Tetrahedron

File:Tetrahedron.stl

Tetrahedrons cannot fill space.

Вписанный тетраэдр

Symmetries of the tetrahedron

Tetraeder animation with cube

3-simplex t0 A2

Tetrahedron is a type of polyhedron that has four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. It belongs to the Platonic solids family, a group of polyhedra that are highly symmetrical, being composed of faces with the same shape and size, and identical vertices.

Properties

A tetrahedron is characterized by several unique properties. It is the only Platonic solid that does not have parallel faces. Each vertex of a tetrahedron is connected to the other three vertices by an edge, and the angles between these edges determine the shape of the tetrahedron. If all the edge lengths are equal, the tetrahedron is called a regular tetrahedron, and its faces are equilateral triangles. In a regular tetrahedron, the angle between any two faces is approximately 70.53 degrees.

The volume V of a tetrahedron, given its edge length a, can be calculated using the formula: \[V = \frac{a^3}{6\sqrt{2}}\] This formula highlights the efficiency of the tetrahedron's shape in enclosing space relative to its surface area.

Applications

Tetrahedrons have applications in various fields such as chemistry, where the molecular shape of many compounds, such as methane (CH4), is tetrahedral. In architecture and structural engineering, tetrahedral shapes are valued for their strength and are used in the design of trusses and frameworks. The tetrahedron is also found in computer graphics and 3D modeling, where it is used in the mathematical representation of physical forms and in the development of mesh and grid structures.

Dual Polyhedron

The dual of a tetrahedron is another tetrahedron. This duality means that a tetrahedron can be inscribed in another tetrahedron, where the vertices of one correspond to the faces of the other and vice versa. This property is unique among the Platonic solids, as the dual of any other Platonic solid is a different solid.

Symmetry

The symmetry group of a regular tetrahedron is the tetrahedral symmetry group, denoted as T_d. This group includes rotations and reflections that map the tetrahedron onto itself. The regular tetrahedron has 12 rotational symmetries and 12 reflection symmetries, making for a total of 24 symmetries.

See Also

• Platonic solid
• Polyhedron
• Geometry
• Symmetry in mathematics