The four-color theorem is associated with the magnetic properties of crystals.

Sometimes purely theoretical, mathematical abstractions find a surprising correspondence in living nature. Perhaps the most famous among them are fractals. But a group of mathematicians, physicists and chemists from the United States, South Korea and Japan managed to find another remarkable example. They proved that the well-known four-color theorem accurately describes the structure of some crystals.

The four-color theorem states that any map located on a sphere or on a plane can be colored with four colors so that any two regions that have a common part of the boundary are colored in different colors.

For many years, cartographers have used this theorem to prepare geographical maps. However, in recent decades, it is of greater interest not so much for cartographers, as for mathematicians, because of the complexity of the proof. It was proved only in 1976, 124 years after the theorem was formulated in 1852. It is said that this is the first major mathematical theorem, proved using a computer. Apparently, her evidence without using a computer has not yet been offered.
')
Returning to the discovery of an interdisciplinary group of scientists, they studied the properties of the Fe _x TaS ₂ crystal. This is a layered structure belonging to the class of transition metal dichalcogenides (TMD), this can be detected under a microscope in metal alloys and magnets. It is the layered structure of TMD that defines the physical macro-properties of the material. In Fe _x TaS _2, thin layers of TaS ₂ alternate with Fe ions, forming a massive crystal lattice.

Scientists have studied the properties of two different types of lattice with different amounts of iron ions included in the structure: for x = 1/4 and 1/3. It turned out that these two lattices exhibit completely different physical properties: the crystals differ in the size of the nodes, in the sequence of layers, and in the number of types of layers in the grid. In the case of Fe _1/4 TaS ₂ , layers of four types were found in the lattice, and in the case of Fe _1/3 TaS ₂ , six types were found. The study authors concluded that the crystal lattice is subject to mathematical conditions. In the first case, this is a four-color theorem, and in the second, a 6-valent graph.

In any case, each node in the crystal lattice never comes into contact with a node of the same type, but only with other nodes, as is the case with colors on a geographic map.

For a 6-valent graph, the crystal lattice can be “painted” in two passes: first with two (for example, dark and light), and then with three colors (for example, red, green and blue, as in the illustration). In such a location in the neighborhood never appear areas of one color, and the first and second characteristics are taken into account. That is, light red cannot coexist with dark red or any other light.



Of course, no one colors crystals, but it is the paint theorem that allows you to intuitively understand the logic by which the physical structure is formed. If you understand how properties are formed, you can create new materials that will find application in electronics, optics, and many other areas.

via phys.org