Tetrahedron, a term with etymological origin in the Greek language, is a concept used in the area of geometry. To understand what the notion refers to, it is important to know the meaning of polyhedron: a solid body of finite volume that has flat faces.
With that in mind, we can move on to the definition of a tetrahedron. It is a polyhedron that has four faces. These data imply that the tetrahedrons are convex polyhedra, since all the segments that connect two of their points are inside the polyhedron. The properties of the tetrahedron make its faces, on the other hand, triangular. At each vertex, therefore, three of the faces meet. When all of these faces are equilateral triangles (that is, triangles with three equal sides), the tetrahedron is classified as regular. In other words: a regular tetrahedron is a tetrahedron that has four equilateral triangles as faces.
In each tetrahedron, the segments connecting the vertices with the intersection points that belong to the medians of the opposite face are concurrent at a point. Likewise, the midpoints of opposite pairs of edges are also concurrent at the same point. Another peculiarity of the tetrahedrons is that the planes perpendicular to the edges according to their midpoints pass through the same point, while the lines perpendicular to their circumcenter to the faces are simultaneous in the center of the sphere that is circumscribed to the polyhedron in question. Symmetry is one of the particular properties of the tetrahedron, as explained below. The number of lines of symmetry of a regular tetrahedron is four, and they all have a rotational order of three. It must be remembered that a line of symmetry is a line about which a figure can rotate without changing its visual appearance; Regarding the turn order, it is the number of times we must turn the smallest angle to complete a turn, that is, to reach 360 °. As for the plane axes of symmetry, that is, a line that divides any geometric shape into two parts, so that the opposite points are at the same distance from it, the tetrahedron has six, and it is they that form between each edge and the midpoint of its opposite. We also have conjugation, a property of the regular tetrahedron that proposes it as the only «self-conjugated» platonic solid, that is, conjugated with itself, and this can be verified with the equation b = a/4, where a is the edge of a tetrahedron and b represents what we get by conjugating it.
To understand another of the particular properties of the tetrahedron, it is necessary to explain the concept of orthogonal projection, which is achieved by drawing lines perpendicular to the plane in which it is made, regardless of the angle of the projected figure. In the case of regular tetrahedrons, the application of this type of projection can give us one of two figures: * a triangle: this occurs if one of its faces is parallel to the plane of projection, since the other three (which are also triangles ) cannot be perceived from the point of view of the plane, which will simply take the three extreme points of the tetrahedron, which in this case are three vertices of one of its triangles; * a quadrilateral: when two opposite edges of the original figure are parallel to the projection plane, a square is obtained, whose side is equivalent to dividing the length of the edge by the square root of two.
