For what dimensions does confinement make vertices? ridges?
suppose sheet has m
material dimensions
–E. Kramer, B. DiDonna, 1990Õs
d ³ 2m, eg fiber in circle
each material axis can curve in
an independent direction:
confinement: all lines in material must curve to fit in confined
volume
... no stretching; no
singularities
d = 2m - 1, eg sheet in sphere
only m-1 directions available
for curving
one direction through each point must be straight line, ...unconfinable
for confinement, the sheet must stretch (at least) at isolated
points: vertices
d = 2m - 2, 3, 4... k
each point has k uncurved directions, forming flat
k-space extending to the boundary ... unconfinable
for confinement, sheet must be punctured with k-1-dimensional vertices
d = m + 1
only one curving direction is possible. 2-dimensional
planes in the material look
like
vertices create ridges