HOME
The Info List - Tetrahedron



--- Advertisement ---


In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX.

The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".

Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets .

For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces.

CONTENTS

* 1 Regular tetrahedron

* 1.1 Formulas for a regular tetrahedron * 1.2 Isometries of the regular tetrahedron * 1.3 Orthogonal projections of the regular tetrahedron * 1.4 Cross section of regular tetrahedron * 1.5 Spherical tiling

* 2 Other special cases

* 2.1 Isometries of irregular tetrahedra

* 3 General properties

* 3.1 Volume

* 3.1.1 Heron-type formula for the volume of a tetrahedron * 3.1.2 Volume divider * 3.1.3 Non-Euclidean volume

* 3.2 Distance between the edges * 3.3 Properties analogous to those of a triangle * 3.4 Geometric relations * 3.5 A law of sines for tetrahedra and the space of all shapes of tetrahedra * 3.6 Law of cosines for tetrahedra * 3.7 Interior point * 3.8 Inradius * 3.9 Faces

* 4 Integer tetrahedra * 5 Related polyhedra and compounds

* 6 Applications

* 6.1 Numerical analysis * 6.2 Chemistry * 6.3 Electricity and electronics * 6.4 Games * 6.5 Color space * 6.6 Contemporary art * 6.7 Popular culture * 6.8 Geology * 6.9 Structural engineering * 6.10 Aviation

* 7 Tetrahedral graph * 8 See also * 9 References * 10 External links

REGULAR TETRAHEDRON

A REGULAR TETRAHEDRON is one in which all four faces are equilateral triangles . It is one of the five regular Platonic solids , which have been known since antiquity.

In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges. Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and a thin volume of empty space is left, where the five edge angles do not quite meet.

Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb , which is a tessellation.

The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.

FORMULAS FOR A REGULAR TETRAHEDRON

The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin: ( 1 , 0 , 1 2 ) and ( 0 , 1 , 1 2 ) {displaystyle left(pm 1,0,-{frac {1}{sqrt {2}}}right)quad {mbox{and}}quad left(0,pm 1,{frac {1}{sqrt {2}}}right)}

Another set of coordinates are based on an alternated cube or DEMICUBE with edge length 2. This form has Coxeter diagram and Schläfli symbol h{4,3}. The tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube. Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1) Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1) Regular tetrahedron ABCD and its circumscribed sphere

For a regular tetrahedron of edge length _a_:

Face area A 0 = 3 4 a 2 {displaystyle A_{0}={frac {sqrt {3}}{4}}a^{2},}

Surface area A = 4 A 0 = 3 a 2 {displaystyle A=4,A_{0}={sqrt {3}}a^{2},}

Height of pyramid h = 6 3 a = 2 3 a {displaystyle h={frac {sqrt {6}}{3}}a={sqrt {frac {2}{3}}},a,}

Edge to opposite edge distance l = 1 2 a {displaystyle l={frac {1}{sqrt {2}}},a,}

Volume V = 1 3 A 0 h = 2 12 a 3 = a 3 6 2 {displaystyle V={frac {1}{3}}A_{0}h={frac {sqrt {2}}{12}}a^{3}={frac {a^{3}}{6{sqrt {2}}}},}

Face-vertex-edge angle arccos ( 1 3 ) = arctan ( 2 ) {displaystyle arccos left({frac {1}{sqrt {3}}}right)=arctan left({sqrt {2}}right),}

(approx. 54.7356°)

Face-edge-face angle , i.e., "dihedral angle" arccos ( 1 3 ) = arctan ( 2 2 ) {displaystyle arccos left({frac {1}{3}}right)=arctan left(2{sqrt {2}}right),} (approx. 70.5288°)

Edge central angle , known as the _tetrahedral angle_ arccos ( 1 3 ) = 2 arctan ( 2 ) {displaystyle arccos left(-{frac {1}{3}}right)=2arctan left({sqrt {2}}right),} (approx. 109.4712°)

Solid angle at a vertex subtended by a face arccos ( 23 27 ) {displaystyle arccos left({frac {23}{27}}right)} (approx. 0.55129 steradians )

Radius of circumsphere R = 6 4 a = 3 8 a {displaystyle R={frac {sqrt {6}}{4}}a={sqrt {frac {3}{8}}},a,}

Radius of insphere that is tangent to faces r = 1 3 R = a 24 {displaystyle r={frac {1}{3}}R={frac {a}{sqrt {24}}},}

Radius of midsphere that is tangent to edges r M = r R = a 8 {displaystyle r_{mathrm {M} }={sqrt {rR}}={frac {a}{sqrt {8}}},}

Radius of exspheres r E = a 6 {displaystyle r_{mathrm {E} }={frac {a}{sqrt {6}}},}

Distance to exsphere center from the opposite vertex d V E = 6 2 a = 3 2 a {displaystyle d_{mathrm {VE} }={frac {sqrt {6}}{2}}a={sqrt {frac {3}{2}}}a,}

With respect to the base plane the slope of a face (2√2) is twice that of an edge (√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 ).

For a regular tetrahedron with side length _a_, radius _R_ of its circumscribing sphere, and distances _di_ from an arbitrary point in 3-space to its four vertices, we have d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = ( d 1 2 + d 2 2 + d 3 2 + d 4 2 4 + 2 R 2 3 ) 2 ; 4 ( a 4 + d 1 4 + d 2 4 + d 3 4 + d 4 4 ) = ( a 2 + d 1 2 + d 2 2 + d 3 2 + d 4 2 ) 2 . {displaystyle {begin{aligned}{frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{frac {16R^{4}}{9}}\4left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}right) margin-bottom: -0.302ex; width:62.945ex; height:12.176ex;" alt="{displaystyle {begin{aligned}{frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{frac {16R^{4}}{9}}\4left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}right)"> The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) 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 that map the tetrahedra 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, , (*332), isomorphic to the symmetric group , _S_4. They can be categorized as follows:

* T, +, (332) is isomorphic to alternating group , _A_4 (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

ORTHOGONAL PROJECTIONS OF THE REGULAR TETRAHEDRON

The regular _tetrahedron_ has two special orthogonal projections , one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A2 Coxeter plane .

Orthogonal projection CENTERED BY FACE/VERTEX EDGE

IMAGE

Projective symmetry

CROSS SECTION OF REGULAR TETRAHEDRON

_ A central cross section of a regular tetrahedron_ is a square .

The two skew perpendicular opposite edges of a _regular tetrahedron_ define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle . When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a square . The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges . A tetragonal disphenoid viewed orthogonally to the two green edges.

This property also applies for tetragonal disphenoids when applied to the two special edge pairs.

SPHERICAL TILING

The tetrahedron can also be represented as a spherical tiling , and projected onto the plane via a stereographic projection . This projection is conformal , preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

ORTHOGRAPHIC PROJECTION STEREOGRAPHIC PROJECTION

OTHER SPECIAL CASES

Tetrahedral symmetry subgroup relations Tetrahedral symmetries shown in tetrahedral diagrams

An ISOSCELES TETRAHEDRON, also called a disphenoid , is a tetrahedron where all four faces are congruent triangles. A SPACE-FILLING TETRAHEDRON packs with congruent copies of itself to tile space, like the disphenoid tetrahedral honeycomb .

In a trirectangular tetrahedron the three face angles at one vertex are right angles . If all three pairs of opposite edges of a tetrahedron are perpendicular , then it is called an orthocentric tetrahedron . When only one pair of opposite edges are perpendicular, it is called a SEMI-ORTHOCENTRIC TETRAHEDRON. An ISODYNAMIC TETRAHEDRON is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent , and an ISOGONIC TETRAHEDRON has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron.

ISOMETRIES OF IRREGULAR TETRAHEDRA

The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C3, +), and (S4, ) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.

TETRAHEDRON NAME Edge equivalence diagram DESCRIPTION

SYMMETRY

SCHöN. COX. ORB. ORD.

REGULAR TETRAHEDRON _ Four EQUILATERAL triangles It forms the symmetry group T_d, isomorphic to the symmetric group , _S_4. A regular tetrahedron has Coxeter diagram _ and Schläfli symbol {3,3}.

T_d _T_ + *332 332 24 12

TRIANGULAR PYRAMID _ An EQUILATERAL triangle base and three equal ISOSCELES triangle sides It 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 the symmetric group , _S_3. A triangular pyramid has Schläfli symbol {3}∨( ).

_C_3v C3 + *33 33 6 3

MIRRORED SPHENOID _ Two equal SCALENE triangles with a common base edge 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, also isomorphic to the cyclic group , Z2.

_C_s =_C_1h =_C_1v * 2

Irregular tetrahedron (No symmetry) Four unequal triangles

Its only isometry is the identity, and the symmetry group is the trivial group . An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ).

C1 + 1 1

DISPHENOIDS (Four equal triangles)

TETRAGONAL DISPHENOID

Four equal ISOSCELES triangles

It has 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. A tetragonal disphenoid has Coxeter diagram _ and Schläfli symbol s{2,4}.

D_2d S4 2*2 2× 8 4

RHOMBIC DISPHENOID

Four equal SCALENE triangles

It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group _V_4 or Z22, present as the point group _D_2. A rhombic disphenoid has Coxeter diagram _ and Schläfli symbol sr{2,2}.

D_2 + 222 4

GENERALIZED DISPHENOIDS (2 pairs of equal triangles)

DIGONAL DISPHENOID

_ Two pairs of equal ISOSCELES 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 the Klein four-group _V_4. A digonal disphenoid has Schläfli symbol { }∨{ }.

_C_2v _C_2 + *22 22 4 2

PHYLLIC DISPHENOID

Two pairs of equal SCALENE or ISOSCELES 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 _C_2 isomorphic to the cyclic group , Z2.

_C_2 + 22 2

GENERAL PROPERTIES

VOLUME

The volume of a tetrahedron is given by the pyramid volume formula: V = 1 3 A 0 h {displaystyle V={frac {1}{3}}A_{0},h,}

where _A_0 is the area of the base and _h_ is 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 = (_a_1, _a_2, _a_3), B = (_b_1, _b_2, _b_3), C = (_c_1, _c_2, _c_3), and D = (_d_1, _d_2, _d_3), the volume is 1/6det (A − D, B − D, 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 = ( a d ) ( ( b d ) ( c d ) ) 6 . {displaystyle 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 = a ( b c ) 6 , {displaystyle 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 A · (B × 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 that shares three converging edges with it.

The absolute value of the scalar triple product can be represented as the following absolute values of determinants: 6 V = a b c {displaystyle 6cdot V={begin{Vmatrix}mathbf {a} &mathbf {b} width:19.516ex; height:2.843ex;" alt="6cdot V={begin{Vmatrix}mathbf {a} &mathbf {b} "> or 6 V = a b c {displaystyle 6cdot V={begin{Vmatrix}mathbf {a} \mathbf {b} \mathbf {c} end{Vmatrix}}} where a = ( a 1 , a 2 , a 3 ) {displaystyle mathbf {a} =(a_{1},a_{2},a_{3}),} is expressed as a row or column vector etc.

Hence 36 V 2 = a 2 a b a c a b b 2 b c a c b c c 2 {displaystyle 36cdot 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} width:30.219ex; height:9.509ex;" alt="36cdot 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} "> where a b = a b cos {displaystyle mathbf {a} cdot mathbf {b} =abcos {gamma }} etc.

which gives V = a b c 6 1 + 2 cos cos cos cos 2 cos 2 cos 2 , {displaystyle V={frac {abc}{6}}{sqrt {1+2cos {alpha }cos {beta }cos {gamma }-cos ^{2}{alpha }-cos ^{2}{beta }-cos ^{2}{gamma }}},,}

where _α_, _β_, _γ_ are the plane angles occurring in vertex D. The angle _α_, is the angle between the two edges connecting the vertex D to the vertices B and C. The angle _β_, does so for the vertices A and C, while _γ_, is defined by the position of the vertices A and B.

Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant : 288 V 2 = 0 1 1 1 1 1 0 d 12 2 d 13 2 d 14 2 1 d 12 2 0 d 23 2 d 24 2 1 d 13 2 d 23 2 0 d 34 2 1 d 14 2 d 24 2 d 34 2 0 {displaystyle 288cdot V^{2}={begin{vmatrix}0&1&1&1&1\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2} margin-bottom: -0.3ex; width:36.277ex; height:17.509ex;" alt="288cdot V^{2}={begin{vmatrix}0&1&1&1&1\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2} _u_ opposite to _U_ and so on), then volume = ( a + b + c + d ) ( a b + c + d ) ( a + b c + d ) ( a + b + c d ) 192 u v w {displaystyle {text{volume}}={frac {sqrt {,(-a+b+c+d),(a-b+c+d),(a+b-c+d),(a+b+c-d)}}{192,u,v,w}}}

where a = x Y Z b = y Z X c = z X Y d = x y z X = ( w U + v ) ( U + v + w ) x = ( U v + w ) ( v w + U ) Y = ( u V + w ) ( V + w + u ) y = ( V w + u ) ( w u + V ) Z = ( v W + u ) ( W + u + v ) z = ( W u + v ) ( u v + W ) . {displaystyle {begin{aligned}a&={sqrt {xYZ}}\b&={sqrt {yZX}}\c&={sqrt {zXY}}\d&={sqrt {xyz}}\X&=(w-U+v),(U+v+w)\x&=(U-v+w),(v-w+U)\Y&=(u-V+w),(V+w+u)\y&=(V-w+u),(w-u+V)\Z&=(v-W+u),(W+u+v)\z width:31.807ex; height:32.509ex;" alt="{begin{aligned}a&={sqrt {xYZ}}\b&={sqrt {yZX}}\c&={sqrt {zXY}}\d&={sqrt {xyz}}\X&=(w-U+v),(U+v+w)\x&=(U-v+w),(v-w+U)\Y&=(u-V+w),(V+w+u)\y&=(V-w+u),(w-u+V)\Z&=(v-W+u),(W+u+v)\z">:pp.89–90

Non-Euclidean Volume

For tetrahedra in hyperbolic space or in three-dimensional spherical geometry , the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula . However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.

DISTANCE BETWEEN THE EDGES

Any two opposite edges of a tetrahedron lie on two skew lines , and the distance between the edges is defined as the distance between the two skew lines. Let _d_ be the distance between the skew lines formed by opposite edges A and B − C as calculated here . Then another volume formula is given by V = d ( a ( b c ) ) 6 . {displaystyle V={frac {d(mathbf {a} times mathbf {(b-c)} )}{6}}.}

PROPERTIES ANALOGOUS TO THOSE OF A TRIANGLE

The tetrahedron has many properties analogous to those of a triangle, including 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.

Gaspard Monge found a center that exists in every tetrahedron, now known as the MONGE POINT: 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. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron .

An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a _median_ and a line segment joining the midpoints of two opposite edges is called a _bimedian_ of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the _centroid_ of the tetrahedron. In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino\'s theorem ). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the _Euler line_ of the tetrahedron that is analogous to the Euler line of a triangle.

The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the TWELVE-POINT SPHERE and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute _Euler points_, one third of the way from 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.

The center _T_ of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies one third of the way from the Monge point _M_ 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 midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.

The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.

There is a relation among the angles made by the faces of a general tetrahedron given by 1 cos ( 12 ) cos ( 13 ) cos ( 14 ) cos ( 12 ) 1 cos ( 23 ) cos ( 24 ) cos ( 13 ) cos ( 23 ) 1 cos ( 34 ) cos ( 14 ) cos ( 24 ) cos ( 34 ) 1 = 0 {displaystyle {begin{vmatrix}-1&cos {(alpha _{12})}&cos {(alpha _{13})}&cos {(alpha _{14})}\cos {(alpha _{12})}&-1&cos {(alpha _{23})}&cos {(alpha _{24})}\cos {(alpha _{13})}&cos {(alpha _{23})}&-1&cos {(alpha _{34})}\cos {(alpha _{14})}&cos {(alpha _{24})}&cos {(alpha _{34})} width:48.698ex; height:13.176ex;" alt="{begin{vmatrix}-1&cos {(alpha _{12})}&cos {(alpha _{13})}&cos {(alpha _{14})}\cos {(alpha _{12})}&-1&cos {(alpha _{23})}&cos {(alpha _{24})}\cos {(alpha _{13})}&cos {(alpha _{23})}&-1&cos {(alpha _{34})}\cos {(alpha _{14})}&cos {(alpha _{24})}&cos {(alpha _{34})} (−1, −1, +1); (−1, +1, −1); (+1, −1, −1).

This yields a tetrahedron with edge-length 2√2, centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube . The stella octangula .

The volume of this tetrahedron is one-third the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the compound of two tetrahedra or stella octangula .

The interior of the stella octangula is an octahedron , and 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, five 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.

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

However, several irregular tetrahedra are known, of which copies can tile space, for instance the disphenoid tetrahedral honeycomb . The complete list remains an open problem.

If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different 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 LAW OF SINES FOR TETRAHEDRA AND THE SPACE OF ALL SHAPES OF TETRAHEDRA

Main article: Trigonometry of a tetrahedron

A corollary of the usual law of sines is that in a tetrahedron with vertices _O_, _A_, _B_, _C_, we have sin O A B sin O B C sin O C A = sin O A C sin O C B sin O B A . {displaystyle sin angle OABcdot sin angle OBCcdot sin angle OCA=sin angle OACcdot sin angle OCBcdot 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 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.

Three angles are the angles of some triangle if and only if their sum is 180° (π radians). 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 180°. 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, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.

LAW OF COSINES FOR TETRAHEDRA

Main article: Trigonometry of a tetrahedron

Let {_P_1 ,_P_2, _P_3, _P_4} be the points of a tetrahedron. Let Δ_i_ be the area of the face opposite vertex _Pi_ and let _θij_ be the dihedral angle between the two faces of the tetrahedron adjacent to the edge _PiPj_.

The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: i 2 = j 2 + k 2 + l 2 2 ( j k cos i l + j l cos i k + k l cos i j ) {displaystyle Delta _{i}^{2}=Delta _{j}^{2}+Delta _{k}^{2}+Delta _{l}^{2}-2(Delta _{j}Delta _{k}cos theta _{il}+Delta _{j}Delta _{l}cos theta _{ik}+Delta _{k}Delta _{l}cos theta _{ij})}

INTERIOR POINT

Let _P_ be any interior point of a tetrahedron of volume _V_ for which the vertices are _A_, _B_, _C_, and _D_, and for which the areas of the opposite faces are _F_a, _F_b, _F_c, and _F_d. Then :p.62,#1609 P A F a + P B F b + P C F c + P D F d 9 V . {displaystyle PAcdot F_{mathrm {a} }+PBcdot F_{mathrm {b} }+PCcdot F_{mathrm {c} }+PDcdot F_{mathrm {d} }geq 9V.}

For vertices _A_, _B_, _C_, and _D_, interior point _P_, and feet _J_, _K_, _L_, and _M_ of the perpendiculars from _P_ to the faces, :p.226,#215 P A + P B + P C + P D 3 ( P J + P K + P L + P M ) . {displaystyle PA+PB+PC+PDgeq 3(PJ+PK+PL+PM).}

INRADIUS

Denoting the inradius of a tetrahedron as _r_ and the inradii of its triangular faces as _r__i_ for _i_ = 1, 2, 3, 4, we have :p.81,#1990 1 r 1 2 + 1 r 2 2 + 1 r 3 2 + 1 r 4 2 2 r 2 , {displaystyle {frac {1}{r_{1}^{2}}}+{frac {1}{r_{2}^{2}}}+{frac {1}{r_{3}^{2}}}+{frac {1}{r_{4}^{2}}}leq {frac {2}{r^{2}}},}

with equality if and only if the tetrahedron is regular.

FACES

The sum of the areas of any three faces is greater than the area of the fourth face. :p.225,#159

INTEGER TETRAHEDRA

There exist tetrahedra having integer-valued edge lengths, face areas and volume. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces have areas of 7005436800000000000♠436800 and the other two have areas of 7004471200000000000♠47120, while the volume is 7007620928000000000♠62092800. :p.107

A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48. :p. 107

RELATED POLYHEDRA AND COMPOUNDS

A regular tetrahedron can be seen as a triangular pyramid .

REGULAR PYRAMIDS

TRIANGULAR SQUARE PENTAGONAL HEXAGONAL HEPTAGONAL OCTAGONAL...

REGULAR EQUILATERAL ISOSCELES

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform _digonal antiprism _, where base polygons are reduced digons .

FAMILY OF UNIFORM ANTIPRISMS _N_.3.3.3

POLYHEDRON

TILING

CONFIG. V2.3.3.3 3.3.3.3 4.3.3.3 5.3.3.3 6.3.3.3 7.3.3.3 8.3.3.3 9.3.3.3 10.3.3.3 11.3.3.3 12.3.3.3 ...∞.3.3.3

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual _digonal trapezohedron _, containing 6 vertices, in two sets of colinear edges.

FAMILY OF TRAPEZOHEDRA V._N_.3.3.3

POLYHEDRON

TILING

CONFIG. V2.3.3.3 V3.3.3.3 V4.3.3.3 V5.3.3.3 V6.3.3.3 V7.3.3.3 V8.3.3.3 ...V10.3.3.3 ...V12.3.3.3 ...V∞.3.3.3

A truncation process applied to the tetrahedron produces a series of uniform polyhedra . Truncating edges down to points produces the octahedron as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.

FAMILY OF UNIFORM TETRAHEDRAL POLYHEDRA

SYMMETRY : , (*332) +, (332)

{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}

DUALS TO UNIFORM POLYHEDRA

V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,_n_}, continuing into the hyperbolic plane .

*_N_32 SYMMETRY MUTATION OF REGULAR TILINGS: {3,_N_}

SPHERICAL EUCLID. COMPACT HYPER. PARACO. NONCOMPACT HYPERBOLIC

3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures .

*_N_32 SYMMETRY MUTATION OF REGULAR TILINGS: {_N_,3}

SPHERICAL EUCLIDEAN COMPACT HYPERB. PARACO. NONCOMPACT HYPERBOLIC

{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}

* Compounds of tetrahedra

*

Two tetrahedra in a cube *

Compound of five tetrahedra *

Compound of ten 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.

APPLICATIONS

NUMERICAL ANALYSIS

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

In numerical analysis , complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations . These methods have wide applications in practical applications in computational fluid dynamics , aerodynamics , electromagnetic fields , civil engineering , chemical engineering , naval architecture and engineering , and related fields.

CHEMISTRY

The ammonium ion is tetrahedral Main article: Tetrahedral molecular geometry

The tetrahedron shape is seen in nature in covalently bonded molecules. All sp3-hybridized atoms are surrounded by atoms (or lone electron pairs ) at the four corners of a tetrahedron. For instance in a methane molecule (CH 4) or an ammonium ion (NH+ 4), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called _ Tetrahedron _. The central angle between any two vertices of a perfect tetrahedron is arccos(−1/3), or approximately 109.47°.

Water , H 2O, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds.

Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.

However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.

ELECTRICITY AND ELECTRONICS

Main articles: Electricity and Electronics

If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.

Since silicon is the most common semiconductor used in solid-state electronics , and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.

GAMES

4-sided die Main article: Game

The Royal Game of Ur , dating from 2600 BC, was played with a set of tetrahedral dice.

Especially in roleplaying , this solid is known as a 4-sided die , one of the more common polyhedral dice , with the number rolled appearing around the bottom or on the top vertex. Some Rubik\'s Cube -like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix .

COLOR SPACE

Main article: Color space

Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).

CONTEMPORARY ART

Main article: Contemporary art

The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps . It was shown at the light art biennale Austria 2010.

It is used as album artwork, surrounded by black flames on _The End of All Things to Come _ by Mudvayne .

POPULAR CULTURE

Stanley Kubrick originally intended the monolith in _2001: A Space Odyssey _ to be a tetrahedron, according to Marvin Minsky , a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.

In Season 6, Episode 15 of _ Futurama _, named "Möbius Dick", the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew.

In the 2013 film _Oblivion _ the large structure in orbit above the Earth is of a tetrahedron design and referred to as the Tet.

GEOLOGY

Main article: Geology

The tetrahedral hypothesis , originally published by William Lowthian Green to explain the formation of the Earth, was popular through the early 20th century.

STRUCTURAL ENGINEERING

A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes .

AVIATION

At some airfields , a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction.

TETRAHEDRAL GRAPH

TETRAHEDRAL GRAPH

VERTICES 4

EDGES 6

RADIUS 1

DIAMETER 1

GIRTH 3

AUTOMORPHISMS 24

CHROMATIC NUMBER 4

PROPERTIES Hamiltonian , regular , symmetric , distance-regular , distance-transitive , 3-vertex-connected , planar graph

The skeleton of the tetrahedron (the vertices and edges) form a graph , with 4 vertices, and 6 edges. It is a special case of the complete graph , K4, and wheel graph , W4. It is one of 5 Platonic graphs , each a skeleton of its Platonic solid .

3-fold symmetry

SEE ALSO

* Boerdijk–Coxeter helix * Caltrop * Demihypercube and simplex – _n_-dimensional analogues * Hill tetrahedron * Pentachoron – 4-dimensional analogue * Schläfli orthoscheme * Tetra Pak * Tetrahedral kite * Tetrahedral number * Tetrahedron packing * Triangular dipyramid – constructed by joining two tetrahedra along one face * Trirectangular tetrahedron * Synergetics

REFERENCES

* ^ _A_ _B_ Weisstein, Eric W. "Tetrahedron". _ MathWorld _. * ^ _A_ _B_ _C_ _D_ _E_ _F_ Coxeter, Harold Scott MacDonald ; _Regular Polytopes _, Methuen and Co., 1948, Table I(i) * ^ Köller, Jürgen, "Tetrahedron", Mathematische Basteleien, 2001 * ^ "Angle Between 2 Legs of a Tetrahedron", Maze5.net * ^ _A_ _B_ Valence Angle of the Tetrahedral Carbon Atom W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145 * ^ Park, Poo-Sung. " Regular polytope distances", Forum Geometricorum 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf * ^ Sections of a Tetrahedron * ^ " Simplex Volumes and the Cayley-Menger Determinant", MathPages.com * ^ Kahan, William M.; "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", pp. 16–17 * ^ Weisstein, Eric W. "Tetrahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Tetrahedron.html * ^ Altshiller-Court, N. "The tetrahedron." Ch. 4 in _Modern Pure Solid Geometry_: Chelsea, 1979. * ^ Murakami, Jun; Yano, Masakazu (2005), "On the volume of a hyperbolic and spherical tetrahedron", _Communications in Analysis and Geometry_, 13 (2): 379–400, ISSN 1019-8385 , MR 2154824 , doi :10.4310/cag.2005.v13.n2.a5 * ^ Havlicek, Hans; Weiß, Gunter (2003). "Altitudes of a tetrahedron and traceless quadratic forms" (PDF). _American Mathematical Monthly _. 110 (8): 679–693. JSTOR 3647851 . doi :10.2307/3647851 . * ^ Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54 * ^ Outudee, Somluck; New, Stephen. _The Various Kinds of Centres of Simplices_ (PDF). Dept of Mathematics, Chulalongkorn University, Bangkok. * ^ Audet, Daniel (May 2011). "Déterminants sphérique et hyperbolique de Cayley-Menger" (PDF). Bulletin AMQ. * ^ Senechal, Marjorie (1981). "Which tetrahedra fill space?". _ Mathematics Magazine _. Mathematical Association of America. 54 (5): 227–243. JSTOR 2689983 . doi :10.2307/2689983 * ^ Rassat, André; Fowler, Patrick W. (2004). "Is There a "Most Chiral Tetrahedron"?". _Chemistry: A European Journal_. 10 (24): 6575–6580. doi :10.1002/chem.200400869 * ^ Lee, Jung Rye (June 1997). "The Law of Cosines in a Tetrahedron". _J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math_. * ^ _A_ _B_ _C_ _D_ _Inequalities proposed in “Crux Mathematicorum ”_, . * ^ _A_ _B_ Wacław Sierpiński , _Pythagorean Triangles_, Dover Publications, 2003 (orig. ed. 1962). * ^ "Angle Between 2 Legs of a Tetrahedron" – Maze5.net * ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). _Croatica Chemica Acta_. 75 (2): 633–649. Retrieved 2006-09-15. * ^ Záležák, Tomáš (18 October 2007); "Resistance of a regular tetrahedron" (PDF), retrieved 25 Jan 2011 * ^ Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). _HP Technical Report_. HPL-98-95: 1–32. * ^ Lightart-Biennale Austria 2010 * ^ "Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron". Web of Stories. Retrieved 20 February 2012. * ^ Green, William Lowthian (1875). _Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography_. Part I. London: E. Stanford. OCLC 3571917 . * ^ Holmes, Arthur (1965). _Principles of physical geology_. Nelson. p. 32. * ^ Hitchcock, Charles Henry (January 1900). Winchell, Newton Horace, ed. " William Lowthian Green and his Theory of the Evolution of the Earth\'s Features". _The American Geologist_. XXV. Geological Publishing Company. pp. 1–10. * ^ Federal Aviation Administration (2009), _Pilot\'s Handbook of Aeronautical Knowledge_, U. S. Government Printing Office, p. 1