyzing and enter the number.
- Count the valley folds (concave grooves) at the same vertex and enter that number.
- Enter the total number of crease lines across the entire crease pattern.
- Enter the paper thickness in millimeters (standard kami is ~0.06–0.08 mm; copy paper ~0.10 mm).
- Check the Maekawa pass (1 = pass) — the absolute difference between mountain and valley folds must equal 2.
- Check the Kawasaki pass and the flat-foldable result to confirm whether the pattern can lie flat.
What each input means
- Mountain folds at vertex
- Number of mountain (convex) folds meeting at the vertex you are analyzing.
- Valley folds at vertex
- Number of valley (concave) folds meeting at the same vertex.
- Total crease lines in pattern
- Total number of crease lines across the entire crease pattern.
- Paper thickness (mm)
- Thickness of the paper in millimeters (standard kami ≈ 0.06-0.08 mm, copy paper ≈ 0.1 mm).
What each result means
- Maekawa's theorem (1=pass)
- Passes if |mountain - valley| = 2. Required for flat-foldability at a single vertex.
- Kawasaki's check (1=pass)
- Passes if total crease count at vertex is even (necessary condition for Kawasaki's theorem).
- Flat-foldable (1=yes)
- 1 if both Maekawa and Kawasaki conditions are met, 0 otherwise.
- |Mountain - Valley|
- Absolute difference between mountain and valley fold counts. Must equal 2.
- Total folds at vertex
- Sum of mountain and valley folds at the analyzed vertex.
- Estimated center layers
- Approximate paper layers at the center of the folded model.
- Max thickness (mm)
- Estimated maximum thickness at the thickest point of the folded model.
- Crease density (per cm²)
- Crease line density assuming a 15 cm standard square.
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersMountain folds at vertex = 4, Valley folds at vertex = 2, Total crease lines in pattern = 20, Paper thickness (mm) = 0.1 = 4 input(s) provided
- Calculate Maekawa's theorem1 = 1
- Calculate Kawasaki's check1 = 1
- Calculate Flat-foldable1 = 1
- Calculate |Mountain - Valley||Mountain - Valley|2 = 2
Figures and sources
- Maekawa's theorem (mountain − valley fold count = ±2 at a flat-foldable vertex) (1966) — General case first proved by S. Murata (1966); independently rediscovered by Jacques Justin (1986) and popularly named for origami artist Jun Maekawa
- Kawasaki's theorem (alternating vertex angle sum = 180° at a flat-foldable vertex) (1979) — Independently discovered in the late 1970s/early 1980s by Toshikazu Kawasaki, Jacques Justin, and Kôdi Husimi, whose special-case result appeared in his 1979 book The Geometry of Origami
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 must mountain minus valley folds equal exactly 2, not any even number?
This is Maekawa's theorem, a proven result for any flat-foldable interior vertex: the absolute difference between mountain and valley fold counts is always exactly 2, never 0, 4, or any other value. First proved in general form by S. Murata in 1966 and later popularized under origami artist Jun Maekawa's name, the theorem is checked here directly from your mountain and valley fold counts (maekawaDiff = |mountainFolds − valleyFolds|), and any value other than 2 means that vertex cannot fold flat as given.
Why does passing the Kawasaki check here not guarantee the pattern is flat-foldable?
The full Kawasaki theorem — independently discovered around the late 1970s and early 1980s by Toshikazu Kawasaki, Jacques Justin, and Kôdi Husimi — requires the actual crease angles around the vertex, taken in alternating order, to sum to 180 degrees, but this calculator only has fold counts as inputs, not angles. It substitutes a necessary-but-not-sufficient proxy: checking that the total number of creases at the vertex is even, since an odd count can never satisfy the true alternating-angle condition. A pattern can pass this proxy and still fail the real Kawasaki test once actual angles are measured.
Why does the realistic thickness estimate grow so much more slowly than 2^n?
The simple accordion model doubles thickness with every single crease, which is only true for a design where every fold lands directly on top of the previous layer — real origami models spread folds across the paper so layers overlap partially rather than stacking perfectly. The calculator's realistic estimate instead scales roughly as creases^1.3 (capped at 1000x paper thickness) to better reflect how actual folded models build up layers.
Should I trust this as a full validation of my crease pattern?
No — it checks only a single vertex you specify, not every interior vertex in the full pattern, and even at that one vertex it can't verify the true Kawasaki angle condition without angle inputs. Treat a pass here as a quick sanity check that hasn't disqualified the design yet, not as proof the complete pattern will fold flat.
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