Tending to the Facial Surfaces of a Mathematical Biology Head-Scratcher: Why Does the Head of the Sea Turtle Natator depressus Resemble a Convex Zygomorphic Dodecahedron?

Two convex polyhedra that markedly resemble the head of the flatback sea turtle hatchling are identified. The first example is a zygomorphic tetragonal dodecahedron, while the other, an even better matching structure, is a related tetradecahedron, herein speculated to arise from this particular dodecahedron via known mechanisms gleaned from studies of the behavior of foams. A segmented, biomorphic, convex polyhedral model to address cephalic topology is thus presented stemming from solid geometry, anatomical observations, and a recently computed densest local packing arrangement of fifteen slightly oblate spheroids in which fourteen oblate spheroids surround a central such spheroid. This particular array of oblate spheroids shares salient structural features with the aforementioned dodecahedron. Successful testing of the model has been achieved by converting this array of fifteen oblate spheroids constructed with putty to the cephaloid dodecahedron in a process involving ventral elongation induced by stretching in the anterior direction along the anteroposterior axis (convergent extension). During convergent extension, the two left most anterolateral oblate spheroids that are in direct contact with the ventral spheroid of the array merge into a single lateral facet of the incipient dodecahedron, while the corresponding two right such oblate spheroids do the same. Thus, the fourteen outer oblate spheroids of the array give rise to the twelve facets of the finalized dodecahedron, while the central oblate spheroid remnant assumes an interior dodecahedral position. The hypothetical dodecahedron to tetradecahedron transformation entails the collapse of a tetravalent vertex (which is known to occur in foams as part of a T1 transition) followed by bilateral facet splitting. Remarkably, a model stipulating that convexity is to be retained in connection with this sequence of steps necessitates that the starting dodecahedral template undergoes modification to become a tetradecahedron in possession of precisely the highly ordered feature found at the top of the head of numerous specimens of the flatback sea turtle hatchling, namely, a fused medial pentagon-heptagon pair in the form of a pentagonal frontal scute and heptagonal frontoparietal scute. Such a possible new instance of geometric biomorphy, taken together with the correct anticipation of the cephalic pentagon-heptagon pair, might serve to instill further confidence in renewed efforts to shed light on morphogenesis with foam embryo models.