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