# Dictionary Definition

tetrahedron n : any polyhedron having four plane
faces [also: tetrahedra (pl)]

# User Contributed Dictionary

## English

### Etymology

From tetra-, "four" + hedron, "seat".### Pronunciation

- /tɛtrəˈhiːdrən/
- /tE.tr@."hi.dr@n/

### Noun

- a polyhedron with four faces; the regular tetrahedron, the faces of which are equal equilateral triangles, is one of the Platonic solids.

#### Related terms

#### Translations

polyhedron

- Catalan: tetràedre
- Czech: čtyřstěn
- Finnish: tetraedri
- French: tetraèdre
- German: Tetraeder
- Hungarian: tetraéder
- Italian: tetraedro
- Japanese: 四面体, 三角錐
- Portuguese: tetraedro
- Spanish: tetraedro
- Swedish: tetraeder

# Extensive Definition

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four
triangular faces, three
of which meet at each vertex.
A regular tetrahedron is one in which the four triangles are
regular, or "equilateral", and is one of the Platonic
solids.

The tetrahedron is one kind of pyramid, the second most common
type; a pyramid has a flat base, and triangular faces above it, but
the base can be of any polygonal shape, not just square or
triangular.

Like all convex
polyhedra, a tetrahedron can be folded from a single sheet of
paper.

## Formulas for regular tetrahedron

For a regular tetrahedron of edge length a: Note
that with respect to the base plane the slope of a face ( 2 \sqrt ) is
twice that of an edge ( \sqrt ), corresponding to the fact that the
horizontal distance covered from the base to the apex
along an edge is twice that along the median of
a face. In other words, if C is the centroid of the base, the
distance from C to a vertex of the base is twice that from C to the
midpoint of an edge of the base. This follows from the fact that
the medians of a triangle intersect at its centroid, and this point
divides each of them in two segments, one of which is twice as long
as the other (see
proof).

## Volume of any tetrahedron

The volume of any tetrahedron is given by the pyramid volume formula:- V = \frac Ah \,

where A is the area of the base and h the height
from the base to the apex. This applies for each of the four
choices of the base, so the distances from the apexes to the
opposite faces are inversely proportional to the areas of these
faces.

For a tetrahedron with vertices a = (a1, a2, a3),
b = (b1, b2, b3), c = (c1, c2, c3), and d = (d1, d2, d3), the
volume is (1/6)·|det(a−b,
b−c, c−d)|, or any other combination of pairs
of vertices that form a simply connected graph. This
can be rewritten using a dot product
and a cross
product, yielding

- V = \frac .

If the origin of the coordinate system is chosen
to coincide with vertex d, then d = 0, so

- V = \frac ,

where a, b, and c represent three edges that meet
at one vertex, and \mathbf \cdot (\mathbf \times \mathbf) is a
scalar
triple product. Comparing this formula with that used to
compute the volume of a parallelepiped, we
conclude that the volume of a tetrahedron is equal to 1/6 of the
volume of any parallelepiped which shares with it three converging
edges.

It should be noted that the triple scalar can be
represented by the following determinants:

- 6 \cdot \mathbf =\begin

- Hence

- 36 \cdot \mathbf =\begin

- which gives

- \mathbf= \frac \sqrt \,

If we are given only the distances between the
vertices of any tetrahedron, then we can compute its volume using
the formula:

- 288 \cdot V^2 =

If the determinant's value is negative this means
we can not construct a tetrahedron with the given distances between
the vertices.

## Distance between the edges

Any two opposite edges of a tetrahedron lie on two skew lines. If the closest pair of points between these two lines are points in the edges, they define the distance between the edges; otherwise, the distance between the edges equals that between one of the endpoints and the opposite edge.## Three dimensional properties of a generalized tetrahedron

As with triangle geometry, there is a similar set
of three dimensional geometric properties for a tetrahedron. A
generalised tetrahedron has an insphere, circumsphere, medial
tetrahedron and exspheres. It has respective centers such as
incenter, circumcenter, excenters, Spieker
center and points such as a centroid. However there is,
generally, no orthocenter in the sense of intersecting altitudes.
There is an equivalent sphere to the triangular nine
point circle that is the circumsphere of the medial
tetrahedron. However its circumsphere does not, generally, pass
through the base points of the altitudes of the reference
tetrahedron.

To resolve these inconsistencies, a substitute
center known as the Monge point that always exists for a
generalized tetrahedron is introduced. This point was first
identified by Gaspard
Monge. For tetrahedra where the altitudes do intersect, the
Monge point and the orthocenter coincide. The Monge point is define
as the point where the six midplanes of a tetrahedron intersect. A
midplane is defined as a plane that is orthogonal to an edge
joining any two vertices that also contains the centroid of an
opposite edge formed by joining the other two vertices.

An orthogonal line dropped from the Monge point
to any face is coplanar with two other orthogonal lines to the same
face. The first is an altitude dropped from a corresponding vertex
to the chosen face. The second is an orthogonal line to the chosen
face that passes through the orthocenter of that face. This
orthogonal line through the Monge point lies mid way between the
altitude and the orthocentric orthogonal line.

The Monge point, centroid and circumcenter of a
tetrahedron are colinear and form the Euler line of the
tetrahedron. However, unlike the triangle, the centroid of a
tetrahedron lies at the midpoint of its Monge point and
circumcenter.

There is an equivalent sphere to the triangular
nine point circle for the generalized tetrahedron. It is the
circumsphere of its medial tetrahedron. It is a twelve point sphere
centered at the circumcenter of the medial tetrahedron. By
definition it passes through the centroids of the four faces of the
reference tetrahedron. It passes through four substitute Euler
points that are located at a distance of 1/3 of the way from M, the
Monge point, toward each of the four vertices. Finally it passes
through the four base points of orthogonal lines dropped from each
Euler point to the face not containing the vertex that generated
the Euler point.

If T represents this twelve point center then it
also lies on the Euler line, unlike its triangular counterpart, the
center lies 1/3 of the way from M, the Monge point towards the
circumcenter. Also an orthogonal line through T to a chosen face is
coplanar with two other orthogonal lines to the same face. The
first is an orthogonal line passing through the corresponding Euler
point to the chosen face. The second is an orthogonal line passing
through the centroid of the chosen face. This orthogonal line
through the twelve point center lies mid way between the Euler
point orthogonal line and the centroidal orthogonal line.
Furthermore, for any face, the twelve point center lies at the mid
point of the corresponding Euler point and the orthocenter for that
face.

The radius of the twelve point sphere is 1/3 of
the circumradius of the reference tetrahedron.

If OABC forms a generalized tetrahedron with a
vertex O as the origin and vectors \mathbf, \mathbf \, and \mathbf
\, represent the positions of the vertices A, B and C with respect
to O, then the radius of the insphere is given by:

- r= \frac \,

and the radius of the circumsphere is given
by:

- R= \frac \,

which gives the radius of the twelve point
sphere:

- r_T= \frac \,

where:

- 6V= |\mathbf \cdot (\mathbf \times \mathbf)| \,

The vector position of various centers are given
as follows:

The centroid

- \mathbf = \frac \,

The circumcenter

- \mathbf= \frac \,

The Monge point

- \mathbf = \frac \,

The Euler line relationships are:

- \mathbf = \mathbf + \frac (\mathbf-\mathbf)\,

- \mathbf = \mathbf + \frac (\mathbf-\mathbf)\,

It should also be noted that:

- \mathbf \cdot \mathbf = \frac \quad\quad \mathbf \cdot \mathbf = \frac \quad\quad \mathbf \cdot \mathbf = \frac \,

and:

## Geometric relations

A tetrahedron is a 3-simplex. Unlike in the case of
other Platonic solids, all vertices of a regular tetrahedron are
equidistant from each other (they are in the only possible
arrangement of four equidistant points).

A regular tetrahedron can be embedded inside a
cube in
two ways such that each vertex is a vertex of the cube, and each
edge is a diagonal of one of the cube's faces. For one such
embedding, the Cartesian
coordinates of the vertices
are

- (+1, +1, +1);
- (−1, −1, +1);
- (−1, +1, −1);
- (+1, −1, −1).

Inscribing tetrahedra inside the regular compound
of five cubes gives two more regular compounds, containing five
and ten tetrahedra.

Regular tetrahedra cannot tessellate
space by themselves, although it seems likely enough that
Aristotle
reported it was possible. However, two regular tetrahedra can be
combined with an octahedron, giving a rhombohedron which can tile
space.

However, there is at least one irregular
tetrahedron of which copies can tile space. If one relaxes the
requirement that the tetrahedra be all the same shape, one can tile
space using only tetrahedra in various ways. For example, one can
divide an octahedron into four identical tetrahedra and combine
them again with two regular ones. (As a side-note: these two kinds
of tetrahedron have the same volume.)

The tetrahedron is unique among the uniform
polyhedra in possessing no parallel faces.

### Related polyhedra

## Intersecting tetrahedra

An interesting polyhedron can be constructed from
five intersecting tetrahedra. This compound
of five tetrahedra has been known for hundreds of years. It comes
up regularly in the world of origami. Joining the twenty
vertices would form a regular dodecahedron. There are
both left-handed and
right-handed
forms which are mirror
images of each other.

## The isometries of the regular tetrahedron

The vertices of a cube can be grouped into two groups
of four, each forming a regular tetrahedron (see above, and also
animation, showing one of the two tetrahedra in the cube). The
symmetries of a regular tetrahedron correspond to half of those of
a cube: those which map the tetrahedrons to themselves, and not to
each other.

The tetrahedron is the only Platonic solid that
is not mapped to itself by point inversion.

The regular tetrahedron has 24 isometries,
forming the symmetry
group Td, isomorphic to S4. They
can be categorized as follows:

- T, isomorphic to alternating
group A4 (the identity and 11 proper rotations) with the
following conjugacy
classes (in parentheses are given the permutations of the
vertices, or correspondingly, the faces, and the
unit quaternion representation):
- identity (identity; 1)
- rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)
- rotation by an angle of 180° such that an edge maps to the opposite edge: 3 ((1 2)(3 4), etc.; i,j,k)

- reflections in a plane perpendicular to an edge: 6
- reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (x is mapped to −x): the rotations correspond to those of the cube about face-to-face axes

## The isometries of irregular tetrahedra

The isometries of an irregular tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed.- An equilateral triangle base and isosceles (and non-equilateral) triangle sides gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group C3v, isomorphic to S3.
- Four congruent isosceles (non-equilateral) triangles gives 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group D2d.
- Four congruent scalene triangles gives 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group V4 ≅ Z22, present as the point group D2.
- Two pairs of isomorphic isosceles (non-equilateral) triangles. This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is C2v, isomorphic to V4.
- Two pairs of isomorphic scalene triangles. This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group C2 isomorphic to Z2.
- Two unequal isosceles triangles with a common base. This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group Cs isomorphic to Z2.
- No edges equal, so that the only isometry is the identity, and the symmetry group is the trivial group.

## A law of sines for tetrahedra and the space of all shapes of tetrahedra

A corollary of the usual law of
sines is that in a tetrahedron with vertices O,
A, B, C, we have

- \sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\,

One may view the two sides of this identity as
corresponding to clockwise and counterclockwise orientations of the
surface.

Putting any of the four vertices in the role of O
yields four such identities, but in a sense at most three of them
are independent: If the "clockwise" sides of three of them are
multiplied and the product is inferred to be equal to the product
of the "counterclockwise" sides of the same three identities, and
then common factors are cancelled from both sides, the result is
the fourth identity. One reason to be interested in this
"independence" relation is this: It is widely known that three
angles are the angles of some triangle if and only if their sum is
a half-circle. What condition on 12 angles is necessary and
sufficient for them to be the 12 angles of some tetrahedron?
Clearly the sum of the angles of any side of the tetrahedron must
be a half-circle. Since there are four such triangles, there are
four such constraints on sums of angles, and the number of degrees
of freedom is thereby reduced from 12 to 8. The four relations
given by this sine law further reduce the number of degrees of
freedom, not from 8 down to 4, but only from 8 down to 5, since the
fourth constraint is not independent of the first three. Thus the
space of all shapes of tetrahedra is 5-dimensional.

## Computational uses

Complex shapes are often broken down into a
mesh of irregular
tetrahedra in preparation for finite
element analysis and
computational fluid dynamics studies.

## Applications and uses

Chemistry- The tetrahedron shape is seen in nature in covalent bonds of molecules. For instance in a methane molecule (CH4) the four hydrogen atoms lie in each corner of a tetrahedron with the carbon atom in the centre. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. The ammonium ion is another example.
- Angle from the center to any two vertices is \arccos, or approximately 109.47°.http://1073741824.org/index.cgi/TetrahedronAngles, http://mathcentral.uregina.ca/QQ/database/QQ.09.00/nishi1.html

Games

- Especially in roleplaying, this solid is known as a d4, one of the more common polyhedral dice.
- Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.

## See also

- caltrop
- Császár polyhedron
- Szilassi polyhedron
- tetrahedral kite
- triangular dipyramid - constructed by joining two tetrahedra along one face
- tetrahedral number
- tetrahedral molecular geometry
- Tetra-Pak
- The inertia tensor of a tetrahedron

## References

## External links

- F. M. Jackson and
- The Uniform Polyhedra
- Tetrahedron: Interactive Polyhedron Model
- K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Paper Models of Polyhedra Many links
- An Amazing, Space Filling, Non-regular Tetrahedron that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle.
- Tetrahedron Core Network Application of a tetrahedral structure to create resilient partial-mesh data network

tetrahedron in Bulgarian: Тетраедър

tetrahedron in Catalan: Tetràedre

tetrahedron in Czech: Čtyřstěn

tetrahedron in Danish: Tetraeder

tetrahedron in German: Tetraeder

tetrahedron in Esperanto: Kvaredro

tetrahedron in Spanish: Tetraedro

tetrahedron in Basque: Tetraedro

tetrahedron in Persian: چهاروجهی

tetrahedron in French: Tétraèdre

tetrahedron in Hebrew: ארבעון

tetrahedron in Croatian: Tetraedar

tetrahedron in Italian: Tetraedro

tetrahedron in Japanese: 三角錐

tetrahedron in Korean: 사면체

tetrahedron in Lithuanian: Tetraedras

tetrahedron in Malay (macrolanguage):
Tetrahedron

tetrahedron in Dutch: Viervlak

tetrahedron in Norwegian: Tetraeder

tetrahedron in Polish: Czworościan

tetrahedron in Portuguese: Tetraedro

tetrahedron in Russian: Правильный
тетраэдр

tetrahedron in Simple English: Tetrahedron

tetrahedron in Slovenian: Tetraeder

tetrahedron in Serbian: Тетраедар

tetrahedron in Finnish: Tetraedri

tetrahedron in Swedish: Tetraeder

tetrahedron in Tamil: நான்முக முக்கோணகம்

tetrahedron in Thai: ทรงสี่หน้า

tetrahedron in Ukrainian: Тетраедр

tetrahedron in Contenese: 四面體

tetrahedron in Chinese: 四面體