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.[2]
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.[3]
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 simplestdeltahedron, a polyhedron all of whose faces are equilateral triangles; there are seven other convex deltahedra.[4]
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.[5]
Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups.[6] 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.[8] 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.[6] 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.
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.[10]
In general, a tetrahedron is athree-dimensional object with four faces, six edges, and four vertices. It can be considered aspyramid whenever one of its faces can be considered as thebase. Its1-skeleton can be generally seen as agraph bySteinitz's theorem, known astetrahedral graph, one of thePlatonic graphs. It iscomplete graph because every pair of its vertices has a unique edge. In a plane, this graph can be regarded as a triangle in which three vertices connect to its fourth vertex in the center, known as theuniversal vertex; hence, the tetrahedral graph is awheel graph.[11]
The tetrahedron is one of the polyhedra that does not havespace diagonal; the other polyhedra with such a property areCsászár polyhedron andSchonhardt polyhedron.[12] It is also known as3-simplex, the generalization of a triangle in multi-dimension. It isself-dual, meaning itsdual polyhedron is a tetrahedron itself.[13] Many other properties of tetrahedra are explicitly described in the following sections.
A simple way to obtain the volume of a tetrahedron is given by the formula for the volume:where is thebase' area and is the height from the base to the apex. This applies to 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.[14]
Alinear algebra approach is an alternative way by the given vertices in terms ofvectors as:In terms of adeterminant, the volume of a tetrahedron is, one-sixth of anyparallelepiped's volume sharing three converging edges with it.[15]
Similarly by the given vertices, another approach is by theabsolute value of the scalar triple product, representing the absolute values of determinants. Hence
Here,, and The variables,, and denotes eachnorm of a vector,, and respectively. This giveswhere theGreek lowercase letters denotes the plane angles occurring in vertex: the angle is an angle between the two edges connecting the vertex to the vertices and; the angle does so for the vertices and; while the angle is defined by the position of the vertices and. Considering that, then
Given the distances between the vertices of a tetrahedron the volume can be computed using theCayley–Menger determinant:where the subscripts represent the vertices, and is the pairwise distance between them, 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.
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:[16]whereThe above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.[16]
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,
Six edge-lengths of Tetrahedron
whereand
Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedronbisects the volume of the tetrahedron.[17]
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.[18] 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.[19]
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.[20] 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. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class oforthocentric tetrahedron.
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, a tetrahedron has four medians and three bimedians. These seven line segments are allconcurrent at a point called thecentroid of the tetrahedron.[21] In addition,the four medians are divided in a 3:1 ratio by the centroid.[22] 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.[23]
The center 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 towards the circumcenter. Additionally, an orthogonal line through 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[24]
where is the angle between the faces and.
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,, at which the solid angles subtended by the faces are equal, having a common value ofsr, and at which the angles subtended by opposite edges are equal.[25] A solid angle of 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 sr, lies inside the tetrahedron. Because the sum of distances from to the vertices is a minimum, coincides with thegeometric median of the vertices. If the solid angle at one of the vertices measures exactly sr, then and coincide with. However, if a tetrahedron has a vertex with solid angle greater than sr, still corresponds to, but lies outside the tetrahedron.
A corollary of the usuallaw of sines is that in a tetrahedron with vertices,,,, is given byOne 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 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. Hence, common factors are cancelled from both sides, and 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? 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.[26]
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:[27]
Let be any interior point of a tetrahedron of volume for which the vertices are,,, and, and for which the areas of the opposite faces are,,, and. Then,[28]
For vertices,,, and, interior point, and feet,,, and of the perpendiculars from to the faces, and suppose the faces have equal areas, then:[29]
Denoting the inradius of a tetrahedron as and theinradii of its triangular faces as for, then:[30]with equality if and only if the tetrahedron is regular.
If,, and denote the area of each faces, the value of 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, then:
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[31][32]
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:[33]
Unlike obtuse triangle, the circumcenter may not always lie outside of an obtuse tetrahedron (i.e. when one or moredihedral angles are greater than), nor does it necessarily lie inside for an acute tetrahedron (i.e. when all dihedral angles are less than)[34]
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.[35]
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.[36]
TheRoyal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice. Especially inroleplaying, this solid is known as a4-sided die, one of the more commonpolyhedral dice, with the number rolled appearing around the bottom or on the top vertex. SomeRubik's Cube-like puzzles are tetrahedral, such as thePyraminx andPyramorphix.
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 when a viewer who saw footage of it did not recognize what it was, and he did not want anything in the film that regular people did not understand.[37]
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.[38]
An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.
The tetrahedron is used incaltrops to provide anarea denial weapon. This is due to their nature of having a sharp corner that always points upwards.[42] Large steel tetrahedra have been used asanti-tank measures,[43] or concrete tetrahedra astetrapods to break down waves at coastlines.[44]
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.[45]
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.
The vertices of a regular tetrahedron applies as the minimum solution known for in bothThomson problem (concerning the minimum-energy configuration of charged particles on a sphere) and for theTammes problem (of constructing aspherical code maximizing the smallest distance among the points).[49]
If six equalresistors aresoldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.[50] 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).[51]
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.
^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.
^Uehara, Ryuhei (2020). "Fig. 4.6: All nets of a regular tetrahedron and a regular octahedron by edge-unfolding".Introduction to Computational Origami: The World of New Computational Geometry. Singapore: Springer. p. 64.doi:10.1007/978-981-15-4470-5.ISBN978-981-15-4469-9.MR4215620.
^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.
^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.
Bottema, O. (1969). "A Theorem of Bobillier on the Tetrahedron".Elemente der Mathematik.24: 6–10.
Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018). "Historical notes on star geometry in mathematics, art and nature". In Emmer, Michele; Abate, Marco (eds.).Imagine Math 6: Between Culture and Mathematics. Springer International Publishing. pp. 197–211.doi:10.1007/978-3-319-93949-0_17.hdl:10447/325250.ISBN978-3-319-93948-3.
Schneider, Teseo; Hu, Yixin; Gao, Xifeng; Dumas, Jérémie; Zorin, Denis; Panozzo, Daniele (2022). "A Large-Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method".ACM Transactions on Graphics.41 (3):1–14.arXiv:1903.09332.doi:10.1145/3508372.
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