Hexagonal Voronoi pattern detected in the microstructural design of the echinoid skeleton

Abstract

Repeated polygonal patterns are pervasive in natural forms and structures. These patterns provide inherent structural stability while optimizing strength-per-weight and minimizing construction costs. In echinoids (sea urchins), a visible regularity can be found in the endoskeleton, consisting of a lightweight and resistant micro-trabecular meshwork (stereom). This foam-like structure follows an intrinsic geometrical pattern that has never been investigated. This study aims to analyse and describe it by focusing on the boss of tubercles—spine attachment sites subject to strong mechanical stresses—in the common sea urchin Paracentrotus lividus . The boss microstructure was identified as a Voronoi construction characterized by 82% concordance to the computed Voronoi models, a prevalence of hexagonal polygons, and a regularly organized seed distribution. This pattern is interpreted as an evolutionary solution for the construction of the echinoid skeleton using a lightweight microstructural design that optimizes the trabecular arrangement, maximizes the structural strength and minimizes the metabolic costs of secreting calcitic stereom. Hence, this identification is particularly valuable to improve the understanding of the mechanical function of the stereom as well as to effectively model and reconstruct similar structures in view of future applications in biomimetic technologies and designs.

A Devonian crinoid with a diamond microlattice

“…Echinoderms elaborate a hierarchically organized lightweight calcite endoskeleton composed of numerous plates with a distinct trabecular microstructure (stereom). A number of different geometrical stereom patterns with unique mechanical designs have been described [11,[20][21][22]]. The most striking example, which has attracted the interest of engineers, is a dual-scale, damage-tolerant diamond-TPMS recently discovered in the skeleton of the knobby starfish P. nodosus [13].…”

Echinoid skeleton: an insight on the species-specific pattern of the Paracentrotus lividus plate and its microstructural variability

“…The low value of the ratio between C 11 and C 22 confirms the anisotropic behaviour and specialization to withstand stresses along the horizontal direction. The tubercle galleried stereom constitutes the tubercle boss and is characterized by aligned trabeculae, which are recently discovered to be arranged as a Voronoi construction with a prevalence of hexagonal polygons, and a regularly organized seed distribution [23]. However, this stereom is different from the one located at the suture for number of nodes, segments, curved length, chord length, radius, thickness, number of pores and pore area as reported by two-sample Wilcoxon Mann-Whitney test results.…”

“…These stereom types are assembled into highly different combinations creating species-specific structural patterns. These patterns can lead to specific mechanical behaviours, and several studies are now emerging to understand their hidden functional meaning and to effectively model and reconstruct similar structures in view of future applications in biomimetic technologies and designs [22,23].…”

Composite material in the sea urchin Cidaris rugosa : ordered and disordered micrometre-scale bicontinuous geometries

“…Since then, authors have designated the 'rectilinear' stereom type, found in several species of Cidaroids [8], to the Primitive surface [7,12,26]; however a three-dimensional quantitative descriptive of this stereom type is lacking. Recently, a related ordered bicontinuous form, the so-called Diamond surface, was discovered in the ossicles of the starfish Protoreaster nodosus [3] and in the fossilized skeleton of a 385 Myr crinoid Haplocrinites boitardi [12], sparking a renewed interest into the calcite microstructures of echinoderms, including their potential in biomimetics [4,5,27,28].…”