Pentagonal trapezohedron
A pentagonal trapezohedron is an isohedral polyhedron composed of 10 congruent kite-shaped quadrilateral faces, with 12 vertices and 20 edges, serving as the dual of the pentagonal antiprism.[1] The structure features two polar vertices where sets of five kites meet, interleaved with a rotational twist of 36 degrees (180°/5), distinguishing it from a simple bipyramid.[1] This polyhedron exhibits dihedral symmetry of order 4n (20 for n=5), making all faces equivalent under the symmetry group, and it can be constructed by connecting two displaced regular pentagons with apical points in a twisted configuration.[1] As a member of the trapezohedron family, its faces are not true trapezoids but rather symmetric kites, with edges alternating between shorter and longer lengths.[1] The pentagonal trapezohedron is one of the few non-Platonic solids employed in practical applications, notably as the shape for 10-sided dice (d10) in role-playing games, where the kite faces allow for fair rolling and numbered vertices.[2]Definition and Classification
Formal definition
A pentagonal trapezohedron is a convex isohedral polyhedron composed of 10 congruent kite-shaped quadrilateral faces, 20 edges, and 12 vertices. It belongs to the family of trapezohedra, which are duals of antiprisms, and specifically arises as the dual of the pentagonal antiprism. Each kite face features three interior angles measuring 108° and one measuring 36°, reflecting the underlying 5-fold symmetry derived from regular pentagonal geometry.[3][1] As an isohedral polyhedron, it is face-transitive, meaning there exists a symmetry of the figure mapping any face to any other face. It is also convex, ensuring all interior angles are less than 180° and line segments between points lie within the polyhedron. The Euler characteristic confirms its topology as a genus-0 surface: V - E + F = 12 - 20 + 10 = 2.[4][5] The term "trapezohedron" originates from the Greek words for "trapezium" (a quadrilateral) and "hedron" (base or seat), referring to the trapezium-like (kite-shaped) faces observed in crystallographic forms. The prefix "pentagonal" indicates the 5-fold rotational symmetry inherent to its structure.[6]Place in trapezohedra family
Trapezohedra form an infinite family of isohedral polyhedra that are the duals of uniform n-gonal antiprisms for integers n ≥ 3, where each member has 2n kite-shaped faces, 2(n+1) vertices, and 4n edges.[1] These polyhedra are characterized by their staggered arrangement of faces, with n kites converging at each of two opposite vertices, and the remaining edges forming two n-gonal zigzags.[1] The family encompasses a range of symmetries belonging to the dihedral groups D_{nd}, reflecting the rotational and reflectional properties inherited from the antiprisms.[1] The pentagonal trapezohedron corresponds to the case n=5 in this family, serving as the dual of the uniform pentagonal antiprism and featuring 10 congruent kite faces.[7] It is also referred to as the pentagonal deltohedron due to its composition of deltoidal (kite) faces or as an antitegum in some classifications of dual polyhedra.[8] In comparison, the n=3 trapezohedron, known as the trigonal trapezohedron or rhombohedron, has 6 faces and can achieve cubic symmetry in the special case where all edges are equal, as in the cube.[1] The n=6 case, the hexagonal trapezohedron, possesses 12 faces with D_{6d} symmetry, paralleling the face count of the regular dodecahedron while maintaining the characteristic kite faces and dihedral symmetry of the family.[1] Unlike rhombi, the kite faces of trapezohedra are generally irregular quadrilaterals with two pairs of adjacent equal sides, becoming rhombi only for specific n such as 3 under equal-edge conditions.[1]Construction
As dual of pentagonal antiprism
The pentagonal trapezohedron can be constructed as the dual polyhedron of the pentagonal antiprism, a uniform polyhedron consisting of two parallel regular pentagonal bases rotated by 36° (or $180^\circ / 5) relative to each other and connected by a band of 10 equilateral triangular faces.[9][10] To form the dual, place a vertex at the centroid of each of the antiprism's 12 faces: one vertex at the center of each pentagonal base and one at the center of each triangular face.[11] Next, connect these dual vertices with edges wherever the corresponding primal faces of the antiprism share an edge; this yields 20 edges in total, matching the antiprism's edge count.[7] The faces of the dual polyhedron are then defined by the sets of dual vertices that correspond to the faces adjacent to each vertex of the antiprism: since each of the antiprism's 10 vertices is incident to four faces (one pentagon and three triangles), the dual has 10 quadrilateral faces, each a kite shaped by the symmetric arrangement of these adjacencies.[3] The two remaining dual vertices, positioned at the centers of the antiprism's pentagonal bases, each connect to the five dual vertices from the adjacent triangular faces, forming two polar 5-valent vertices that cap the structure.[7] This 36° twist in the antiprism's bases ensures the kite faces align in opposing pairs with alternating orientations, characteristic of the trapezohedral form, while producing a total of 12 vertices as noted in its topological properties.[9][10]Cartesian coordinates
The vertices of a pentagonal trapezohedron can be expressed in Cartesian coordinates with the two apical vertices positioned at (0, 0, \pm h), where h is the height parameter representing the distance from the center to each apex. The remaining ten vertices form two regular pentagons in parallel planes at heights \pm z, twisted relative to each other by an angle \theta = 36^\circ (or \pi/5 radians) to ensure the kite-shaped faces. Specifically, the upper pentagon vertices are at (r \cos(2\pi k / 5 + \theta), r \sin(2\pi k / 5 + \theta), z) for k = 0, 1, 2, 3, 4, and the lower pentagon vertices at (r \cos(2\pi k / 5), r \sin(2\pi k / 5), -z) for the same k, where r is the radial distance from the axis and z is the half-height of the equatorial bands. These parameters r, z, and h are chosen to satisfy the geometric constraints of the polyhedron, such as the edge lengths and face planarity.[12] In a common normalization derived from the dual pentagonal antiprism with unit edge length, the short edges (connecting vertices within each kite face) have length a = (\sqrt{5} - 1)/2 \approx 0.618 and the long edges (connecting to the apices) have length b = (1 + \sqrt{5})/2 = \phi \approx 1.618, where \phi is the golden ratio. In this scaling, the height parameter is h = \sqrt{5 + 2\sqrt{5}}/2 \approx 1.5388, corresponding to the circumradius to the apical vertices (the farthest vertices). The distance from the center to the equatorial vertices is \sqrt{3}/2 \approx 0.866. Alternative normalizations include scaling for unit short edge length (a = 1) or unit inscribed sphere radius (inradius = 1), which adjust r, z, h proportionally while preserving the ratios involving \phi. For instance, the full height between apices is $2h \approx 3.0777 in the above normalization.[13][14] An example set of coordinates in the antiprism-unit normalization places the apices at (0, 0, \pm 1.5388); the upper ring at approximately $0.8507 \cdot (\cos(72^\circ k + 36^\circ), \sin(72^\circ k + 36^\circ), 0.1624) for k = 0 to $4 (yielding equatorial distance \approx 0.866); and the lower ring at $0.8507 \cdot (\cos(72^\circ k), \sin(72^\circ k), -0.1624), where r = \sqrt{(5 + \sqrt{5})/10} \approx 0.8507 and z = 1 / (2 \sqrt{5 + 2\sqrt{5}}) \approx 0.1624. These positions facilitate direct computational modeling, such as in 3D graphics or finite element analysis.[15]Geometric Properties
Topological features
The pentagonal trapezohedron is a polyhedron comprising 12 vertices, 20 edges, and 10 kite-shaped quadrilateral faces.[1] Its vertex configuration consists of 10 trivalent vertices, each incident to three edges, and 2 pentavalent vertices at the apices, each incident to five edges.[3] This distribution arises from its role as the dual of the pentagonal antiprism, where the trivalent vertices correspond to the triangular faces of the antiprism and the pentavalent vertices to its pentagonal bases.[10] The 20 edges form a connected graph that is topologically uniform, meaning all edges are equivalent in terms of their connectivity within the polyhedron's skeleton, regardless of any metric distinctions such as length.[13] Each of the 10 kite faces shares each of its four edges with an adjacent face, resulting in a 4-regular face adjacency graph with 10 vertices.[3] As the dual of the uniform polyhedron indexed U77 (the pentagonal antiprism), the pentagonal trapezohedron is an irregular isohedral polyhedron without a standard Schläfli symbol.[16] Combinatorially, it can be decomposed as a twisted bipyramid constructed over a pentagonal base, where the two apical pyramids are rotated relative to each other by 36 degrees.[14]Metric properties
The pentagonal trapezohedron features two distinct edge lengths in its standard scaling, derived from the geometry of its dual pentagonal antiprism with unit edge length. The ten short edges measure \frac{\sqrt{5} - 1}{2} \approx 0.618, while the ten long edges measure \frac{1 + \sqrt{5}}{2} \approx 1.618.[13][17] Each of the ten kite-shaped faces has an area of \frac{\sqrt{10 + 2\sqrt{5}}}{4} \approx 0.9511. The total surface area is therefore \frac{5}{2} \sqrt{10 + 2\sqrt{5}} \approx 9.510.[17] The volume is given by \frac{5(3 + \sqrt{5})}{12} \approx 2.1817.[13][17] The inradius (radius of the inscribed sphere) is \frac{\sqrt{5(5 + 2\sqrt{5})}}{10} \approx 0.6882, the midradius (or edge-scribed radius, tangent to the midpoints of the edges) is \frac{1 + \sqrt{5}}{4} \approx 0.8090, and the circumradius (radius of the circumscribed sphere) is \frac{\sqrt{5 + 2\sqrt{5}}}{2} \approx 1.5388. These radii reflect the distances from the center to the respective tangential or spherical contacts.[13]| Radius Type | Formula | Approximate Value |
|---|---|---|
| Inradius | \frac{\sqrt{5(5 + 2\sqrt{5})}}{10} | 0.6882 |
| Midradius | \frac{1 + \sqrt{5}}{4} | 0.8090 |
| Circumradius | \frac{\sqrt{5 + 2\sqrt{5}}}{2} | 1.5388 |