The tetrahedron is thethree-dimensional case of the more general concept of aEuclideansimplex, and may thus also be called a3-simplex.
The tetrahedron is one kind ofpyramid, which is a polyhedron with a flatpolygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is atriangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Like allconvex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two suchnets.[1]
For any tetrahedron there exists a sphere (called thecircumsphere) on which all four vertices lie, and another sphere (theinsphere)tangent to the tetrahedron's faces.[2]
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 anda thin volume of empty space is left, where the five edge angles do not quite meet.
Aregular tetrahedron is a tetrahedron in which all four faces areequilateral triangles. In other words, all of its faces are the same size and shape (congruent) and all edges are the same length. The regular tetrahedron is the simplestconvexdeltahedron, a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra.[3]
The regular tetrahedron is also one of the five regularPlatonic solids, a set of polyhedrons in which all of their faces areregular polygons.[4] Known since antiquity, the Platonic solid is named after the Greek philosopherPlato, who associated those four solids with nature. The regular tetrahedron was considered as the classical element offire, because of his interpretation of its sharpest corner being most penetrating.[5]
The regular tetrahedron is self-dual, meaning itsdual is another regular tetrahedron. Thecompound figure comprising two such dual tetrahedra form astellated octahedron orstella octangula. Its interior is anoctahedron, 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 tetrahedron is yet related to another two solids: Bytruncation the tetrahedron becomes atruncated tetrahedron. The dual of this solid is thetriakis tetrahedron, a regular tetrahedron with four triangular pyramids attached to each of its faces. i.e., itskleetope.
Consider a regular tetrahedron with edge length. Its height is.[6] Its surface area is four times the area of an equilateral triangle:[7]The volume is one-third of the base times the height, the general formula for a pyramid;[7] this can also be found by dissecting a cube into a tetrahedron and four triangular pyramids.[8]
Itsdihedral angle—the angle formed by two planes in which adjacent faces lie—is[7]Its vertex–center–vertex angle—the angle between lines from the tetrahedron center to any two vertices—is denoted thetetrahedral angle.[9] It is the angle betweenPlateau borders at a vertex. Its value in radians is the length of the circular arc on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. In chemistry, it is also known as thetetrahedral bond angle.
Regular tetrahedron ABCD and its circumscribed sphere
The radii of itscircumsphere,insphere,midsphere, andexsphere are:[7]For a regular tetrahedron with side length and circumsphere radius, the distances from an arbitrary point in 3-space to its four vertices satisfy the equations:[10]
With respect to the base plane theslope of a face (2√2) is twice that of an edge (√2), corresponding to the fact that thehorizontal distance covered from the base to theapex along an edge is twice that along themedian of a face. In other words, ifC is thecentroid of the base, the distance fromC to a vertex of the base is twice that fromC 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 (seeproof).
One way to construct a regular tetrahedron is by using the followingCartesian coordinates, defining the four vertices of a tetrahedron with edge length 2, centered at the origin, and two-level edges:
Expressed symmetrically as 4 points on theunit sphere, centroid at the origin, with lower face parallel to the plane, the vertices are:with the edge length of.
A regular tetrahedron can be embedded inside acube 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, theCartesian coordinates of the vertices areThis yields a tetrahedron with edge-length, centered at the origin. For the other tetrahedron (which isdual 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, a polyhedron that is byalternating a cube. This form hasCoxeter diagram andSchläfli symbol.
The vertices of acube can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. Thesymmetries 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 not mapped to itself bypoint inversion.
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
It has rotational tetrahedral symmetry. This symmetry is isomorphic toalternating group—the identity and 11 proper rotations—with the followingconjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and theunit quaternion representation):
identity (identity; 1)
2 conjugacy classes corresponding to positive and negative rotations about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together(4 (1 2 3), etc., and4 (1 3 2), etc.;1 ±i ±j ±k/2).
rotation by an angle of 180° such that an edge maps to the opposite edge:3 ((1 2)(3 4), etc.;i,j,k)
reflections in a plane perpendicular to an edge: 6
reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (x is mapped to −x): the rotations correspond to those of the cube about face-to-face axes
The regular tetrahedron has two specialorthogonal projections, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A2Coxeter plane.
A central cross section of aregular tetrahedron is asquare.
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 arectangle.[11] 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 asquare. 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 becomewedges.
A tetragonal disphenoid viewed orthogonally to the two green edges.
This property also applies fortetragonal disphenoids when applied to the two special edge pairs.
The tetrahedron can also be represented as aspherical tiling (ofspherical triangles), and projected onto the plane via astereographic projection. This projection isconformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Infour dimensions, all the convexregular 4-polytopes with tetrahedral cells (the5-cell,16-cell and600-cell) can be constructed as tilings of the3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
Tetrahedral symmetries shown in tetrahedral diagrams
Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.
If all three pairs of opposite edges of a tetrahedron areperpendicular, then it is called anorthocentric tetrahedron. When only one pair of opposite edges are perpendicular, it is called asemi-orthocentric tetrahedron.In atrirectangular tetrahedron the three face angles atone vertex areright angles, as at the corner of a cube.
Anisodynamic tetrahedron is one in which thecevians that join the vertices to theincenters of the opposite faces areconcurrent.
Anisogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with theinscribed sphere of the tetrahedron.
A space-filling tetrahedral disphenoid inside a cube. Two edges havedihedral angles of 90°, and four edges have dihedral angles of 60°.
Adisphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.
A cube dissected into six characteristic orthoschemes.
A3-orthoscheme is a tetrahedron where all four faces areright triangles. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct adisphenoid with right triangle or obtuse triangle faces.
Anorthoscheme is an irregularsimplex that is theconvex hull of atree in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another name for it isbirectangular tetrahedron. It is also called aquadrirectangular tetrahedron because it contains four right angles.[12]
Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups.[13] For example, the special case of a 3-orthoscheme with equal-length perpendicular edges ischaracteristic of the cube, which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length√2 and one of length√3, so all its edges are edges or diagonals of the cube. The cube can be dissected into six such 3-orthoschemes four different ways, with all six surrounding the same√3 cube diagonal. The cube can also be dissected into 48smaller instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of the cube is an example of aHeronian tetrahedron.
Every regular polytope, including the regular tetrahedron, has itscharacteristic orthoscheme. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron is subdivided into 24 instances of its characteristic tetrahedron by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths,, around its exterior right-triangle face (the edges opposite thecharacteristic angles 𝟀, 𝝉, 𝟁),[a] plus,, (edges that are thecharacteristic radii of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is,,, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a60-90-30 triangle which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges,,, a right triangle with edges,,, and a right triangle with edges,,.
Aspace-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space.[15] The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed (one of each at each cube face), and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense. (The characteristic orthoscheme of the cube is one of theHill tetrahedra, a family of space-filling tetrahedra. All space-filling tetrahedra arescissors-congruent to a cube.)
A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in thedisphenoid tetrahedral honeycomb. Regular tetrahedra, however, cannot fill space by themselves (moreover, it is not scissors-congruent to any other polyhedra which can fill the space, seeHilbert's third problem). Thetetrahedral-octahedral honeycomb fills space with alternating regular tetrahedron cells and regularoctahedron cells in a ratio of 2:1.
For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in akaleidoscope. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. (TheCoxeter-Dynkin diagram of the generated polyhedron contains threenodes representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a singlegenerating point which is multiplied by mirror reflections into the vertices of the polyhedron.)
Among the Goursat tetrahedra which generate 3-dimensionalhoneycombs we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustratedabove.[13] The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can bedissected into characteristic tetrahedra of the cube.
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a3-dimensional 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.
Anequilateral triangle base and three equalisosceles 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 groupC3v, isomorphic to thesymmetric group,S3. A triangular pyramid has Schläfli symbol {3}∨( ).
C3v C3
[3] [3]+
*33 33
6 3
Mirrored sphenoid
Two equalscalene 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 groupCs, also isomorphic to thecyclic group,Z2.
Cs =C1h =C1v
[ ]
*
2
Irregular tetrahedron (No symmetry)
Four unequal triangles
Its only isometry is the identity, and the symmetry group is thetrivial group. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ).
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 groupD2d. A tetragonal disphenoid has Coxeter diagram and Schläfli symbol s{2,4}.
It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is theKlein four-groupV4 orZ22, present as the point groupD2. A rhombic disphenoid has Coxeter diagram and Schläfli symbol sr{2,2}.
D2
[2,2]+
222
4
Generalized disphenoids (2 pairs of equal 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 isC2v, isomorphic to theKlein four-groupV4. A digonal disphenoid has Schläfli symbol { }∨{ }.
C2v C2
[2] [2]+
*22 22
4 2
Phyllic disphenoid
Two pairs of equalscalene orisosceles 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 groupC2 isomorphic to thecyclic group,Z2.
Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics. One of the commonly used subdivision methods is theLongest Edge Bisection (LEB), which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB.
Asimilarity class is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing the formation of highly irregular elements that could compromise simulation results.
The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal to, the iterated LEB produces no more than 37 similarity classes.[17]
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume:where is thebase' area and 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 apices to the opposite faces are inversely proportional to the areas of these faces. Another way is by dissecting a triangular prism into three pieces.[18]
Given the vertices of a tetrahedron in the following:The volume of a tetrahedron can be ascertained in terms of adeterminant,[19] or any other combination of pairs of vertices that form a simply connectedgraph. Comparing this formula with that used to compute the volume of aparallelepiped, we conclude that the volume of a tetrahedron is equal to1/6 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:
orwhereare expressed as row or column vectors.
Hence
where
where,, and, which gives
whereα,β,γ are the plane angles occurring in vertexd. The angleα, is the angle between the two edges connecting the vertexd to the verticesb andc. The angleβ, does so for the verticesa andc, whileγ, is defined by the position of the verticesa andb.
If we do not require thatd = 0 then
Given the distances between the vertices of a tetrahedron the volume can be computed using theCayley–Menger determinant:
where the subscriptsi,j ∈ {1, 2, 3, 4} represent the vertices{a,b,c,d} anddij 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 calledTartaglia's formula, is essentially due to the painterPiero della Francesca in the 15th century, as a three-dimensional analogue of the 1st centuryHeron's formula for the area of a triangle.[20]
Let,, and be the lengths of three edges that meet at a point, and,, and be those of the opposite edges. The volume of the tetrahedron is:[21]whereThe above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.[21]
Six edge-lengths of Tetrahedron
The volume of a tetrahedron can be ascertained by using the Heron formula. Suppose,,,., and are the lengths of the tetrahedron's edges as in the following image. Here, the first three form a triangle, with opposite, opposite, and opposite. Then,whereand
Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedronbisects the volume of the tetrahedron.[22]
For tetrahedra inhyperbolic space or in three-dimensionalelliptic geometry, thedihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by theMurakami–Yano formula, after Jun Murakami and Masakazu Yano.[23] However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.
Any two opposite edges of a tetrahedron lie on twoskew lines, and the distance between the edges is defined as the distance between the two skew lines. Let be the distance between the skew lines formed by opposite edges and as calculatedhere. Then another formula for the volume of a tetrahedron is given by
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.[24]
Gaspard Monge found a center that exists in every tetrahedron, now known as theMonge 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 oforthocentric 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 thecentroid of the opposite face is called amedian and a line segment joining the midpoints of two opposite edges is called abimedian of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are allconcurrent at a point called thecentroid of the tetrahedron.[25] In addition the four medians are divided in a 3:1 ratio by the centroid (seeCommandino's theorem). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define theEuler line of the tetrahedron that is analogous to theEuler line of a triangle.
Thenine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is thetwelve-point sphere and besides the centroids of the four faces of the reference tetrahedron, it passes through four substituteEuler 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.[26]
The centerT 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 pointM towards the circumcenter. Also, an orthogonal line throughT 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[27]
whereαij is the angle between the facesi andj.
Thegeometric median of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle.Lorenz Lindelöf found that, corresponding to any given tetrahedron is a point now known as an isogonic center,O, at which the solid angles subtended by the faces are equal, having a common value of πsr, and at which the angles subtended by opposite edges are equal.[28] A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr,O lies inside the tetrahedron, and because the sum of distances fromO to the vertices is a minimum,O coincides with thegeometric median,M, of the vertices. In the event that the solid angle at one of the vertices,v, measures exactly π sr, thenO andM coincide withv. If however, a tetrahedron has a vertex,v, with solid angle greater than π sr,M still corresponds tov, butO lies outside the tetrahedron.
A tetrahedron is a 3-simplex. 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 3-dimensional space, for an example inelectromagnetism cf.Thomson problem).
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. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be added to make a cube, which has 8 vertices.
Inscribing tetrahedra inside the regularcompound of five cubes gives two more regular compounds, containing five and ten tetrahedra.
Regular tetrahedra cannottessellate space by themselves, although this result seems likely enough thatAristotle claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving arhombohedron that can tile space as thetetrahedral-octahedral honeycomb.
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 theuniform polyhedra in possessing no parallel faces.
A law of sines for tetrahedra and the space of all shapes of tetrahedra
A corollary of the usuallaw of sines is that in a tetrahedron with verticesO,A,B,C, we have
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 ofO 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 ofdegrees 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.[30]
Let,,, be the points of a tetrahedron. Let be the area of the face opposite vertex and let be the dihedral angle between the two faces of the tetrahedron adjacent to the edge. Thelaw of cosines for a tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation:[31]
LetP be any interior point of a tetrahedron of volumeV for which the vertices areA,B,C, andD, and for which the areas of the opposite faces areFa,Fb,Fc, andFd. Then[32]: p.62, #1609
For verticesA,B,C, andD, interior pointP, and feetJ,K,L, andM of the perpendiculars fromP to the faces, and suppose the faces have equal areas, then[32]: p.226, #215
Denoting the inradius of a tetrahedron asr and theinradii of its triangular faces asri fori = 1, 2, 3, 4, we have[32]: p.81, #1990
with equality if and only if the tetrahedron is regular.
IfA1,A2,A3 andA4 denote the area of each faces, the value ofr is given by
.
This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. Since the four subtetrahedra fill the volume, we have.
Denote the circumradius of a tetrahedron asR. Leta,b,c be the lengths of the three edges that meet at a vertex, andA,B,C the length of the opposite edges. LetV be the volume of the tetrahedron. Then[33][34]
The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron.With this definition, the circumcenterC of a tetrahedron with verticesx0,x1,x2,x3 can be formulated as matrix-vector product:[35]
In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron.Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron.
There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are calledHeronian tetrahedra. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces areisosceles triangles with areas of436800 and the other two are isosceles with areas of47120, while the volume is124185600.[36]
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.[37]
A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dualdigonaltrapezohedron, containing 6 vertices, in two sets of colinear edges.
A truncation process applied to the tetrahedron produces a series ofuniform polyhedra. Truncating edges down to points produces theoctahedron 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.
An interesting polyhedron can be constructed fromfive intersecting tetrahedra. Thiscompound of five tetrahedra has been known for hundreds of years. It comes up regularly in the world oforigami. Joining the twenty vertices would form a regulardodecahedron. There are bothleft-handed andright-handed forms, which aremirror images of each other. Superimposing both forms gives acompound of ten tetrahedra, in which the ten tetrahedra are arranged as five pairs ofstellae octangulae. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull.
Thesquare hosohedron is another polyhedron with four faces, but it does not have triangular faces.
TheSzilassi polyhedron and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face. Furthermore, theCsászár polyhedron (itself is the dual of Szilassi polyhedron) and the tetrahedron are the only two known polyhedra in which every diagonal lies on the sides.
At someairfields, 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.[38]
The tetrahedron shape is seen in nature incovalently bonded molecules. Allsp3-hybridized atoms are surrounded by atoms (orlone electron pairs) at the four corners of a tetrahedron. For instance in amethane molecule (CH 4) or anammonium 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 calledTetrahedron. Thecentral angle between any two vertices of a perfect tetrahedron is arccos(−1/3), or approximately 109.47°.[39]
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.
Quaternaryphase diagrams of mixtures of chemical substances are represented graphically as tetrahedra.
However, quaternary phase diagrams incommunication engineering are represented graphically on a two-dimensional plane.
There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such aswhite phosphorus allotrope[40] and tetra-t-butyltetrahedrane, known derivative of the hypotheticaltetrahedrane.
If six equalresistors aresoldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.[41]
Sincesilicon is the most commonsemiconductor used insolid-state electronics, and silicon has avalence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on howcrystals of silicon form and what shapes they assume.
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).[42]
Stanley Kubrick originally intended themonolith in2001: A Space Odyssey to be a tetrahedron, according toMarvin Minsky, a cognitive scientist and expert onartificial intelligence who advised Kubrick on theHAL 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.[46]
The tetrahedron with regular faces is a solution to an old puzzle asking to form four equilateral triangles using six unbroken matchsticks. The solution places the matchsticks along the edges of a tetrahedron.[47]
^ab(Coxeter 1973) uses the greek letter 𝝓 (phi) to represent one of the threecharacteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent thegolden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
^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. Archived from the original on 27 February 2009.{{cite book}}: CS1 maint: bot: original URL status unknown (link)
^Lindelof, L. (1867). "Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes".Acta Societatis Scientiarum Fennicae.8 (Part 1):189–203.
^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.PMID15558830.
^Lévy, Bruno; Liu, Yang (2010), "Lp centroidal Voronoi tessellation and its applications",ACM Transactions on Graphics,29 (4): 119:1–119:11,doi:10.1145/1778765.1778856
^"Problem 930"(PDF), Solutions,Crux Mathematicorum,11 (5):162–166, May 1985
^Wacław Sierpiński,Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however that Sierpiński repeats an erroneous calculation of the volume of the Heronian tetrahedron example above.
^Klein, Douglas J. (2002)."Resistance-Distance Sum Rules"(PDF).Croatica Chemica Acta.75 (2):633–649. Archived fromthe original(PDF) on 10 June 2007. Retrieved15 September 2006.
