Tetrahedron
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.
The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle, so a tetrahedron is also known as a "triangular pyramid".
Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.
For any tetrahedron there exists a sphere on which all four vertices lie, and another sphere tangent to the tetrahedron's faces.
Regular tetrahedron
A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. In other words, all of its faces are the same size and shape and all edges are the same length. The regular tetrahedron is the simplest deltahedron, a polyhedron all of whose faces are equilateral triangles; there are seven other convex deltahedra.Irregular tetrahedra
If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron.In a trirectangular tetrahedron the three face angles at one vertex are right angles, as at the corner of a cube.
An isodynamic tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent.
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.
Disphenoid
A disphenoid 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.Orthoschemes
A 3-orthoscheme is a tetrahedron where all four faces are right triangles. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a disphenoid with right triangle or obtuse triangle faces.An orthoscheme is an irregular simplex that is the convex hull of a tree 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 is birectangular tetrahedron. It is also called a quadrirectangular tetrahedron because it contains four right angles.
Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups. For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is characteristic 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 and one of length, 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 cube diagonal. The cube can also be dissected into 48 smaller instances of this same characteristic 3-orthoscheme. The characteristic tetrahedron of the cube is an example of a Heronian tetrahedron.
Every regular polytope, including the regular tetrahedron, has its characteristic 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 ? = 2, its characteristic tetrahedron's six edges have lengths,, around its exterior right-triangle face, plus,, . 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 a 60-90-30 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,,.
Space-filling tetrahedra
A space-filling tetrahedron packs with directly congruent or enantiomorphous copies of itself to tile space. The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed, and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense.A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the disphenoid tetrahedral honeycomb. Regular tetrahedra, however, cannot fill space by themselves. The tetrahedral-octahedral honeycomb fills space with alternating regular tetrahedron cells and regular octahedron cells in a ratio of 2:1.
Fundamental domains
An irregular tetrahedron which is the fundamental domain of a symmetry group is an example of a Goursat tetrahedron. The Goursat tetrahedra generate all the regular polyhedra by mirror reflections, a process referred to as Wythoff's kaleidoscopic construction.For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a kaleidoscope. 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.
Among the Goursat tetrahedra which generate 3-dimensional honeycombs we can recognize an orthoscheme, a double orthoscheme, and the space-filling disphenoid illustrated [|above]. The disphenoid is the double orthoscheme face-bonded to its mirror image. Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be dissected into characteristic tetrahedra of the cube.
Subdivision and similarity classes
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 the Longest Edge Bisection , 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.A similarity 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.
General properties
In general, a tetrahedron is a three-dimensional object with four faces, six edges, and four vertices. It can be considered as pyramid whenever one of its faces can be considered as the base. Its 1-skeleton can be generally seen as a graph by Steinitz's theorem, known as tetrahedral graph, one of the Platonic graphs. It is complete 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 the universal vertex; hence, the tetrahedral graph is a wheel graph.The tetrahedron is one of the polyhedra that does not have space diagonal; the other polyhedra with such a property are Császár polyhedron and Schonhardt polyhedron. It is also known as 3-simplex, the generalization of a triangle in multi-dimension. It is self-dual, meaning its dual polyhedron is a tetrahedron itself. Many other properties of tetrahedra are explicitly described in the following sections.
Volume
A simple way to obtain the volume of a tetrahedron is given by the formula for the volume:where is the base' 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.
Algebraic approach
A linear algebra approach is an alternative way by the given vertices in terms of vectors as:In terms of a determinant, the volume of a tetrahedron is, one-sixth of any parallelepiped's volume sharing three converging edges with it.
Similarly by the given vertices, another approach is by the absolute value of the scalar triple product, representing the absolute values of determinants. Hence
Here,, and The variables,, and denotes each norm of a vector,, and respectively. This gives
where the Greek 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 the Cayley–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 called Tartaglia's formula, is essentially due to the painter Piero della Francesca in the 15th century, as a three-dimensional analogue of the 1st century Heron's formula for the area of a triangle.