Tikalon Blog by Dev Gualtieri

Cubic Earth

September 7, 2020

It's easy to find the important things in life, since they're usually taxed. Salt has been taxed through most of history. Salt is essential to life, since it supplies sodium ions that are the means for nerve impulse generation, proper functioning of the heart, and fluid balance. Salting is used as an easy means of food preservation, since the dehydrating action of salt kills most bacteria, fungi and other pathogens. Salt is also a much used food seasonings. I use a pinch of salt in my morning coffee, since it reduces the bitter taste,[1] but this is not recommended for people with hypertension.

A tax on salt in China around 300 BC was the main source of funding for the Great Wall. A salt tax was imposed in France in 1360, and in England in 1693, where it was doubled in just three years, repealed in 1825 and then established under British-ruled India from 1835. The word salary comes from the Latin word for salt, sal, a salarium ("salt money") being paid to Roman soldiers for the purchase of salt.

Left image, crystals of Halite (rock salt) created by slow crystallization from a concentrated salt solution at room temperature. Right image, the crystal structure of sodium chloride (NaCl) with the larger chlorine anions shown in green. While we think of salt in the context of food, most of the 250 million tons of annual production are used in chemical processes. (Left image; and, right image by Benjah-bmm27, both from Wikimedia Commons)

The mineral crystal form of salt (sodium chloride, NaCl) is called Halite. This crystal, also known as rock salt, takes the shape of its underlying crystal structure (see figure). This is true for many minerals. The crystals are cubic, and the crystal structure is cubic; specifically, the halite structure that's found in many other compounds. If you ignore the fact that the crystal is built from two different types of atoms, it's the simplest cubic structure known appropriately as "simple cubic." It can be viewed as interpenetrating face-centered cubic lattices of sodium and chlorine atoms.

A cube is one of the five Platonic solids. Platonic solids are the regular, convex polyhedrons, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, as shown in the table.

tetrahedron
cube
octahedron
dodecahedron
icosahedron

Plato (c.428 BC-c.348 BC), in his c. 360 BC Timaeus, associated each of the four classical elements (earth, air, water, and fire) with one of the regular polyhedra.[2] Earth was associated with the cube, air with the octahedron, fire with the tetrahedron, and water with the icosahedron. Aristotle, perhaps seeing that one of these, the dodecahedron, was not associated, added a fifth element, the Aether. Interestingly, on a macroscopic scale, there aren't too many cubes or squares in nature. You're more likely to see hexagons, a preferred shape, since hexagons can tile two-dimensional space.

While cubes seem to appear only rarely found in nature, a recent article in the Proceedings of the National Academy of Sciences demonstrates that cubes will arise when fragments of natural materials are appropriately averaged. This is a consequence of the isotropic stress conditions that formed them.[3-5] In simple language, it was found that when rocks break apart, the result is a bunch of cubes.[5] This rule seems to apply generally to materials over a size range from microscopic to planetary.[5] The research team had members from the University of Pennsylvania (Philadelphia, Pennsylvania), the Budapest University of Technology and Economics (Budapest, Hungary), and the University of Debrecen (Debrecen, Hungary).[4]

Says Douglas Jerolmack, a geophysicist at the University of Pennsylvania and one of the study authors,

"It turns out that Plato’s conception about the element earth being made up of cubes is, literally, the statistical average model for real earth. And that is just mind-blowing... If you take a three-dimensional polyhedral shape, slice it randomly into two fragments and then slice these fragments again and again, you get a vast number of different polyhedral shapes. But in an average sense, the resulting shape of the fragments is a cube."[4]

This research was inspired by geometrical models developed by Gábor Domokos, a mathematician at the Budapest University of Technology and Economics. In 2006, Domokos and colleagues showed the existence of the gömböc, a gemstone-like shape that has only one stable balance point conjectured in 1995 to exist. Domokos and his colleagues found that natural materials erode toward this shape but never quite reach it.[5] The early geometrical models also predicted the fragmentation of natural rocks into cubes.[4]

Juan_Gris, Portrait of Pablo Picasso, and a gömböc

Cubism or Gömböcism? The left image is a 1912 oil on canvas portrait of Portrait of Pablo Picasso by Juan Gris (1887-1927) done in the Cubist style and presently at the Art Institute of Chicago. On the left is the structure of a Gömböc, a shape described in the text. (Left image, a Wikimedia Commons image from the Google Art Project. Right image, a modified Wikimedia Commons image by vierkantswortel2.)

One possible reason why materials fragment into cuboid shapes is that the fragments must still fit together without any gaps, and the cube is a regular polyhedron with sides of equal length that does that.[3-4] As Jerolmack explains, "It turns out in two dimensions you're about equally likely to get either a rectangle or a hexagon in nature... They're not true hexagons, but they’re the statistical equivalent in a geometric sense. You can think of it like paint cracking; a force is acting to pull the paint apart equally from different sides, creating a hexagonal shape when it cracks."[4]

The research team used a supercomputer to model the fracture of three-dimensional materials under an idealized condition in which it's pulled equally in all directions. This produced polyhedrons that are ostensibly cuboid.[5] The team then searched for real world examples of this tendency, both in existing datasets and fragmentation patterns of rocks they collected.[4] One example, a dolomite mineral deposit near Budapest, Hungary, had objects with the expected cuboid fragments, regardless of whether they had been created naturally, or through dynamiting the deposit.[4-5]

An image of a patterned surface on Mars on the left is analyzed to extract its polygonal shapes (right). The average vertex number is found to be about 4.9, closer to a pentagon than a square. This is a portion of Fig. S6, supplementary information from ref. 3. The image of Mars is a NASA/JPL-Caltech/University of Arizona image.

This fracture into cubes doesn't happen just here, on Earth, but elsewhere in the Solar System.[4] However, at base, this phenomenon is just an approximation. As Jerolmack explains, "The world is a messy place... Nine times out of 10, if a rock gets pulled apart or squeezed or sheared - and usually these forces are happening together - you end up with fragments which are, on average, cubic shapes."[4] Minerals will cleave according to their crystal structure; for example, mica breaks into sheets, and they do not follow the results of the simulations since they do not have isotropic mechanical properties.[5] Says Jerolmack,

“When you pick up a rock in nature, it’s not a perfect cube, but each one is a kind of statistical shadow of a cube... It calls to mind Plato’s allegory of the cave. He posited an idealized form that was essential for understanding the universe, but all we see are distorted shadows of that perfect form."[4]

The data for this research is available at the Center for Open Science (https://osf.io/h2ezc/), and the computer code can be found at GitHub (https://github.com/torokj/Geometric_fragmentation).

Cubic Earth and Kepler's Solar System model from the Mysterium Cosmographicum

Humans have always looked for regularity in nature. On the right is Johannes Kepler's (1571-1630) Platonic solid model of the Solar System from his 1596 Mysterium Cosmographicum. The cubic Earth image was created using Inkscape. The "Blue Marble" Earth image, taken by the crew of Apollo 17, December 7, 1972, is from Wikimedia Commons. On the right is a modified Wikimedia Commons image.

Cubic Earth

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