Mathematics of Paper Folding
The discipline of origami has received considerable mathematical study. Understanding these principles is essential for creating valid crease patterns that can actually be folded flat without tearing or stretching the paper.
|M - V| = 2
At any interior vertex of a flat-foldable crease pattern, the number of mountain folds (M) minus the number of valley folds (V) always equals ±2.
Key Implications:
- Every interior vertex must have an even number of creases
- The regions between creases can always be two-colored
- This is a necessary condition for flat-foldability
INTERACTIVE DEMONSTRATION
|M - V| = |4 - 4| = 0
Invalid. The difference must equal 2, not 0.
θ₁ + θ₃ + θ₅ + ... = θ₂ + θ₄ + θ₆ + ... = 180°
At any interior vertex, the sum of alternating angles around the vertex must equal 180 degrees. This is also known as the Kawasaki-Justin theorem.
Key Implications:
- Determines whether a crease pattern can fold flat at a single vertex
- Necessary but not sufficient for global flat-foldability
- Combined with Maekawa's theorem, provides strong constraints on valid patterns
INTERACTIVE DEMONSTRATION
Adjust the angles around a vertex. For flat-foldability, alternating angles must sum to 180°.
ODD ANGLES (θ1 + θ3 + θ5)
270°
Target: 180°
EVEN ANGLES (θ2 + θ4)
90°
Target: 180°
Invalid. Both alternating sums must equal 180°.
A sheet of paper can never penetrate itself during folding. This constraint determines the valid layer ordering of a folded model and is crucial for determining global flat-foldability.
Computational Complexity
Determining whether a crease pattern can be folded flat (considering all three conditions) is NP-complete, as proven by Marshall Bern and Barry Hayes in 1996.