When researching hexagons, we also find a lot of information about the structure of regular tetrahedrons. Apparently, snowflakes are hexagonal because when frozen, water molecules take on a three-dimensional tetrahedral structure, which becomes hexagonal when flat. A regular tetrahedron is a solid body made up of four equilateral triangles, and is considered one of the regular polyhedrons. Regular polyhedrons are polyhedrons with all faces being regular polygons, and are also called Platonic solids, and there are only five types: the regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron, and regular icosahedron.
https://mathematica.site/web-mag/column/platonic-solid
Other regular polyhedrons are also very interesting, but this time we will look at the regular tetrahedron. Other relatively familiar objects that have a regular tetrahedral structure include ice, diamond, and silicon. Carbon is a single molecule, water is made up of hydrogen and oxygen combined, and silicon is also made up of oxygen combined to form a regular tetrahedron. This structure has very strong bonds, is geometrically free of distortion, is resistant to pressure, and is extremely hard.
Interestingly, the constituent elements of the universe – carbon, hydrogen, oxygen, and silicon – are also the top two constituent elements of the Earth’s crust, oxygen and silicon, the top two constituent elements of the oceans, oxygen and hydrogen, and oxygen and carbon, the top two constituent elements of living organisms.
https://gendai.media/articles/-/137645?page=2
Silicate minerals, which are made up of silicon and oxygen, are the main minerals that make up the Earth’s crust and mantle, and the tetrahedral structure of these minerals may be what makes the Earth so hard. The Earth’s surface is supported by tetrahedrons. Other elements such as methane also have a tetrahedral structure.
This regular tetrahedron has constant angles connecting the center and each vertex of approximately 109°28’16” = 109.5 degrees, which are called Maraldi angles. The angles of a rhombus inscribed in a rectangle with an aspect ratio of 1:√2, called the silver rectangle, are also approximately 109.5 degrees. Maraldi angles also appear in nature, such as the bottom of a honeycomb, soap films, and the angles at which scallop shells open.
https://manabitimes.jp/math/1376
It seems that most crystals are made up of linked polyhedrons, with regular octahedrons and regular tetrahedrons also being seen in abundance. Regular tetrahedral types include the spinel type, which is often magnetic, and the wurtzite and zinc blende types, which have a structure that makes them semiconductors.
Also, most metal crystals seem to be designed to pack as many atoms as possible into one space, and take one of three structures: a hexagonal close-packed structure in which the space is shaped like a hexagonal prism and the atoms are packed as tightly as possible, a body-centered cubic structure in which the space is a cube with atoms at the center and vertices, or a face-centered cubic structure in which atoms are at the faces and vertices. In these structures, gaps are created between the regular tetrahedrons. In the hexagonal close-packed structure and face-centered cubic structure, there are eight tetrahedral gaps, and in the body-centered cubic structure, there are 12 tetrahedral gaps. Other atoms fill in these gaps, forming complex crystals.
https://kimika.net/r3roppousaimitsu.html
The regular tetrahedron, which appears frequently on Earth, seems to form ice, diamonds, and rocks, and to form the gaps within many crystals, both as a molecular structure and as a space. It must be a rational structure, even geometrically. The regular tetrahedron is amazing, and the Earth is amazing.
