Tetrahedron

For the academic journal, see Tetrahedron (journal). Tetrahedron is an international journal publishing full original research papers in the field of Organic chemistry.

Regular Tetrahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F = 4, E = 6 V = 4 (χ = 2)
Faces by sides	4{3}
Schläfli symbol	{3,3}
Wythoff symbol	3 \| 2 3 \| 2 2 2
Coxeter-Dynkin
Symmetry	T_d
References	U₀₁, C₁₅, W₁
Properties	Regular convex deltahedron
Dihedral angle	70. In Geometry, a Platonic solid is a convex Regular polyhedron. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, the Schläfli symbol is a notation of the form {pqr In Geometry, a Wythoff symbol is a short-hand notation created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a List of Symmetry groups on the sphere Spherical symmetry groups are also called Point groups in three dimensions. A regular Tetrahedron has 12 rotational (or orientation-preserving symmetries and a total of 24 symmetries including transformations that combine a reflection and a rotation A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the A deltahedron ( Plural deltahedra) is a Polyhedron whose faces are all Equilateral triangles The name is taken from the Greek In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. 528779° = arccos(1/3)
3. 3. 3 (Vertex figure)	Self-dual (dual polyhedron)
Net

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the In Geometry the net of a Polyhedron is an arrangement of edge-joined Polygons in the plane which can be folded (along edges to become the faces of the polyhedron What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a vertex (plural "vertices" is a special kind of point. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. In Geometry, a Platonic solid is a convex Regular polyhedron.

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. A pyramid is a Building where the upper surfaces are triangular and converge on one point

1 Formulas for regular tetrahedron
2 Volume of any tetrahedron
3 Distance between the edges
4 Three dimensional properties of a generalized tetrahedron
5 Geometric relations
- 5.1 Related polyhedra
6 Intersecting tetrahedra
7 The isometries of the regular tetrahedron
8 The isometries of irregular tetrahedra
9 A law of sines for tetrahedra and the space of all shapes of tetrahedra
10 Computational uses
11 Applications and uses
12 See also
13 References
14 External links

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the

Formulas for regular tetrahedron

For a regular tetrahedron of edge length $a$ :

Surface area	$A=a^2\sqrt{3} \,$
Volume	$V=\begin{matrix}{1\over12}\end{matrix}a^3\sqrt{2} \,$
Height	$h=\sqrt{6}(a/3) \,$
Angle between an edge and a face	$\arctan(\sqrt{2}) \,$ (approx. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically 55°)
Angle between two faces	$\arccos(1/3) = \arctan(2\sqrt{2}) \,$ (approx. 71°)
Angle between the segments joining the center and the vertices	${\pi \over 2} + \arcsin(1/3)\,$ (approx. 109. 471°)
Solid angle at a vertex subtended by a face	$3 \arccos(1/3) - \pi \,$ (approx. The solid angle, Ω, is the angle in three-dimensional space that an object Subtends at a point 0. 55129 steradians)
Radius of circumsphere	$R=\sqrt{6}(a/4) \,$
Radius of insphere that is tangent to faces	$r=\sqrt{6}(a/12) \,$
Radius of midsphere that is tangent to edges	$r_M=\sqrt{2}(a/4) \,$
Radius of exspheres	$r_E=\sqrt{6}(a/6) \,$
Distance to exsphere center from a vertex	$\sqrt{6}(a/2) \,$

Note that with respect to the base plane the slope of a face ( $2 \sqrt{2}$ ) is twice that of an edge ( $\sqrt{2}$ ), 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 Geometry, a circumscribed sphere of a Polyhedron is a Sphere that contains the polyhedron and touches each of the polyhedron's vertices In Geometry, the inscribed sphere or insphere of a Convex Polyhedron is a Sphere that is contained within the polyhedron and In Geometry, the midsphere or intersphere of a Polyhedron is a Sphere which is tangent to every edge of the polyhedron Slope is used to describe the steepness incline gradient or grade of a straight line. In Geometry, a median of a Triangle is a Line segment joining a vertex to the Midpoint of the opposing side 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. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes 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). In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes

Volume of any tetrahedron

The volume of any tetrahedron is given by the pyramid volume formula:

$V = \frac{1}{3} Ah \,$

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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects This can be rewritten using a dot product and a cross product, yielding

$V = \frac { |(\mathbf{a}-\mathbf{d}) \cdot ((\mathbf{b}-\mathbf{d}) \times (\mathbf{c}-\mathbf{d}))| } {6}.$

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

$V = \frac { |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| } {6},$

where a, b, and c represent three edges that meet at one vertex, and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ is a scalar triple product. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. 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. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism

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

$6 \cdot \mathbf{V} =\begin{vmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{vmatrix}$ or $6 \cdot \mathbf{V} =\begin{vmatrix} \mathbf{a} \\ \mathbf{b} \\ \mathbf{c} \end{vmatrix}$ where $\mathbf{a} = (a_1,a_2,a_3) \,$ is expressed as a row or column vector etc.

Hence

$36 \cdot \mathbf{V^2} =\begin{vmatrix} \mathbf{a^2} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{b} & \mathbf{b^2} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} & \mathbf{c^2} \end{vmatrix}$ where $\mathbf{a} \cdot \mathbf{b} = ab\cos{C}$ etc.

which gives

$\mathbf{V}= \frac {abc} {6} \sqrt{1 + 2\cos{A}\cos{B}cos{C}-\cos^2{A}-\cos^2{B}-\cos^2{C}} \,$

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

$288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{21}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{31}^2 & d_{32}^2 & 0 & d_{34}^2 \\ 1 & d_{41}^2 & d_{42}^2 & d_{43}^2 & 0 \end{vmatrix}.$

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. In Geometry, skew lines are two lines that do not intersect but are not Parallel. 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. In Geometry, the Incircle of the Medial triangle of a triangle ABC is the Spieker circle. 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. In Geometry, the nine-point circle is a Circle that can be constructed for any given triangle. However its circumsphere does not, generally, pass through the base points of the altitudes of the reference tetrahedron. ^[1]

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. Gaspard Monge Comte de Péluse ( May 10, 1746 &ndash July 28, 1818) was a French Mathematician and inventor of 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. ^[2]

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 $\mathbf{a}, \mathbf{b} \,$ and $\mathbf{c} \,$ represent the positions of the vertices A, B and C with respect to O, then the radius of the insphere is given by:

$r= \frac {6V} {|\mathbf{b} \times \mathbf{c}| + |\mathbf{c} \times \mathbf{a}| + |\mathbf{a} \times \mathbf{b}| + |(\mathbf{b} \times \mathbf{c}) + (\mathbf{c} \times \mathbf{a}) + (\mathbf{a} \times \mathbf{b})|} \,$

and the radius of the circumsphere is given by:

$R= \frac {|\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})|} {12V} \,$

which gives the radius of the twelve point sphere:

$r_T= \frac {|\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})|} {36V} \,$

where:

$6V= |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \,$

The vector position of various centers are given as follows:

The centroid

$\mathbf{G} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{4} \,$

The circumcenter

$\mathbf{O}= \frac {\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})} {12V} \,$

The Monge point

$\mathbf{M} = \frac {\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})(\mathbf{b} \times \mathbf{c}) + \mathbf{b}\cdot (\mathbf{c} + \mathbf{a})(\mathbf{c} \times \mathbf{a}) + \mathbf{c} \cdot (\mathbf{a} + \mathbf{b})(\mathbf{a} \times \mathbf{b})} {12V} \,$

The Euler line relationships are:

$\mathbf{G} = \mathbf{M} + \frac{1}{2} (\mathbf{O}-\mathbf{M})\,$

$\mathbf{T} = \mathbf{M} + \frac{1}{3} (\mathbf{O}-\mathbf{M})\,$

It should also be noted that:

$\mathbf{a} \cdot \mathbf{O} = \frac {\mathbf{a^2}}{2} \quad\quad \mathbf{b} \cdot \mathbf{O} = \frac {\mathbf{b^2}}{2} \quad\quad \mathbf{c} \cdot \mathbf{O} = \frac {\mathbf{c^2}}{2}\,$

and:

$\mathbf{a} \cdot \mathbf{M} = \frac {\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})}{2} \quad\quad \mathbf{b} \cdot \mathbf{M} = \frac {\mathbf{b} \cdot (\mathbf{c} + \mathbf{a})}{2} \quad\quad \mathbf{c} \cdot \mathbf{M} = \frac {\mathbf{c} \cdot (\mathbf{a} + \mathbf{b})}{2}\,$

Geometric relations

A tetrahedron is a 3-simplex. In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle 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 tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual. Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex

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. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. For one such embedding, the Cartesian coordinates of the vertices are

(+1, +1, +1);

(−1, −1, +1);

(−1, +1, −1);

(+1, −1, −1). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Geometry, a vertex (plural "vertices" is a special kind of point.

For the other tetrahedron (which is dual to the first), reverse all the signs. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the The volume of this tetrahedron is 1/3 the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre See also Polyhedron Merkaba Polyhedron models * Plane (metaphysics An octahedron (plural octahedra is a Polyhedron with eight faces Correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i. e. , rectifying the tetrahedron). In Euclidean geometry, rectification is the process of truncating a Polytope by marking the midpoints of all its edges and cutting off its vertices at those points The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, 5 is the minimum number of tetrahedra required to compose a cube.

Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra. A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre

Regular tetrahedra cannot tessellate space by themselves, although it seems likely enough that Aristotle reported it was possible. In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron which can tile space. See also Rhombohedral - Crystal system

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. A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i

Related polyhedra

Truncated tetrahedron

Two tetrahedra in a cube

Compound of five tetrahedra

Compound of ten tetrahedra

Intersecting tetrahedra

An interesting polyhedron can be constructed from five intersecting tetrahedra. The truncated tetrahedron is an Archimedean solid. It has 4 regular Hexagonal faces 4 regular triangular faces 12 vertices and 18 edges See also Polyhedron Merkaba Polyhedron models * Plane (metaphysics As a compound It can be constructed by arranging five tetrahedra in Icosahedral symmetry ( I) as colored in the upper right model As a compound It can also be seen as the compound of ten tetrahedra with Icosahedral symmetry ( I h As a compound It can be constructed by arranging five tetrahedra in Icosahedral symmetry ( I) as colored in the upper right model This compound of five tetrahedra has been known for hundreds of years. A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre It comes up regularly in the world of origami. (from oru meaning "folding" and kami meaning "paper" is the ancient Japanese Art of Paper folding. Joining the twenty vertices would form a regular dodecahedron. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal There are both left-handed and right-handed forms which are mirror images of each other. Someone who is right-handed will prefer to use this hand for everyday activities such as writing, maintaining personal hygiene, Cooking and so forth "Mirror Image" is an episode of the Television series The Twilight Zone.

The isometries of the regular tetrahedron

The proper rotations and reflections in the symmetry group 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). A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. 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 T_d, isomorphic to S₄. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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. In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions ; (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. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

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 C_3v, 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 D_2d.
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. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2
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 C_2v, 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 C_s 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

Image:tetra.png

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. The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a plane

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. A mesh is a flat semi-permeable barrier made of connected strands of Metal, Fiber, or other flexible/ductile material Computational fluid dynamics (CFD is one of the branches of Fluid mechanics that uses Numerical methods and algorithms to solve and analyze problems that involve

Applications and uses

The ammonium ion is tetrahedral

Chemistry

The tetrahedron shape is seen in nature in covalent bonds of molecules. Ammonium is also an old name for the Siwa Oasis in western Egypt. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties 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. Methane is a Chemical compound with the molecular formula. It is the simplest Alkane, and the principal component of Natural gas. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. Tetrahedron is an international journal publishing full original research papers in the field of Organic chemistry. The ammonium ion is another example. Ammonium is also an old name for the Siwa Oasis in western Egypt.
Angle from the center to any two vertices is $\arccos{\left(-\tfrac{1}{3}\right)}$ , or approximately 109. 47°. [1], [2]

Electronics

If each edge of a tetrahedron were to be replaced by a one ohm resistor, the resistance between any two vertices would be 1/2 ohm. Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical The ohm (symbol Ω) is the SI unit of Electrical impedance or in the Direct current case Electrical resistance, |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | ^[3]

Games

Especially in roleplaying, this solid is known as a d4, one of the more common polyhedral dice. A game is a structured activity, usually undertaken for Enjoyment and sometimes also used as an Educational tool In roleplaying, participants adopt and act out the Role of characters, or parts that may have personalities motivations and backgrounds different from For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French
Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix. The Rubik's Cube is a Mechanical puzzle invented in 1974 by Hungarian Sculptor and Professor of Architecture Ernő Rubik The Pyraminx is a Tetrahedron -shaped puzzle similar to the Rubik's Cube. The Pyramorphix is a Tetrahedron -shaped puzzle similar to the Rubik's Cube.

References

^ Havlicek, H. A caltrop (also known as Caltrap, galtrop,or in Japanese: Makibishi or Tetsubishi. In Geometry, the Császár polyhedron (ˈtʃaːsaːr is a nonconvex Polyhedron, topologically a Torus, with 14 triangular faces. The Szilassi polyhedron is a nonconvex Polyhedron, topologically a Torus, with seven hexagonal faces A tetrahedral kite is a multicelled rigid Box kite composed of Tetrahedrally shaped cells For the related molecular geometry see Trigonal bipyramid molecular geometry In Geometry, the triangular dipyramid (or Bipyramid) is the first A tetrahedral number, or triangular pyramidal number, is a Figurate number that represents a Pyramid with a triangular base and three sides called a In a Tetrahedral molecular geometry a central Atom is located at the center with four Substituents that are located at the corners of a Tetrahedron. Tetra Pak is a multinational Food processing and Packaging company of Swedish origin & Weiß, G. (2003), Altitudes of a tetrahedron and traceless quadratic forms, Amer. Math. Monthly 110, 679-693. , <http://www.geometrie.tuwien.ac.at/havlicek/publications.html>
^ Outudee, Somluck & New, Stephen, The Various Kinds of Centres of Simpices, Dept of Maths. , Chulalongkorn University, Bangkok, <http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf>
^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649.

External links

F. M. Jackson and Eric W. Weisstein, Tetrahedron at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Eric W. Weisstein, Monge point at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Eric W. Weisstein, Euler points at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
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. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M
Tetrahedron Core Network Application of a tetrahedral structure to create resilient partial-mesh data network

Dictionary

-noun

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

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