  Twisted ribbons subjected to a tension exhibit a remarkably rich morphology,
from smooth and wrinkled helicoids, to cylindrical or faceted patterns. These
shapes are intimately related to the instability of the natural, helicoidal
symmetry of the system, which generates both longitudinal and transverse
stresses, thereby leading to buckling of the ribbon. In this paper, we focus on
the tessellation patterns made of triangular facets. Our experimental
observations are described within an "asymptotic isometry" approach that brings
together geometry and elasticity. The geometry consists of parametrized
families of surfaces, isometric to the undeformed ribbon in the singular limit
of vanishing thickness and tensile load. The energy, whose minimization selects
the favored structure among those families, is governed by the tensile work and
bending cost of the pattern. This framework describes the coexistence lines in
a morphological phase diagram, and determines the domain of existence of
faceted structures.

Soft Condensed Matter;Materials Science