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Encyclopedia > Tetrahedron
Regular Tetrahedron
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 Image:CDW_ring.png Image:CDW_3.png Image:CDW_dot.png Image:CDW_3.png Image:CDW_dot.png
Image:CDW_hole.png Image:CDW_2.png Image:CDW_hole.png Image:CDW_2.png Image:CDW_hole.png
Symmetry Td
References U01, C15, W1
Properties Regular convex deltahedron
Dihedral angle 70.528779° = arccos(1/3)
Tetrahedron
3.3.3
(Vertex figure)

Self-dual
(dual polyhedron)
Tetrahedron
Net

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. Tetrahedron is an international journal publishing full original research papers in the field of organic chemistry. ... Image File history File links Tetrahedron. ... Image File history File links Tetrahedron. ... In geometry, a Platonic solid is a convex regular polyhedron. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_hole. ... // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... H.S.M. Coxeter. ... This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Look up Convex set in Wiktionary, the free dictionary. ... The triaugmented triangular prism, a convex deltahedron A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... Image File history File links Tetrahedron_vertfig. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, duality has numerous meanings. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... Image File history File links No higher resolution available. ... Categories: Polyhedra | Stub ... For the game magazine, see Polyhedron (magazine). ... For other uses, see Triangle (disambiguation). ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... 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. For other meanings, see pyramid (disambiguation). ...

Contents

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. Look up Convex set in Wiktionary, the free dictionary. ...


Formulas for regular tetrahedron

For a regular tetrahedron of edge length a:

Surface area A=a^2sqrt{3} ,
Volume V=begin{matrix}{1over12}end{matrix}a^3sqrt{2} ,
Height h=sqrt{6}(a/3) ,
Angle between an edge and a face arctan(sqrt{2}) ,
(approx. 55°)
Angle between two faces arccos(1/3) = arctan(2sqrt{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. 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 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). Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... For other uses, see Volume (disambiguation). ... A solid angle is the three dimensional analog of the ordinary angle. ... In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedrons vertices. ... A sphere containing on its surface one point from each edge of a semiregular or regular polyhedron. ... This article is about the mathematical term. ... In geometry, an apex is a descriptive label for a visual singular highest or most distant point or vertex in an isosceles triangle, pyramid or cone, usually contrasting with the opposite side called the base. ... The triangle medians and the centroid. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...


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(ab, bc, cd)|, 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 In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... For the cross product in algebraic topology, see Künneth theorem. ...

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. 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. In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...


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} = abcos{C}    etc.
which gives
 mathbf{V}= frac {abc} {6} sqrt{1 + 2cos{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. 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. In geometry, skew lines are two lines in Euclidean space that do not intersect but are not parallel. ...


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.[1] In geometry, the nine point circle is a circle that can be constructed for any given triangle. ...


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. Gaspard Monge. ...


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} quadquad mathbf{b} cdot mathbf{O} = frac {mathbf{b^2}}{2} quadquad 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} quadquad mathbf{b} cdot mathbf{M} = frac {mathbf{b} cdot (mathbf{c} + mathbf{a})}{2} quadquad mathbf{c} cdot mathbf{M} = frac {mathbf{c} cdot (mathbf{a} + mathbf{b})}{2},

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 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...


A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual. This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... Polyhedra for which the dual polyhedron is a congruent figure. ...


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 Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...

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

For the other tetrahedron (which is dual to the first), reverse all the signs. 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. 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). 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. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... 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. ...


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 which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...


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. In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ... For other uses, see Aristotle (disambiguation). ... The n-sided trapezohedron or deltohedron is the dual polyhedron of a regular n-sided antiprism. ...


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 with regular polygons as faces and identical vertices. ...


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. This polyhedral compound is a symmetric arangement of five tetrahedra. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ... This article is about paper folding. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... People who are left-handed are more dextrous with their left hand than with their right hand: they will probably also use their left hand for tasks such as personal care, cooking, and so on. ... A person who is right-handed is more dextrous with their right hand than with their left hand: they will write with their right hand, and probably also use this hand for tasks such as personal care, cooking, and so on. ... A mirror image is a mirror based duplicate of a single image. ...


The isometries of the regular tetrahedron

The proper rotations and reflections in the symmetry group 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). 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. A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Tetraeder_animation_with_cube. ...


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: The symmetry group of an object (e. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

  • 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

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 share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ... Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ...

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. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

  • 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 V4Z22, 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.

This article is about the mathematical group. ...

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

Image:tetra.png Image File history File links No higher resolution available. ...


A corollary of the usual law of sines is that in a tetrahedron with vertices O, A, B, C, we have In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ...

 sinangle OABcdotsinangle OBCcdotsinangle OCA = sinangle OACcdotsinangle OCBcdotsinangle 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. The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...


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. For other uses, see Mesh (disambiguation). ... Visualization of how a car deforms in an asymmetrical crash using finite element analysis. ... A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ...


Applications and uses

The ammonium ion is tetrahedral
The ammonium ion is tetrahedral

Chemistry Image File history File links Download high-resolution version (1100x1077, 157 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Ammonium ... Image File history File links Download high-resolution version (1100x1077, 157 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Ammonium ... A ball-and-stick model of the ammonium cation Ammonium is also an old name for the Siwa Oasis in western Egypt. ... For other uses, see Chemistry (disambiguation). ...

  • 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{left(-tfrac{1}{3}right)}, or approximately 109.47°.[1], [2]

Electronics Covalent bonding is a form of chemical bonding characterized by the sharing of one or more pairs of electrons between atoms, in order to produce a mutual attraction, which holds the resultant molecule together. ... Methane is a chemical compound with the molecular formula . ... Tetrahedron is an international journal publishing full original research papers in the field of organic chemistry. ... A ball-and-stick model of the ammonium cation Ammonium is also an old name for the Siwa Oasis in western Egypt. ... Surface mount electronic components Electronics is the study of the flow of charge through various materials and devices such as semiconductors, resistors, inductors, capacitors, nano-structures and vacuum tubes. ...

  • 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.[3]

Games The ohm (symbol: Ω) is the SI unit of electric resistance. ... Resistor symbols (American) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... For other uses, see Game (disambiguation). ...

In roleplaying, participants adopt and act out the role of characters, or parts, that may have personalities, motivations, and backgrounds different from their own. ... Two standard six-sided pipped dice with rounded corners. ... Rolling dice Dice (the plural of the word die, probably from the Latin dare: to give) are, in general, small polyhedral objects with the faces marked with numbers or other symbols, thrown in order to choose one of the faces randomly. ... Variations of Rubiks Cubes (from left to right: Rubiks Revenge, the original design of Rubiks Cube, Professors Cube, & Pocket Cube, also known as Mini-Cube). Rubiks Cube is a mechanical puzzle invented in 1974[1] by Hungarian sculptor and professor of architecture Ernő Rubik. ... Pyraminx in its solved state The Pyraminx (aka Pyramix) is a tetrahedron-shaped puzzle similar to the Rubiks Cube. ... The Pyramorphix in its solved state. ...

See also

Caltrop used by the Office of Strategic Services. ... The Szilassi polyhedron. ... A tetrahedral kite is a multicelled rigid box kite composed of tetrahedrally shaped cells. ... In geometry, the triangular dipyramid is a polyhedron made entirely out of 6 faces, which are all equilateral triangles, 9 edges, and 5 vertexes. ... A pyramid with side length 5 contains 35 spheres. ... In a tetrahedral molecular geometry a central atom is located at the center with four substituents located at the corners of a tetrahedron. ... Tetrapak logo Tetra Pak is a multinational food packaging company of Swedish origin. ...

References

  1. ^ Havlicek, H. & 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> 
  2. ^ 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> 
  3. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-15. 

Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 258th day of the year (259th in leap years) in the Gregorian calendar. ...

External links

Wikimedia Commons has media related to:
Tetrahedron
Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... For the game magazine, see Polyhedron (magazine). ... A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A hexahedron is a polyhedron with six faces. ... A heptahedron is a polyhedron having seven sides, or faces. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... // History note: all information thus far taken from http://www. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... A tetradecahedron is a polyhedron with 14 faces. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

  Results from FactBites:
 
Tetrahedron - LoveToKnow 1911 (305 words)
This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system.
The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids)., and the trigonal pyramid of the hexagonal system, are examples of non-regular tetrahedra (see Crystallography).
This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
What The World May Come To (2116 words)
The flat surfaces of the tetrahedron are the Indian Ocean, the Pacific Ocean, and the Atlantic Ocean.
We should naturally expect, with a flat surface of the tetrahedron on top and with the point stretching south, that there would be an ocean at the North Pole and a continent at the South; and such is actually the case, as we know from the discoveries of Nansen, Peary, Scott, and Shackleton.
The terrestrial tetrahedron - that is, the earth on which we live - is not, however, perfectly symmetrical in shape, America being a little too far east to be midway between Asia with Australasia on one side and Europe with Africa on the other.
  More results at FactBites »

 
 

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