Thin elastic sheets buckle to form a wide variety of morphologies that can be broadly categorised intotwo groups: smoothly wrinkled shapes, and sharply faceted shapes. According to common knowledge, the former is energetically dominated by external sources of work (like a substrate or boundary tension), while the latter is dominated by sources of work arising from the sheet’s intrinsic elasticity and geometry. In this thesis, we analyse a buckled morphology that shows characteristics of both these aforementioned categories. We call this intermediate category “faceted wrinkling”. Using numerical finite-element simulations, we study a minimal two-dimensional system: a circular annulus contracted at the inner boundary by fraction ∆, so that it buckles into a radial wrinkling pattern that shows sharp zig-zag faceting at the inner boundary. In our first result, we argue that this morphology results from the fact that the wrinkling is asymptotically isometric, i.e. its stretching energy approaches zero relative to its bending energy. To this end, we compare our numerically generated solutions to an Ansatz zero-thickness solution made up of alternating triangles and cones that is developable, and hence isometric, by design. We find this isometric cone-triangle Ansatz to agree with simulations over a wide range of values of system size, thickness, and wrinkle wavenumber and amplitude. In our second result, we address the mechanism that selects the wrinkle wavelength λ in such a pure-bending configuration. Usually, wavelength selection in elastic wrinkling occurs through macroscopic competition between the sheet’s bending energy and some external source of deformation work, like a (real or effective) substrate. What could select λ in the absence of any competition to the bending energy? Using our numerical simulations, we argue that competition between stretching and bending energies at mesoscopic scales leads to the selection of a wavelength scale sensitive to both the width w and thickness t of the sheet: λ∗ ∼ w2/3t1/3∆−1/6. This scale λ∗ corresponds to an arrest criterion for wrinkle coarsening in the sheet starting from any wavelength finer than λ∗: λ ≲ λ∗. However, the sheet can support coarser wavelengths, λ ≳ λ∗, since there is no penalty to their existence. Since this wavelength selection mechanism depends on the initial λ, it is path-dependent (or hysteretic).