In her fifth paper on cyclols (1937), Wrinch identified the conditions under which two planar cyclol fabrics could be joined to make an angle between their planes while respecting the chemical bond angles. She identified a mathematical simplification, in which the non-planar six-membered rings of atoms can be represented by planar "median hexagon"s made from the midpoints of the chemical bonds. This "median hexagon" representation made it easy to see that the cyclol fabric planes can be joined correctly if the dihedral angle between the planes equals the tetrahedral bond angle δ = arccos(-1/3) ≈ 109.47°.
A large variety of closed polyhedra meeting this criterion can be constructed, of which the simplest are the truncated tetrahedron, the truncated octahedron, and the octahedron, which are Platonic solids or semiregular polyhedra. Considering the first series of "closed cyclols" (those modeled on the truncated tetrahedron), Wrinch showed that their number of amino acids increased quadratically as 72n, where n is the index of the closed cyclol C. Thus, the C cyclol has 72 residues, the C cyclol has 288 residues, etc. Preliminary experimental support for this prediction came from Max Bergmann and Carl Niemann, whose amino-acid analyses suggested that proteins were composed of integer multiples of 288 amino-acid residues (n=2). More generally, the cyclol model of globular proteins accounted for the early analytical ultracentrifugation results of Theodor Svedberg, which suggested that the molecular weights of proteins fell into a few classes related by integers.
The cyclol model was consistent with the general properties then attributed to folded proteins. (1) Centrifugation studies had shown that folded proteins were significantly denser than water (~1.4 g/mL) and, thus, tightly packed; Wrinch assumed that dense packing should imply regular packing. (2) Despite their large size, some proteins crystallize readily into symmetric crystals, consistent with the idea of symmetric faces that match up upon association. (3) Proteins bind metal ions; since metal-binding sites must have specific bond geometries (e.g., octahedral), it was plausible to assume that the entire protein also had similarly crystalline geometry. (4) As described above, the cyclol model provided a simple chemical explanation of denaturation and the difficulty of cleaving folded proteins with proteases. (5) Proteins were assumed to be responsible for the synthesis of all biological molecules, including other proteins. Wrinch noted that a fixed, uniform structure would be useful for proteins in templating their own synthesis, analogous to the Watson-Francis Crick concept of DNA templating its own replication. Given that many biological molecules such as sugars and sterols have a hexagonal structure, it was plausible to assume that their synthesizing proteins likewise had a hexagonal structure. Wrinch summarized her model and the supporting molecular-weight experimental data in three review articles.