Fig. 2: Polyhedral structure of thermodynamic constraints.
From: A polyhedral structure controls programmable self-assembly
a, A constraint plane corresponding to a structure s, defined by \({{\bf{M}}}_{s}\cdot \hat{{\bf{\xi }}}=0\), separates the parameter space into two half-spaces. The grey half-space in which \({{\bf{M}}}_{s}\cdot \hat{{\bf{\xi }}} > 0\) is physically forbidden in the limit of high λ, which restricts the allowed limiting directions to the space \({{\bf{M}}}_{s}\cdot \hat{{\bf{\xi }}}\le 0\). b, The intersection of all the allowed half-spaces forms a convex polyhedral cone. Redundant constraint planes may ‘touch’ the cone, but the cone is unaffected by their presence. For clarity, this figure sketches a two-dimensional slice through a higher-dimensional space. c, A designable structure corresponds to every (d – 1)-dimensional face of the constraint cone. Lower-dimensional faces correspond to designable sets of structures. Structures corresponding to redundant constraints are not designable. A three-dimensional cartoon of the constraint cone is shown in the lower-left inset. d, Relationship between the faces of the constraint cone, and therefore, the designable sets can be visualized as a Hasse diagram, as sketched in the cartoon example here. Nodes in the diagram correspond to faces/designable sets and edges indicate a containment relation: a df-dimensional set is contained in a (df − 1)-dimensional set if they are connected in the diagram.
