Tetrahedron

The ( also Austrian v. a: of ) tetrahedra [ tetrahedron dər ː ] ( from Greek Tetraedron " tetrahedron "), even tetrahedron or four flat, is a body with four triangular faces. It is the only convex (three-dimensional ) polyhedron ( polyhedron ) with four faces.

The word is, however, rarely used in this general sense. Most is meant for the general tetrahedron is called depending on symmetry as a three-sided pyramid, triangular pyramid Disphenoid or three-dimensional simplex with the regular tetrahedron (or equilateral ) tetrahedron.

Regular tetrahedron

The regular tetrahedron ( regular tetrahedron ) is one of the five Platonic solids, or more precisely a polyhedron with

• Four ( congruent ) equilateral triangles as faces
• Six ( equally divided) edges and
• Four corners where three surfaces meet

The regular tetrahedron is an equilateral three -sided pyramid ( with an equilateral triangle as the base ).

Symmetry

Due to its high symmetry - all corners, edges and surfaces are equal to each other like - is the regular tetrahedron is a regular polyhedron. It has

• Four three-fold rotation axes ( through the corners and the centers of the opposite side faces ),
• Three fourfold rotation inversion axes and thus three -fold axes of rotation ( through the midpoints of opposite edges ) and
• Six planes of symmetry (in each case by an edge and perpendicular (normal ) to the opposite edge).

Overall, the symmetry group of the tetrahedron - the tetrahedral group - 24 elements. She is the symmetric group S4 ( the point group Td after Schoenflies or by Hermann- Mauguin 43m ) and causes all 4! = 24 permutations of the corners or the sides. It is subset of the octahedral group ( group of dice ).

In detail, the tetrahedral group

As well as

The even permutations form a subgroup of the tetrahedral group, the alternating group called ( the point group T and 23, respectively ). Sometimes the term tetrahedral group is used only for this exclusion of the reflections.

Other properties

Relation to octahedron, cube, Archimedean bodies

By connecting the centroids obtained again a tetrahedron. It is said therefore: The tetrahedron is dual to itself. The side length of the new tetrahedron is one third of the original side length.

With the help of these two tetrahedra body can be constructed which also have the tetrahedral group as a symmetry group. Thus, for example, receives

• The truncated tetrahedron with four hexagons and four triangles ( see Archimedean bodies)
• The octahedron with 4 4 = 8 triangles and 6 corners ( higher symmetry ) as the intersection of two tetrahedra,
• The star tetrahedron ( an octahedron with 8 attached tetrahedra ) as the union of two tetrahedra
• The cube with 4 4 = 8 corners ( and with higher symmetry ) as the convex hull of this star body.

See also the example below.

Ambient Cube

The tetrahedron can be inscribed in a cube so that its corners are at the same time cube corners and its six edges diagonals of the cube faces. ( The eight corners of the cube to form two disjoint sets of four corners, which correspond to the two possible orientations of the tetrahedron. ) The volume of this cube is three times the volume of the tetrahedron.

Dually, the tetrahedron can be an octahedron circumscribed so that four of the octahedral faces lie in the boundary faces of the tetrahedron and the six vertices of the octahedron are the centers of the six tetrahedron edges. ( The eight faces of the octahedron form two disjoint sets, corresponding to the two positions for the the octahedron circumscribed tetrahedron. )

Angle

The angle between two boundary surfaces of the regular tetrahedron ( in the drawing ) is from 70.53 ° ( rounding accuracy as with the following information to two decimal places ). Each edge forms an angle with the opposite surface () of 54.74 °. The links between the tetrahedron center and two corners close to an angle a, which corresponds to 109.47 °. The last-mentioned angle ( ) is called the tetrahedral angle and plays an important role in chemistry, for example the geometry of the methane molecule. The sizes of the specified angle can be determined by application of trigonometric functions. One looks to the average figure of the tetrahedron with one of its six planes of symmetry. The result is exact:

To calculate the tetrahedral angle see Article Obtuse angle.

Cross-section

The regular tetrahedron can be cut into two parts, that the sectional area is a square. The parts are congruent to each other.

Is the section plane by means of a regular tetrahedron in parallel with one of the four side faces, the cross-section results in an equilateral triangle.

Is the section plane by means of a regular tetrahedron in parallel with two opposite edges, the cross-section results in a rectangle. If the cutting plane additionally of these two edges at the same distance, so it shares the remaining four edges exactly in half, then cut the image is a square. The square has a side length which is exactly half as long as the length of an edge of the tetrahedron.

Example

The embedding of the tetrahedron in a cube provides an easy way to construct a regular tetrahedron. We denote the vertices of the cube to the base A, B, C and D and the overlying corner points E, F, G and H, to form A, C, F, H and B, D, E and G in each case, the corners of a tetrahedron. If one considers, for example, in a spatial Cartesian coordinate system the cube whose vertices are the coordinates of 1 and -1 have, we obtain the angles for the first tetrahedron

• A ( 1,1, -1), C (-1, -1, -1 ), F ( -1,1,1 ), and H (1, -1,1 ).

The edges are: AC, AF, AH, CF, CH and FH. The side faces are triangles ACF, ACH, AFH and CFH.

The second tetrahedron has the corners

• B ( -1,1, -1 ), D ( 1, -1, -1), D (1,1,1 ) and G (-1, -1,1 ).

The average of these two tetrahedra is that of the points ( 1,0,0 ), ( -1,0,0 ), ( 0,1,0 ) (0, -1,0 ), ( 0,0,1 ) and (0,0, -1) specific octahedron. Their union is a star body with 8 tips ( in each corner of the cube one). Its convex hull is therefore of the dice.

Formulas

Applications

Although the tetrahedron is not a tiling stone of the room, it occurs (see above) in the cubic crystal system.

In chemistry, the tetrahedron plays a major role in the spatial arrangement of atoms in compounds. Simple molecular shapes can be predicted with the VSEPR model. Thus, the four hydrogen atoms in the methane molecule are arranged tetrahedrally around the carbon atom, since then the bond angle becomes largest. The carbon atoms in the diamond lattice are arranged tetrahedrally, each atom is surrounded by four other atoms. The carbon atom is then the orbital model to sp3 hybridization.

The tetrahedron was eponymous for the Tetra Pak due to its original shape.

In many pen -and- paper role-playing games tetrahedra are used as four-sided dice (W4 ).

Other technical applications are inspired by the structure that results from the oriented tetrahedron from the center to the four corners of the room stretches:

• Tetrapods, which are used in coastal areas as a breakwater
• So-called crow's feet, a defensive weapon that is used by police and military against cars to have burst their tires.

Unit cell of diamond lattice

A crow's foot of the Office of Strategic Services

Dimensional framework of tetrahedra

General tetrahedron ( three-dimensional simplex)

A tetrahedron in a general sense, ie a body with four side faces, is always a three -sided pyramid, ie with a triangle base and three triangles as faces, and therefore also has four corners and six edges. Since he has the smallest of a body in space possible number of corners and sides, it is called in the jargon ( three-dimensional ) simplex or 3 - simplex. The two-dimensional simplices are triangles.

• Each 3- simplex has a circumscribed sphere and an inscribed ball.
• The center is the intersection of the lines connecting the corners and the centroids of the opposite triangles and shares this in a 3:1 ratio.
• Every 3- simplex is the convex hull of its four corners.

In a tetrahedron can be described by a point and three vectors for the neighboring points. Denoting these vectors, so the volume of the tetrahedron calculated with

Calculation of any tetrahedron

A tetrahedron has 6 edges. A triangle is determined by specifying three sides. Each additional edge can be chosen freely ( within certain limits ). So chairs 6 independent information about the size of edges and / or points in front, one can calculate from each other missing edges and / or angle.

Analogies in higher dimensions

The analogs of the tetrahedron in any dimension are referred to as ( -dimensional ) simplices. The -dimensional simplex has corners and is bounded by simplices of dimension ( as facets ). A zero-dimensional simplex is a point, a one-dimensional simplex is a line, a two-dimensional simplex is a triangle. The four-dimensional equivalent of the tetrahedron Pentachoron, has 5 vertices, 10 edges, 10 triangles as faces and 5 three-dimensional tetrahedron as facets.

Coordinate description of a regular simplex:

For example, for the result here is an equilateral triangle that is spanned by the points in three-dimensional space.