*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 |

"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]

*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).

*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.*

- P. A. S. Breslin and G. K. Beauchamp, "Salt enhances flavour by suppressing bitterness," Nature, vol. 387 (June 5, 1997), pp. 563ff., https://doi.org/10.1038/42388.
- Plato, "Timaeus," Benjamin Jowett, Trans., at the The Internet Classics Archive by Daniel C. Stevenson.
- Gábor Domokos, Douglas J. Jerolmack, Ferenc Kun and János Török, Plato's cube and the natural geometry of fragmentation," Proceedings of the National Academy of Sciences, July 17, 2020, DOI: 10.1073/pnas.2001037117.
- Plato was right. Earth is made, on average, of cubes, University of Pennsylvania Press Release, July 20, 2020.
- Adam Mann, "From rocks to icebergs, the natural world tends to break into cubes," Science, July 27, 2020, doi:10.1126/science.abe0397.