Slide 1

Òenergy condensationÓ in matter

Elastic focusing: sharp structure from smooth forcing

Singularities in thin sheets  I

Conditions for crumpling: thin elastic manifold

Bending vs stretching when thickness h ¨ small

Confining an unstretchable object

When does curvature create stretching in d dimensions?

For what dimensions does confinement make vertices? ridges?

Isometric (unstretchable) limit accounts for singularities

Slide 11

Energy of an isolated vertex is logarithmic

Òkite shapeÓ explains emergent width scale w of ridge

numerics confirm scaling of curvature profile with thickness

virial theorem confirms stretching-bending competition

implications of stretching ridges

growth of ridge energy with opening angle a

Induced edge singularities

Induced edges are weaker than real ridges

Unstretchable d-cone: sets stage for d-cone puzzles

D-cone core: what limits the focusing?

Simulation tests scaling of Rc

Slide 23

Gaussian charge optimisation confirms ridge scaling

	Stretching energy can be expressed as an integral of gaussian curvature analogous to electrostatic energy.
	A stretching ridge has negative gaussian curvature.
	It must be compensated by an equal amount of positive gaussian curvature on the adjacent flanks
	Optimising the width w of this ÒchargeÓ distribution gives
	w ~ X (X/h)-1/3  ,  confirming other methods.

Curvature constraints from Gauss Bonnet theorem