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Triangular bipyramid

A triangular bipyramid, also known as a trigonal dipyramid, is a polyhedron formed by joining two regular tetrahedra along one of their faces or, equivalently, by attaching two triangular pyramids base-to-base.^[1]^[2] It features six equilateral triangular faces, nine edges of equal length, and five vertices—three forming an equatorial triangle and two apical vertices.^[2]^[3] This polyhedron is one of the eight convex deltahedra, which are strictly convex polyhedra bounded exclusively by equilateral triangles, and it is the only such deltahedron with six faces.^[1]^[4] It is also classified as the twelfth Johnson solid (J_{12}), part of the 92 strictly convex polyhedra enumerated by Norman Johnson in 1966 that possess regular polygonal faces and equal edge lengths but are neither Platonic solids, Archimedean solids, prisms, nor antiprisms.^[1]^[2] As an isohedron, it is face-transitive, meaning all faces are equivalent under the polyhedron's symmetries, though its vertices are of two types: the three equatorial vertices each incident to four faces and the two apical vertices each incident to three faces.^[1]^[2] The triangular bipyramid exhibits dihedral symmetry of order 12, belonging to the point group D_3h, which includes rotations and reflections preserving its structure.^[2] It serves as the dual polyhedron of the triangular prism, where faces of one correspond to vertices of the other.^[5] For a triangular bipyramid with edge length s, the surface area is approximately 2.598s² and the volume is approximately 0.236s³, reflecting its compact, symmetrical form.^[6]

Definition and Construction

General Description

A triangular bipyramid is a hexahedron with six equilateral triangular faces, nine edges, and five vertices, constructed by joining two regular tetrahedra along a shared triangular face.^[1]^[7] It is also known by the alternative names triangular dipyramid and trigonal bipyramid.^[1]^[7] The triangular bipyramid is classified as a convex deltahedron, with all faces being equilateral triangles, an isohedral polyhedron where the symmetry group acts transitively on the congruent faces, and a composite polyhedron due to its formation from two tetrahedra.^[1]^[7] It was recognized as Johnson solid J12 in Norman Johnson's 1966 enumeration of the 92 strictly convex polyhedra having regular faces but not qualifying as Platonic, Archimedean, prisms, or antiprisms.^[8]^[9]

Construction Methods

A primary method to construct a regular triangular bipyramid involves attaching two regular tetrahedra face-to-face along one shared triangular face, resulting in a polyhedron where the glued face is internalized and the outer surfaces form six equilateral triangular faces.^[10]^[11] This assembly ensures that the three vertices of the shared face serve as the equatorial triangle, with the two remaining apex vertices positioned symmetrically on opposite sides of this plane.^[12] An alternative construction treats the triangular bipyramid as a specific case of a bipyramid over a triangular base: start with an equilateral triangular polygon as the equator, then add two apical vertices, one above and one below the plane, and connect each apex to all three equatorial vertices via edges of equal length to form the six triangular faces.^[12] In this approach, the equatorial triangle acts as the shared base, and the apices are placed such that all edges are congruent for the regular variant.^[11] While the above methods yield the regular triangular bipyramid with uniform edge lengths, the construction can be generalized to irregular bipyramids by using a non-equilateral triangular base or offsetting the apical positions asymmetrically, though the focus here remains on the symmetric, regular case.^[12] Cartesian coordinates may be employed for precise vertex placement in computational models, as detailed in the geometric properties section.

Geometric Properties

Combinatorial Structure

The triangular bipyramid is a polyhedron consisting of 6 triangular faces, 9 edges, and 5 vertices.^[13] These elements satisfy Euler's formula for convex polyhedra, where the Euler characteristic \chi = V - E + F = 5 - 9 + 6 = 2.^[14] This confirms its topological structure as a genus-0 surface homeomorphic to a sphere.^[15] In terms of vertex configuration, the triangular bipyramid features two apical vertices, each of degree 3 and connected exclusively to the three equatorial vertices, and three equatorial vertices, each of degree 4 and linked to the two apices as well as to their two adjacent equatorial neighbors.^[13] This arrangement arises from joining two tetrahedra along a shared triangular face.^[3] All faces of the triangular bipyramid are congruent triangles, rendering the polyhedron face-transitive, or isohedral, meaning any face can be mapped to any other via a symmetry of the figure.^[16] As the 12th Johnson solid (J12), it exemplifies a convex polyhedron with regular polygonal faces that is neither a Platonic solid nor an Archimedean solid.^[13]

Metric Properties and Formulas

The regular triangular bipyramid, with all edges of equal length a, can be positioned in three-dimensional Cartesian space with its equatorial triangle in the xy-plane centered at the origin. For an edge length of a = 2, the coordinates of the vertices are the three equatorial points (1, -1/\sqrt{3}, 0), (-1, -1/\sqrt{3}, 0), and (0, 2/\sqrt{3}, 0), along with the two apical vertices at (0, 0, \sqrt{8/3}) and (0, 0, -\sqrt{8/3}).^[3] These coordinates ensure all edges are of length 2 and can be scaled linearly by a/2 for general edge length a. The surface area of the regular triangular bipyramid consists of six equilateral triangular faces, each with area (\sqrt{3}/4) a^2. Thus, the total surface area is A = 6 \times (\sqrt{3}/4) a^2 = (3\sqrt{3}/2) a^2 \approx 2.598 a^2.^[17]^[18] The volume of the regular triangular bipyramid can be derived by viewing it as the union of two triangular pyramids sharing the equatorial face, with no interior overlap. Each pyramid has a base equilateral triangle of area A_b = (\sqrt{3}/4) a^2 and height h equal to the perpendicular distance from an apex to the equatorial plane. Using the scaled coordinates, h = a \sqrt{2/3}. The volume of one such pyramid is V_p = (1/3) A_b h = (1/3) (\sqrt{3}/4) a^2 \cdot a \sqrt{2/3} = (\sqrt{2}/12) a^3, so the total volume is V = 2 V_p = (\sqrt{2}/6) a^3 \approx 0.236 a^3.^[19]^[13]

Symmetry

Symmetry Group

The symmetry group of the triangular bipyramid is the dihedral point group D_{3h} of order 12.^[13]^[20] This group encompasses a principal 3-fold rotation axis (C_3) passing through the two apical vertices, three perpendicular 2-fold rotation axes (C_2) in the equatorial plane that bisect the equatorial edges, a horizontal reflection plane (\sigma_h) coinciding with the equatorial triangle, and three vertical reflection planes (\sigma_v) each containing the apical vertices and one equatorial vertex.^[21]^[20] The full set of 12 elements includes the identity, two non-trivial 3-fold rotations, three 2-fold rotations (6 proper rotations total), and six improper isometries comprising the horizontal reflection, three vertical reflections, and two improper 3-fold rotations (S_3).^[20] The triangular bipyramid possesses prismatic symmetry under D_{3h}, reflecting its construction as a dual to the triangular prism.^[13] However, the polyhedron is not vertex-transitive, as the apical vertices (each adjacent to three faces) differ from the equatorial vertices (each adjacent to four faces).^[18]^[22] In irregular variants, such as those with unequal apical heights or distorted edges, the symmetry lowers to subgroups of D_{3h}, such as C_{3v} (retaining the 3-fold axis and vertical mirrors but lacking the horizontal mirror and 2-fold axes) or further to C_3.^[20]

Dihedral Angles

In the regular triangular bipyramid, where all six faces are equilateral triangles of equal edge length, there are two distinct dihedral angles corresponding to the types of edges. The dihedral angle between the two triangular faces sharing an equatorial edge—such as the faces connecting the northern apex to one equatorial edge and the southern apex to the same edge—is \arccos\left(-\frac{7}{9}\right) \approx 141.06^\circ. This obtuse angle arises from the geometry of the two regular tetrahedra that compose the bipyramid, with each apex and the equatorial triangle forming a regular tetrahedron; the equatorial dihedral is determined by the orientation of the corresponding lateral faces from each tetrahedron relative to the shared internal equatorial plane.^[7] The dihedral angle between two triangular faces sharing an apical edge—such as two faces both connected to the northern apex and adjacent equatorial vertices—is \arccos\left(\frac{1}{3}\right) \approx 70.53^\circ. This acute angle matches the dihedral angle of a regular tetrahedron, as the three faces meeting at each apex replicate the tetrahedral configuration locally.^[7] The D_{3h} symmetry of the regular triangular bipyramid preserves these dihedral angles under the group's rotations and reflections. In irregular triangular bipyramids, where the equatorial edges or apical heights differ, the dihedral angles vary continuously; for instance, increasing the separation between apices tends to widen the equatorial dihedral toward $180^\circ, while the apical dihedrals adjust based on the specific edge lengths, computable via the cosine of the angle between face normals derived from vector cross products.^[7]

Special Cases

As a Right Bipyramid

A right triangular bipyramid is formed by joining two regular tetrahedra base-to-base such that the line connecting the two apices passes perpendicularly through the centroid of the shared equilateral triangular base, ensuring axial symmetry along this line.^[23] This configuration distinguishes it from oblique bipyramids, in which the apices are displaced laterally from the centroid, resulting in asymmetric edge lengths and face distortions even if the equatorial base remains equilateral.^[24] In the right form with an equilateral triangular equator of side length a, all nine edges are equal to a, and the six triangular faces are congruent equilateral triangles, achieving full regularity as the Johnson solid J₁₂.^[18] The apical height h—the distance from each apex to the equatorial plane—is given by h = \frac{\sqrt{6}}{3} a.^[3] For visualization, the vertices of a right triangular bipyramid with edge length a = 2 can be placed at coordinates (\pm 1, -\frac{1}{\sqrt{3}}, 0), (0, \frac{2}{\sqrt{3}}, 0) in the equatorial plane, and (0, 0, \pm \sqrt{\frac{8}{3}}) for the apices, confirming the right alignment with the origin at the centroid.^[3] Scaling these coordinates by \frac{a}{2} yields the positions for arbitrary edge length a.

As a Johnson Solid

The triangular bipyramid is identified as Johnson solid J₁₂, the twelfth entry in mathematician Norman W. Johnson's 1966 enumeration of 92 strictly convex polyhedra whose faces are regular polygons but which are neither Platonic nor Archimedean solids.^[25] This classification emphasizes its construction from six equilateral triangular faces meeting at edges of equal length, forming a non-uniform polyhedron due to the irregular arrangement of faces around its vertices.^[8] Among the Johnson solids, the triangular bipyramid stands out as one of two simple bipyramids—the other being the pentagonal bipyramid (J₁₃)—with all faces equilateral triangles, yet it lacks the full symmetry of Platonic solids because of its unequal vertex figures: two apical vertices where three triangles meet and three equatorial vertices where four triangles meet.^[26] Unlike Archimedean solids or their Catalan duals, Johnson solids like J₁₂ do not possess transitive vertex symmetries or corresponding dual catalanoids with equal edge lengths.^[8] In Johnson's ordered list, J₁₂ follows early entries such as the square pyramid (J₁) and triangular cupola (J₃), preceding more complex elongations and gyroelongations.^[25] Its inclusion highlights the bipyramids' role in completing the catalog of irregular regular-faced polyhedra, where the right triangular bipyramid achieves edge uniformity without attaining Platonic regularity.^[26]

Dual Polyhedron

The dual polyhedron of the triangular bipyramid is the triangular prism, where the six triangular faces of the bipyramid correspond to the six vertices of the prism, and the five vertices of the bipyramid correspond to the five faces of the prism (two equilateral triangles and three quadrilaterals).^[27]^[5] This duality preserves the nine edges of both polyhedra while interchanging the roles of faces and vertices, reflecting the reciprocal structure inherent in convex polyhedra.^[28] In the polar reciprocation process, the vertices of the triangular prism are obtained by taking the points that are the poles with respect to a unit sphere centered at the origin for each face plane of the triangular bipyramid; for a triangular bipyramid with vertices at coordinates such as (\pm 1, 1/\sqrt{3}, 0), (0, -2/\sqrt{3}, 0), and (0, 0, \pm \sqrt{8/3}) (scaled for edge length 2), the resulting dual coordinates yield the prism's structure.^[3]^[29] The triangular prism, as the dual, exhibits properties such as rectangular lateral faces in its uniform realization, where these faces are squares to achieve vertex-transitivity, making it one of the Archimedean solids.^[27] Both polyhedra share the D_{3h} point group symmetry.^[28] The kleetope of the triangular bipyramid is formed by attaching a pyramid to each of its six triangular faces, resulting in a polyhedron whose skeleton is the Goldner–Harary graph, a maximal planar graph with 11 vertices and 27 edges that is the smallest known non-Hamiltonian simplicial polyhedron.^[30] This construction adds one vertex per original face, connecting it to the three vertices of that face, yielding a triangulation with all faces remaining triangles.^[30] The elongated triangular bipyramid, designated as Johnson solid J14, is obtained by attaching a triangular prism to the equatorial triangle of the triangular bipyramid, producing a polyhedron with 8 vertices, 15 edges, 9 faces (6 equilateral triangles and 3 squares), and D_{3h} symmetry. This augmentation increases the number of faces while maintaining convexity and regular polygonal faces, distinguishing it from the parent bipyramid by introducing square faces along the elongation axis. Gyroelongated forms extend the triangular bipyramid by inserting a triangular antiprism between its apical pyramids, creating a polyhedron with additional triangular faces bridging the rotated bases; for equilateral triangles, adjacent faces become coplanar, merging into 6 rhombi with 60°–120° angles. This construction generalizes to higher n-gonal bipyramids as Johnson solids (e.g., J17 for squares), but the triangular case yields a degenerate form where the antiprism's rotation leads to face coplanarity rather than distinct triangles. The triangular bipyramid is one of eight convex deltahedra, polyhedra composed exclusively of equilateral triangular faces, alongside the tetrahedron, regular octahedron, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid, and icosahedron.^[4] These eight were fully enumerated and proven exhaustive for convex cases with equilateral triangular faces.

Applications

In Molecular Geometry

In molecular geometry, the triangular bipyramid describes the trigonal bipyramidal arrangement adopted by five-coordinate molecules with no lone pairs on the central atom, as predicted by valence shell electron pair repulsion (VSEPR) theory. This structure features three ligands in an equatorial plane at 120° angles to each other and two axial ligands positioned 90° from the equatorial plane, minimizing repulsion between the five bonding pairs around the central atom.^[31] Representative examples include phosphorus pentafluoride (PF₅), where the central phosphorus atom bonds to five fluorine atoms in this geometry, and phosphorus pentachloride (PCl₅) in the gas phase, which similarly exhibits trigonal bipyramidal coordination with chlorine ligands. Transition metal complexes, such as iron pentacarbonyl (Fe(CO)₅), also display this structure, with the iron center coordinated to five carbonyl groups—three equatorial and two axial.^[31]^[32] Many such molecules are fluxional, undergoing rapid interconversion of axial and equatorial ligand positions via the Berry pseudorotation mechanism, a low-energy pathway that preserves bond integrity and explains NMR observations of equivalent ligands at higher temperatures. This process involves a square pyramidal transition state where one equatorial ligand becomes axial and vice versa.^[33] Deviations from ideal trigonal bipyramidal geometry often arise due to steric repulsions from ligand sizes, as larger ligands increase equatorial-equatorial angles beyond 120° or elongate axial bonds to reduce crowding, consistent with VSEPR principles extended to account for such interactions.^[34]

In Physics and Other Fields

In the Thomson problem, which seeks the minimum-energy configuration of repelling point charges on a sphere, the optimal arrangement for five electrons forms a triangular bipyramid, where three points lie on the equator and two at the poles.^[35] This configuration has been rigorously proven to be unique via computer-assisted methods, minimizing the total Coulomb potential energy.^[36] In historical color theory, German astronomer Tobias Mayer proposed a three-dimensional color order system in a 1758 lecture, extending a triangular base of primary colors (red, yellow, blue) into a double pyramid with white at the upper apex and black at the lower apex, effectively a triangular bipyramid where vertices represent color mixtures.^[37] This model quantified color mixtures by proportions along edges, influencing later systematic approaches to color harmony.^[38] The triangular bipyramid serves as a foundational unit in coupled polyhedral mechanisms for robotics, enabling single-degree-of-freedom translational motion through symmetric linkage designs. For instance, a 2016 study introduced a novel mechanism based on this polyhedron, analyzed for deployable structures and path trajectories, with applications in motion planning for robotic systems.^[39] In computational geometry, the triangular bipyramid can be divided into two tetrahedra (triangular pyramids) by sharing a common face, facilitating polyhedral enclosures and tetrahedralizations for mesh generation and symbolic constraint solving.^[40] This decomposition supports algorithms for volume partitioning and optimization in 3D modeling.^[41] Parallels to molecular coordination geometries exist, where similar axial-equatorial arrangements describe ligand placements around central atoms.

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