Paper Can Do Math: The Geometry of Folding
Every fold is a mathematical operation
Every fold is a mathematical operation
Every origami model uses just two fold types. Click each demo to see how the paper bends!
Click to fold
Click to fold
Side View Comparison (Cross-Section)
Paper folds UP and away from you, creating a ridge like a mountain peak. In crease patterns: solid red line.
Paper folds DOWN toward you, creating a trough like a valley. In crease patterns: dashed blue line.
When you fold one edge of paper onto another edge, the crease always bisects the angle perfectly. Click to fold!
Adjust the angle, then click "Fold Edge to Edge" to see the bisector
To bisect any angle with a ruler and compass takes multiple steps. With paper? Just one fold!
Folding is a reflection. For one edge to land exactly on another, the fold must be equidistant from both—that's the bisector!
When fold lines meet at a point, there's a magic rule: you need exactly 2 more of one fold type than the other for the paper to fold flat!
Click any fold line to switch between Mountain (red) and Valley (blue)
A vertex is where multiple fold lines meet at a single point. Think of it as the "hub" of the folds.
If the difference isn't exactly 2, the paper will collide with itself and can't fold flat!
Here's another rule for flat folding: add up every other angle around a vertex, and you'll always get 180°!
Drag the orange handles to change the angles
When paper folds flat, it's like closing a book. Each "page" needs to close completely — that's why opposite angles balance out!
Skip an angle, add the next. Skip an angle, add the next. The total is always 180° if the pattern can fold flat.
This zigzag fold pattern lets huge surfaces collapse into tiny packages. Drag to fold and unfold!
Solar panels on satellites use Miura-ori to pack huge arrays into tiny rocket fairings, then unfold in orbit!
Unlike regular folds, Miura-ori unfolds with a single pull - all creases move together!
Some maps use this pattern - they unfold and refold easily without wear on the creases.
You've discovered that origami isn't just art - it's geometry in action. Every fold follows precise mathematical rules!
Mountain folds go up (like a tent), valley folds go down (like a valley).
Every fold creates a perfect angle bisector - no ruler needed!
At any vertex, mountains minus valleys always equals +2 or -2.
For flat foldability, alternating angles around a vertex sum to 180°.
This zigzag pattern lets huge surfaces fold into tiny packages - used on satellites!
Put your new knowledge into practice!