In the micro-world polyhedra are found in the form of molecules, viruses and bacteria - protozoa. For example, fullerenes are spherical carbon molecules C60 (Fig.) - "Bricks" of nanoelectronics and superconductors.
Elementary water cells exist in the shape of tetrahedron which contain five molecules H2O interconnected by hydrogen bonds. At the same time each of the water molecules in elementary tetrahedra retains the ability to form hydrogen bonds. Due to them simple tetrahedra can interconnect by vertices, edges or faces, forming a variety of spatial structures.
Hexagonal structure is the basic one among the wide variety of structures existing in nature, when six molecules of water (tetrahedra) are combined into a ring. This type of structure is typical for ice, snow and melt-water. Molecules of methane (CH4) and molecules of ammonia (NH3) also have the shape of tetrahedron.
In nature there can be found some objects with the symmetry of icosahedron, viruses, for example.
This is not accidental that viruses use the exclusiveness of icosahedron. The whole point is in saving the genetic information. You may ask: why is it a regular polyhedron? And why is it icosahedron in particular? A viral particle should turn the metathesis of the host cell upside down; it must force the infected cell to synthesize numerous enzymes and other molecules necessary for the synthesis of new viral particles.
All of these enzymes are to be encoded in the viral nucleic acid. But its number is limited. Therefore there is not much space left for the encoding of its own envelope proteins in the nucleic acid. What does the virus do? It just uses the same piece of the nucleic acid multiple times in order to synthesize a large amount of standard molecules, proteins which merge together in the process of building a viral particle.
As the result the maximum saving of genetic information is reached. There is only one thing to add: according to the laws of mathematics, in order to build a closed envelope of identical elements in the most economical way, one should make an icosahedron which we see in case of viruses.
This is the way how viruses solve the most difficult problem: to find the body with the smallest surface in a given volume and, moreover, consisting of identical and simplest shapes. Viruses, the smallest organisms, are so simple that it is still unclear if they should be classified as a part of living or inanimate nature. Viruses have coped with the geometric problem which took people to solve more than two thousand years! All the so-called "spherical viruses", including the polio virus, are icosahedra, and not spheres, as previously thought.
Bacteriophages (from Greek “phagos” meaning “devourer”; literally - eaters of bacteria) are bacterial viruses which cause the destruction of bacteria and other microorganisms. The particles consist of a hexagonal or rod-shaped head with a diameter of 45-140 nanometers, a thickness of 10-40 nanometers and length of 100-200 nanometers. Bacteriophage attaches its offshoot to the bacterial cell, release enzyme and dissolves the cell wall; then the contents of its head moves into cell through the offshoot, where due to the influence of nucleic acid phage stops the synthesis of bacterial proteins.
Alga Volvox is one of the simplest multicellular organisms. It is a spherical envelope consisting mainly of heptagonal, hexagonal and pentagonal cells. In each "vertex" three cells converge. There are examples with square and octagonal cells, but biologists have noticed that if such "non-standard" cells are absent (less than five or more than seven faces), the number of pentagonal cells always exceeds the amount of heptagonal by twelve. The total number of cells can reach hundreds or even thousands.
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