Concept

Hinge crease — where it appears

A crease derived from a circle packing that lies on the common tangent where two discs touch. It is where a flap folds away from its neighbour, and its position is fixed by the packing rather than chosen.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as ridge crease — the same set of essays touches all of them, so they are one junction rather than several.

the treeleg0.62leg0.62body0.34arm0.50head0.78the basethe axis0.620.620.340.500.78the flaps are the tree's edges, at the tree's lengths, all square to one lineso the base's shadow along the axis is the tree, and nothing else can be designed this waywhich is the restriction the circle argument quietly depends on

Every flap on one axis

The tree method does not design a shape. It designs a base whose flaps all lie along a single line — and that restriction, which is almost never stated aloud, is what makes the circle argument true.

design · Uniaxial bases
what the packing gives5 discs, 4 contactsno two overlapping, checkedhinge creasesone per contact, perpendicularto the line of centresridge creasesalong the axial lines, dividingthe paper between the flapsthe packing is the hard part;this part is a constructioncorner discs of radius 0.28, the middle one 0.427 — every contact measured

From a packing to a crease pattern

The circles say where the flaps are. They do not say where to fold, and the step in between is a construction rather than a search — two families of crease, both determined by the packing, neither of them visible in the picture of the discs.

design · Circle packing
6 flaps at radius 0.1875883 contact graphs among the runs that agree about itone run5 contacts · 5 against the paper's edgeanother run4 contacts · 5 against the paper's edge

Two packings, one radius

A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.

design · Circle packing

Named alongside it

The objects these essays reach for when they reach for this one.

Ridge creaseUniaxial baseCircle packingAxial polygonContact graphDesign hypothesisOptimisationProjectionTree method

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