Think of a soccer ball, covered in both hexagons and pentagons. Domokos crunched the numbers for the surface of a sphere and found that on a globe, Voronoi mosaic cells should average 5. One idea holds that plates are just a byproduct of burbling convection cells deep in the mantle. The observed Voronoi pattern of plates, reminiscent of much smaller mud flats, might support the second argument, Jerolmack said. In three dimensions, meanwhile, exceptions to the cuboid rule were rare enough.
And they too could be produced by simulating unusual, outward-pulling forces. One distinctively non-cubic rock formation lies on the coast of Northern Ireland, where waves lap against tens of thousands of basalt columns. Crucially, those columns and other similar volcanic rock formations are six-sided. Zooming out, the team argues, you could classify most real fractured-rock mosaics using just Platonic rectangles, 2D Voronoi patterns, and then — overwhelmingly — Platonic cubes in three dimensions.
Each of these patterns could tell a geological story. And yes, with the appropriate caveats, you really could say the world is made of cubes. Specifically, perhaps you could take a real fractured field site, count up things like vertices and faces, and then be able to infer something about the geological circumstances responsible.
As for Jerolmack, after first feeling uncomfortable over a possibly coincidental connection to Plato, he has come to embrace it.
After all, the Greek philosopher proposed that ideal geometric forms are central to understanding the universe but always out of sight, visible only as distorted shadows. This article was reprinted on Wired. Get highlights of the most important news delivered to your email inbox. Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.
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Thomas Shahan for Quanta Magazine. The Moeraki Boulders in New Zealand. Daniel Lienert. Fred Scharmen. Counting the Crust Midway through the project, the team met in Budapest and spent three whirlwind days sprinting to incorporate more natural examples. The Quanta Newsletter Get highlights of the most important news delivered to your email inbox. Show comments. That Can Be a Good Thing. Probably the most famous of all is DNA — deoxyribonucleic acid — a double helix of two chains running in opposite directions.
The discoverers of DNA rightly predicted that helixes would be one of the simplest of shapes to spontaneously assemble through inter-molecular forces. They can also flex without buckling, lengthen without breaking, and are capable of rotation without deformation. Stecco All known filamentous viruses are helical most of the rest are icosahedral fig. In proteins, sequences of amino acids can fold into helixes and further twist around each other to form double or triple coiled-coils super-coils fig.
Of the huge number of possible amino acid combinations, protein folding is limited to a set of about different forms because of some basic self-assembly rules, analogous to the laws of chemistry or crystallography, which correspond to an energy minimum. Denton et al. Within the cytoplasm, the prestressed cytoskeletal lattice consists of microtubules — tightly packed helices of globular tubulin protein under compression fig. Ingber ; Brangwynne et al ; Maguire et al Experimental support for the cytoskeleton as a tensegrity structure now seems overwhelming Ingber although some studies have been unable to confirm it.
Heidemann et al ; Ingber et al The cellular cortex is essentially made from triangulated hexagons of the helical protein spectrin fig.
It has also been modelled around an icosahedron. Li et al Deformation of the membrane network may cause turbining of the actin protofilaments through the suspension mechanism, thereby facilitating oxygen transfer from one side of the membrane to the other.
This arrangement of different components at varying size scales is a common feature in biology, where the functions of each one contribute to a higher collective function within a hierarchy of structures figs. The nano-structures of collagen and spider silk have both been described as tensegrities Skelton et al within a hierarchical fibrous structure fig. More than twenty different types have been described in different tissue specific combinations which are particularly able to resist tensional stresses.
At the nano-scale three helical procollagen polymers wind around each other to form a triple helix of tropocollagen. These molecules then arrange laterally in a quasi-hexagonal configuration with cross-linking to form collagen microfibrils, and pack sequentially in a hierarchy to form a subfibril, fibril and collagen fibre fig. Collagens and spider silk are also examples of liquid crystal elastomers — different states, or mesophases of matter, that lie between liquids and solid crystals.
Even though spider web fibres are secured at their ends to what is effectively a continuous compression component, the whole web has been classed as a tensegrity on structural engineering grounds fig. The self-similar geometrical structure of collagenous tissues has been demonstrated in the fractal-like character of their polarization properties, with degenerative-dystrophic changes revealed by alterations in these properties.
Ho et al ; Angelsky et al This observation may be of value in pre-clinical pathological diagnostics, and a correlation between these findings and tissue palpation would be useful research in the future. Fractal analysis is commonly applied to natural structures, where similar shapes and patterns appear at different size scales, linking hierarchies throughout the body. Mandelbrot ; Skelton et al ; Jelinek et al The branching patterns of blood vessels, the bronchial tree and nerve fibres all display this property.
Zamir ; Pelagyia et al ; Phalen et al ; Thomas et al These characteristic shapes are developmental remnants of nonlinear dynamic systems which were sensitive to small changes in the local environment, created instabilities in growth and caused the typical branching pattern which extends from the micro to the macro scale. Goldberger et al The extracellular matrix, surrounding virtually every cell in the body, provides a branching structural framework which extends through the fascia to the whole organism.
It attaches to the cellular cytoskeleton through adhesion molecules in the cell membrane, allowing a transfer of mechanical forces between them and changes in the cytoskeletal tension. Ingber a,b, Multiple intra-cellular signalling pathways are activated as a result which provide multiplexed switching between different states such as cell growth, differentiation or apoptosis. Conversely, local tensional stresses within the cytoskeleton transfer to the extracellular matrix and produce effects on other cells at some distance.
Long-distance transfer of mechanical forces between different tissues could then spatially orchestrate their growth and expansion, allowing complex multi-cellular tissue patterns to emerge through interactions among a hierarchy of different components. Multi-modular hierarchies of form and function can thus be linked, with simplicity evolving into complexity, and the whole system mechanically functioning as a unit.
Stecco p31 described the fascia as a tensioned network which may coordinate the motor system in a way that the central nervous system is incapable of. The icosahedron is particularly useful in modelling the tensegrities of biological structures, because it demonstrates both geodesic dome and tensegrity properties which can be connected to form an infinite variety of shapes figs.
Even the tension and compression elements can be made from interlinked icosahedra, themselves constructed from smaller icosahedra, repeating further in a self-similar structural hierarchy. Biological materials deal with both types of loading by separating them into nano and micro structures within a hierarchy. Gordon ; Lakes ; Puxkandl et al ; Gao et al ; Gupta et al ; Brangwynne et al Inferences that this is ultimately due to their tensegrity construction have been made.
Skelton et al ; Levin ; Ingber In addition, when spheres are added to the outside of an icosahedron they do not close-pack completely, and an instability develops figs. It may now be seen that the distinction between geodesic dome and tensegrity is really just relative to the scale at which they are observed, a corollary which should be added to any definition.
Although geodesic domes appear limited to the cellular size level, tensegrities almost certainly dominate beyond this. At the macro scale the fascia, muscles, ligaments and capsules provide the tension; while the bones and tissue bulk of muscles, organs and fluid-filled vessels resist compression.
Tensegrities within Tensegrities. Levin was the first to describe the higher complexities of the human body in terms of tensegrity, using the analogy of a bicycle wheel, where the compression elements of the central hub and outer rim are held in place by a network of wire spokes in reciprocal tension.
He suggested that the scapula may function as the hub of such a wheel, in effect as a sesamoid bone, and transfer its load to the axial skeleton through muscular and fascial attachments. Levin ,a. The sterno-clavicular joint is not really in a position to accept much compressional load, and the transfer of compression across the gleno-humeral joint has been found to be at maximum only when loaded axially at 90 o abduction.
A humerus hub model would function equally well with the arm in any position. In this respect, zero compression across the femur and meniscii has been observed during arthroscopy in an extended knee joint under axial loading in vivo. Levin b. The pelvis is also like a wheel with the iliac crests, anterior spines, pubis and ischia representing the outer rim; and the sacrum representing the hub, tied in with strong sacro-iliac, sacro-tuberous and sacro-spinous ligaments.
Joint movements display helicoid motion around a variable fulcrum and this is demonstrated in the simple tensegrity elbow in figure 31, a physiological feature not found in most other models. Kushner ; Kjaer As part of a tensegrity structure, each attachment would influence all the others, distributing forces throughout the system and avoiding points of potential weakness; Skelton et al ; Masic et al in contrast to a pure geodesic chain or truss which is vulnerable to buckling fig. Such a mechanism would be an advantage in long-necked animals such as giraffes and dinosaurs, where the load from the head is distributed throughout the neck fig.
Levin The erect spine and bipedal weight bearing capability of humans has traditionally been viewed as a tower of bricks and compressed disc joints, transferring the body weight down through each segment until it reaches the sacrum, but this is a relative rarity amongst vertebrates.
Most other species have little or no use for a compressive vertebral column, which is frequently portrayed as a horizontal truss and cantilever support system. Thompson p; Gordon p As the main difference in vertebrate anatomies is in the detail, it seems reasonable to suppose that they have some structural properties in common.
Levin ; Tensegrities are omni-directional, i. Tension is provided by the dura mater — a tough membrane covering the brain — which regulates bone growth, maintains the separation of vault bones at the adjoining sutures, and integrates the whole structure into a single functional unit. It has been suggested that the brain influences the vault to grow outwards, through the dura mater, rather than physically pushing it out. Scarr Balanced and symmetrical tensegrities automatically assume the configuration that minimizes their stored elastic energy, with changes in shape that require very little control energy; in contrast to classical structures where significant energy is requi red to work against the old equilibrium.
Skelton et al ; Masic et al Their high yield strain allows large shape changes to be accomplished at no loss in stiffness Masic et al a distinctive feature in biological structures. The icosahedron has been described as an intermediary in a potential oscillating system Fuller sec. Levin If the vector equilibrium cuboctahedron fig. Further compression causes the equator to twist and fold, and the structure transform itself through an octahedron, to the tetrahedron, and back again.
The icosahedron is at the lowest energy state within the system, and the point around which changes in shape occur. The ability… to generate elaborate and beautiful forms… comes from a simple but fundamental principle which governs the deep structure of the physical universe: symmetry. The ubiquitous nature of symmetry offers a simple explanation for stable crystal lattices and other regular patterns, because of the balance of forces. Instabilities in the dynamics of living systems also generate complex patterns and shapes within hierarchies of structure, as self-similar shapes fractals scale up through dilation — one of the four principal types of symmetry transformation.
Essentially, the sum of all the asymmetries is symmetrical…Thus, we have patterns of symmetry that reflect the underlying symmetrical nature of the universe, and patterns and forms that are generated by instabilities in the system. That different structures should emerge at different levels in complex organisms is typical of evolutionary selection. Over hundreds of millions of years chance mutations in the genetic code occasionally gave rise to new characteristics which conferred an advantage to the organism, whilst existing traits which remained useful were retained, through natural selection.
This bifurcation in development creates asymmetries which must be balanced in order to permit new higher order symmetries. Stewart p Organization is still part of the self-assembly capacity, but the resulting form may have a very different appearance from its component substructures figs. The tetrahedron is one of the simplest of shapes in 3-D because of its close-packing efficiency, minimal-energy configuration and triangular stability.
The hexagon, and cubic symmetry of these shapes links them to the polyhedron most suited to fulfill structural evolution in biology, which is the icosahedron, possibly part of the substructure predicted by Ramsey. Self-assembling helical proteins carry the tensegrity principle into the nano-structures of the cell and extracellular matrix; and through structural hierarchies to the whole body. Denton et al Stephen Jay Gould Thompson xi.
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