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.[1] The tetrahedron is the threedimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3simplex. 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.[1] 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.[citation needed] 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 Herontype formula for the volume of a tetrahedron
3.1.2
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
4 Integer tetrahedra 5 Related polyhedra and compounds 6 Applications 6.1 Numerical analysis
6.2 Chemistry
6.3
Electricity
7 Tetrahedral graph 8 See also 9 References 10 External links Regular tetrahedron[edit] 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, all faces are the same size and shape (congruent) and all edges are the same length. Five tetrahedra are laid flat on a plane, with the highest 3dimensional 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 in the ratio of two tetrahedra to
one octahedron, they form the alternated cubic honeycomb, which is a
tessellation.
The regular tetrahedron is selfdual, 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[edit]
The following
Cartesian coordinates
( ± 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) Expressed symmetrically as 4 points on the unit sphere, centroid at
the origin, with lower face level, the vertices are:
v1 = ( sqrt(8/9), 0 , 1/3 )
v2 = ( sqrt(2/9), sqrt(2/3), 1/3 )
v3 = ( sqrt(2/9), sqrt(2/3), 1/3 )
v4 = ( 0 , 0 , 1 )
with the edge length of sqrt(8/3).
Still another set of coordinates are based on an alternated cube or
demicube with edge length 2. This form has
Coxeter diagram
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
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[2] A = 4 A 0 = 3 a 2 displaystyle A=4,A_ 0 = sqrt 3 a^ 2 , Height of pyramid[3] 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[2] 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 , Facevertexedge angle arccos ( 1 3 ) = arctan ( 2 ) displaystyle arccos left( frac 1 sqrt 3 right)=arctan left( sqrt 2 right), (approx. 54.7356°) Faceedgeface angle, i.e., "dihedral angle"[2] 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,[4][5] known as the tetrahedral angle, as it is the bond angle in a tetrahedral molecule. It is also the angle between Plateau borders at a vertex. arccos ( − 1 3 ) = 2 arctan ( 2 ) displaystyle arccos left( frac 1 3 right)=2arctan left( sqrt 2 right), (approx. 109.4712°)
Solid angle
arccos ( 23 27 ) displaystyle arccos left( frac 23 27 right) (approx. 0.55129 steradians) Radius of circumsphere[2] 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[2] r = 1 3 R = a 24 displaystyle r= frac 1 3 R= frac a sqrt 24 , Radius of midsphere that is tangent to edges[2] 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 3space to its four vertices, we have[6] 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 &=left( frac d_ 1 ^ 2 +d_ 2 ^ 2 +d_ 3 ^ 2 +d_ 4 ^ 2 4 + frac 2R^ 2 3 right)^ 2 ;\4left(a^ 4 +d_ 1 ^ 4 +d_ 2 ^ 4 +d_ 3 ^ 4 +d_ 4 ^ 4 right)&=left(a^ 2 +d_ 1 ^ 2 +d_ 2 ^ 2 +d_ 3 ^ 2 +d_ 4 ^ 2 right)^ 2 .end aligned Isometries of the regular tetrahedron[edit] The proper rotations, (order3 rotation on a vertex and face, and order2 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
T, [3,3]+, (332) is 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 facetoface axes Orthogonal projections of the regular tetrahedron[edit] 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 [3] [4] Cross section of regular tetrahedron[edit] 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.[7] 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[edit] 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[edit]
Tetrahedral symmetry
Tetrahedral symmetries shown in tetrahedral diagrams An isosceles tetrahedron, also called a disphenoid, is a tetrahedron where all four faces are congruent triangles. A spacefilling 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 semiorthocentric 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[edit] The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3dimensional point group is formed. Two other isometries (C3, [3]+), and (S4, [2+,4+]) 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
Symmetry Schön. Cox. Orb. Ord. Regular Tetrahedron Four equilateral triangles
It forms the symmetry group Td, isomorphic to the symmetric group, S4.
A regular tetrahedron has
Coxeter diagram
Td T [3,3] [3,3]+ *332 332 24 12
Triangular
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 C3v,
isomorphic to the symmetric group, S3. A triangular pyramid has
Schläfli symbol
C3v C3 [3] [3]+ *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 Cs, also isomorphic to the cyclic group, Z2. Cs =C1h =C1v [ ] * 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
D2d. A tetragonal disphenoid has
Coxeter diagram
D2d S4 [2+,4] [2+,4+] 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 fourgroup
D2 [2,2]+ 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
C2v, isomorphic to the
Klein fourgroup
C2v C2 [2] [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 C2 isomorphic to the cyclic group, Z2. C2 [2]+ 22 2 General properties[edit] Volume[edit] 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 A0 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 = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3), and d = (d1, d2, d3), 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 &mathbf c end Vmatrix 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 &mathbf c^ 2 end vmatrix where a ⋅ b = a b cos γ displaystyle mathbf a cdot mathbf b =abcos gamma etc. which gives V = a b c 6 1 + 2 cos α cos β cos γ − cos 2 α − cos 2 β − cos 2 γ , 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 &0end vmatrix where the subscripts i, j ∈ 1, 2, 3, 4 represent the vertices a,
b, c, d and dij is the pairwise distance between them – i.e., the
length of the edge connecting the two vertices. A negative value of
the determinant means that a tetrahedron cannot be constructed with
the given distances. This formula, sometimes called Tartaglia's
formula, is essentially due to the painter
Piero della Francesca
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),(ab+c+d),(a+bc+d),(a+b+cd) 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&=(wU+v),(U+v+w)\x&=(Uv+w),(vw+U)\Y&=(uV+w),(V+w+u)\y&=(Vw+u),(wu+V)\Z&=(vW+u),(W+u+v)\z&=(Wu+v),(uv+W).end aligned
Volume
V = d
( a × ( b − c ) )
6 . displaystyle V= frac d(mathbf a times mathbf (bc) ) 6 .
Properties analogous to those of a triangle[edit]
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.[13]
Gaspard Monge
− 1 cos ( α 12 ) cos ( α 13 ) cos ( α 14 ) cos ( α 12 ) − 1 cos ( α 23 ) cos ( α 24 ) cos ( α 13 ) cos ( α 23 ) − 1 cos ( α 34 ) cos ( α 14 ) cos ( α 24 ) cos ( α 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 ) &1\end vmatrix =0, where αij is the angle between the faces i and j. Geometric relations[edit] A tetrahedron is a 3simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3dimensional space). A tetrahedron is a triangular pyramid, and the regular tetrahedron is selfdual. A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are (+1, +1, +1); (−1, −1, +1); (−1, +1, −1); (+1, −1, −1). This yields a tetrahedron with edgelength 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 3demicube. The stella octangula. The volume of this tetrahedron is onethird 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
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
5dimensional.[18]
Law of cosines
Δ i 2 = Δ j 2 + Δ k 2 + Δ l 2 − 2 ( Δ j Δ k cos θ i l + Δ j Δ l cos θ i k + Δ k Δ l cos θ 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[edit] 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 Fa, Fb, Fc, and Fd. Then[20]: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,[20]: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[edit] Denoting the inradius of a tetrahedron as r and the inradii of its triangular faces as ri for i = 1, 2, 3, 4, we have[20]: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[edit] The sum of the areas of any three faces is greater than the area of the fourth face.[20]:p.225,#159 Integer tetrahedra[edit] There exist tetrahedra having integervalued 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.[21]: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.[21]:p. 107 Related polyhedra and compounds[edit] A regular tetrahedron can be seen as a triangular pyramid. Regular pyramids Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal... Improper 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 selfdual tetrahedron once again. Family of uniform tetrahedral polyhedra Symmetry: [3,3], (*332) [3,3]+, (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. *n32 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 order3 vertex figures. *n32 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 lefthanded and righthanded forms, which are mirror images of each other. Applications[edit] Numerical analysis[edit] An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements. In numerical analysis, complicated threedimensional 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[edit] The ammonium ion is tetrahedral Main article: Tetrahedral molecular geometry
The tetrahedron shape is seen in nature in covalently bonded
molecules. All sp3hybridized 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°.[5][22]
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 twodimensional plane.
Electricity
4sided die 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 4sided die, one
of the more common polyhedral dice, with the number rolled appearing
around the bottom or on the top vertex. Some Rubik's Cubelike puzzles
are tetrahedral, such as the
Pyraminx
Tetrahedral graph Vertices 4 Edges 6 Radius 1 Diameter 1 Girth 3 Automorphisms 24 Chromatic number 4 Properties Hamiltonian, regular, symmetric, distanceregular, distancetransitive, 3vertexconnected, 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.[32] It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. 3fold symmetry See also[edit] Boerdijk–Coxeter helix
Caltrop
Demihypercube
References[edit] ^ 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 Brittin, W. E. (1945). "Valence angle of the tetrahedral carbon
atom". Journal of Chemical Education. 22 (3): 145.
doi:10.1021/ed022p145.
^ Park, PooSung. "
Regular polytope
External links[edit] Wikimedia Commons has media related to Tetrahedron. Weisstein, Eric W. "Tetrahedron". MathWorld.
Free paper models of a tetrahedron and many other polyhedra
An Amazing, Space Filling, Nonregular
Tetrahedron
v t e Polyhedra Listed by number of faces 1–10 faces Monohedron Dihedron Trihedron Tetrahedron Pentahedron Hexahedron Heptahedron Octahedron Enneahedron Decahedron 11–20 faces Hendecahedron Dodecahedron Tridecahedron Tetradecahedron Pentadecahedron Hexadecahedron Heptadecahedron Octadecahedron Enneadecahedron Icosahedron Others Triacontahedron Hexecontahedron Enneacontahedron Skew apeirohedrons v t e Convex polyhedra Platonic solids (regular) tetrahedron cube octahedron dodecahedron icosahedron Archimedean solids (semiregular or uniform) truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron Catalan solids (duals of Archimedean) triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron Dihedral regular dihedron hosohedron Dihedral uniform prisms antiprisms duals: bipyramids trapezohedra Dihedral others pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola pyramidal frusta bifrustum rotunda birotunda Degenerate polyhedra are in italics. v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron
Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
