Developability — where it appears
Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.
Two conditions at a point
Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.
Which way the disc curves
A growing disc either domes or ruffles, and which one it does is not a matter of how much it grew. It is decided by where the growth was — more at the rim opens the sheet, more at the middle closes it — and one number in one formula takes it through both.
A crease carries no curvature
A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.
Where curved creases meet
A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.
What one cut buys
A fold moves paper about and cannot change how much of it surrounds a point. A cut can, and that one difference is the whole of what this site's founding rule is worth. Take a wedge out and the sheet closes into a cone; let one in and it has more paper than the plane will accept.
The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
Solving every face at once
A quadrilateral mesh that folds rigidly has to close round every one of its faces, and the rung that built the general mesh could close one. Four of them at once resisted a descent that drove each free length to its own root, because closing a loop is a condition on several lengths together — and solving them jointly finds a sheet with no two vertices alike that folds, and a surface of them sixteen dimensions wide.
The vertices nobody checks
Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.
Nothing grown has a seam
A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.
The test measures the rim
Flattening the specimen is the right test and the measurement anybody actually makes on a flattened specimen is a boundary one — how far the margin overruns its chord. Total curvature is a boundary quantity too: it equals minus two pi R times the growth profile's slope at the rim, and nothing else about the interior survives into it. So a sheet can be curved everywhere and integrate to nothing, and the test reports it flat.
Named alongside it
The objects these essays reach for when they reach for this one.
Angle deficitGaussian curvatureGrowthKawasaki's theoremConeDifferential growthMaekawa's theoremMetricAssignmentThe big-little-big lemmaBoundaryBoundary vertex