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.

Keywords: Voronoi; echinoids; geometric pattern; stereom; trabecular system.

Figure 1.

Echinoid test, skeletal plate, and tubercle zone. (a) Paracentrotus lividus test and (b) the extracted interambulacral plate. (c) Illustration of the four tubercle regions analysed: bottom, left, right and top. (d) Illustration of the primary spine tubercle and associated soft tissues (muscle and catch apparatus): (TM) mamelon, (TP) platform, (TB) boss and (TA) areola.

Figure 2.

Tubercle architecture and regions examined. (a) SEM micrograph (top view) of Paracentrotus lividus interambulacral plate showing the primary spine tubercle and its stereom microstructural variability. Three stereom types can be recognized: (1) microperforate, (2) galleried, and (3) labyrinthic. In addition, the mamelon of secondary spines (ms) is also shown. The region topographic reference is underlined by a solid line circle in which the pores (p) and trabeculae (t) are indicated (arrows). (b) Micro-CT scan of tubercle boss subsection extracted by P. lividus interambulacral plate showing (a) transversal, (b) sagittal and (c) coronal views.

Figure 3.

Trabecular analysis of Paracentrotus lividus TB stereom. (a) SEM micrograph binarization, (b) skeletonization of the binarized image, and (c) computation of segment-node configuration.

Figure 4.

Pore analysis of Paracentrotus lividus TB stereom. (a) Stereom binarization, (b) pore identification and (c) computation of the Voronoi model from centroids.

Figure 5.

Voronoi divergence analysis of Paracentrotus lividus TB stereom. (a) Skeletonization of the binarized stereo; (b) computation of the Voronoi model; (c) superimposition of the actual stereom (a) and the Voronoi model (c).

Figure 6.

Voronoi polygonal shape. Planar Voronoi model generated from a seed having (a) four, (b) five, (c) six and (d) seven neighbour seeds.

Figure 7.

Non-metric multidimensional scaling (nMDS) ordination of apical and ambital samples for each plate region. Dimensions k = 2; stress = 0.12. ‘Ap’ and ‘Am’ represent the plate types and stand for apical and ambital plate, respectively; bottom, left, right and top represent the plate regions.

Figure 8.

Divergence trend of the skeleton and Voronoi comparisons. ‘Ap’ and ‘Am’ represent the plate types and stand for apical and ambital plate, respectively; bottom, left, right and top represent the plate regions.

Figure 9.

Histograms of number of neighbour seeds or number of TB Voronoi cell sides. Mean values (±SD) for each plate type sample (Ap = apical plate and Am = ambital plate) among its regions (bottom, left, right and top).

Figure 10.

K function estimation results. The blue line represents the estimation of K(h)−πh² for apical (Ap2 left) and ambital (Am2 left) plates. The plotted functions (blue lines) indicate that K(h), varying h, is smaller than πh² and lies below the confidence band (dashed lines) departing from CSR.