Researchers have found that the philosopher Plato may be right after all: the world, on average, breaks down into cubes — and not just in the game Minecraft. A new paper from Hungary and the U.S. takes readers through a simulation of the world's "natural 3D fragments," and conclude that the most said fragments conform to a cube-like shape.

**RELATED: THE BEAUTY OF GEOMETRIC SOLIDS: AN INTRODUCTION**

## Earth is on average made of cubes, Plato was right

Plato is relevant to the discovery because in one of his works — the Timaeus — Plato wrote about the "classical elements," which was his way of categorizing the Earth, fire, wind, and air. This was long before the empirical discovery of individual elements, let alone the fully-developed idea of atomic structure.

"Natural philosophers" — the predecessors of modern-day scientists — used the idea of classical elements, sometimes including a fifth, "aether" element, to explain the fundamental composition of the world.

In his work, Plato took this idea and morphed it into the Platonic solids — a series of 3D shapes where all sides are squares, or equilateral triangles — pulling from 2D polygons. According to Plato's system, the cube was linked with Earth, but his decision to name all regular solids this way lived on into the works of later geometers, like Euclid. For example, the "atom" (or uncuttable) comes from the thought of another ancient Greek philosopher: Democritus.

## Plato's cubes intuitive to real-world material breakdown

While none of these shapes obtain in reality with perfect conformity to their idea, and no one is finding 20-sided water segments or dirt — there is an abundance of minerals that naturally form in cubic crystals — along with numerous rocks that naturally break down into vaguely cubic shapes.

Plato's thought had the right intuition: things break according to the force that breaks them apart — and right angles are the easiest at which to split bodies apart.

"We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth's tectonic plates, has two attractors: 'Platonic' quadrangles and 'Voronoi' hexagons," said the scientists in their paper.

Voronoi polygons are a distinct algorithmic means to break down material. For instance, a drying area of mud or a sheet of ceramic slammed onto concrete will probably split into Voronoi polygons — which Plato called "Earth" squares.

## Plato's cubes beat Voronoi polygons in simulation

However, when scientists carried out simulations for the same idea in three dimensions, Plato's cubes stole the show from Voronoi polygons.

"Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato's forms."

## Ancient Greek philosophy in modern-day science

This finding happened via sampled and computed rock-breaking simulators, along with additional software to process the litany of resulting data using Platonic shapes and Voronoi as "attractor" parameters.

The new research comes on the heels of a 2019 paper regarding Plato's room for error that included one of the same researchers — Gábor Domokos, renowned in the mathematics world for helping to create the first example of a self-righting object, called a gömböc.

Platonic solids may be the topic of this paper, but there's no doubt Plato would have jumped at the idea of the gömböc. Perhaps as modern scientific advancements continue to astound us, the works of natural philosophy and ancient Greek thought will prove closer to the truth than anyone expected since Plato himself.