Fact-checked by Grok 2 weeks ago

Trapezohedron

A trapezohedron, also known as an antidipyramid or deltohedron, is an isohedral polyhedron composed of 2n congruent kite-shaped (deltoid) faces, serving as the dual of an n-gonal antiprism. It features 2(n+1) vertices and 4n edges, with half the faces converging at a top vertex and the other half at a bottom vertex, forming a convex hull from two displaced and rotated n-gons plus axial points. Trapezohedra exhibit face-transitive symmetry, making all faces equivalent under the polyhedron's , and are notable for their applications in both and practical contexts. For instance, the (n=3) is a , with the as a special regular case that is also the dual of the . The tetragonal trapezohedron (n=4) appears in M.C. Escher's lithograph (1948), highlighting its aesthetic properties. In gaming, the (n=5) forms the shape of standard 10-sided dice, ensuring fair rolling due to its isohedral nature and opposite faces summing to 9 on 0–9 dice or to 11 on 1–10 dice. Beyond , trapezohedra are significant in as forms arising from 3-, 4-, or 6-fold axes combined with a 2-fold , commonly observed in minerals like garnets, which often exhibit trapezohedral crystal habits. These polyhedra can be with specific edge lengths, heights, and volumes derived from trigonometric formulas, such as the edge length e_n = 1/(2 - 2\cos(\pi/n)). Star variants, like the pentagrammic concave trapezohedron, extend the family into non-convex forms as duals of crossed antiprisms.

Definition and Terminology

Definition

A trapezohedron is the of an n-gonal , consisting of 2n congruent kite-shaped (deltoid) faces, each a with two pairs of adjacent equal sides. This structure yields the topological invariants of 2n faces, 4n edges, and 2n+2 vertices, which satisfy for convex polyhedra: V - E + F = 2. As a , face-transitive (isohedral) , it features regular vertices where three faces meet at each . The general form exists for integers n \geq 3, where the prefix indicates the gonal aspect (e.g., trigonal for n=3); a special case is the for n=3.

Terminology

The term trapezohedron derives from the New Latin trapezohedron, combining (referring to a quadrilateral with no parallel sides in the sense) and the Greek suffix -hedron meaning "face" or "," reflecting its faces. First recorded around 1810–1820, it entered geometric literature in the early to describe polyhedra with staggered kite-shaped faces. Alternative names include deltohedron, emphasizing the deltoid (kite-like) configuration of its faces, and antidipyramid or antibipyramid, which evoke the structure as two pyramids joined at their bases but rotated relative to each other. Early visual representations of trapezohedral forms appear in Albrecht Dürer's 1514 engraving , where a truncated triangular trapezohedron—known as Dürer's solid—features prominently, sparking enduring interest at the intersection of art and mathematics. This depiction, predating formal geometric nomenclature, influenced later studies by highlighting the polyhedron's aesthetic and structural intrigue in scholarship. The term trapezohedron must be distinguished from deltahedron, which denotes a with all faces as equilateral triangles, whereas the deltohedron specifically involves kite faces without such triangular uniformity. In crystallography, trapezohedron often refers to specific forms like the tetragonal trapezohedron, generated by 4-fold axes combined with perpendicular 2-fold axes, but the broader usage can encompass non-convex variants in mineral structures, differing from the geometric ideal. In contemporary geometry, the family is denoted as an n-gonal trapezohedron, generalizing the structure for any integer n ≥ 3, with particular cases named by symmetry: the tetragonal trapezohedron for n=4 and the for n=5. This nomenclature underscores the polyhedron's role as the dual of an n-gonal , standardizing its classification across mathematical contexts.

Geometric Construction

Faces and Structure

A trapezohedron features 2n congruent kite-shaped faces, each a with two pairs of adjacent equal-length sides and two acute angles paired with two obtuse angles. These faces connect vertices of a zig-zag equatorial 2n-gon to two polar apices, one at each end of the , forming a twisted arrangement that gives the its characteristic structure. In the symmetric case, the kites are isohedral, meaning all faces are congruent and can be mapped onto one another by the polyhedron's symmetries, ensuring uniform face transitiveness. The overall structure arises from joining two n-gonal pyramids base-to-base, with one base rotated relative to the other by an of 180°/n to create a staggered alignment. This rotation transforms the original triangular faces of the pyramids into kites, as adjacent triangles from opposing pyramids combine across the . The resulting equatorial belt forms a non-planar 2n-gon with alternating up and down vertices, and the edges around this equator alternate between shorter and longer lengths, contributing to the polyhedron's and aesthetic . In general, no pairs of faces are , distinguishing the trapezohedron from prisms or other parallel-faced polyhedra.

Vertices, Edges, and Dual Relationship

A trapezohedron possesses 2n + 2 vertices, where n ≥ 3 denotes the order of the underlying . These consist of two polar vertices situated along the principal and 2n equatorial vertices arranged in a zig-zag equatorial band. The polar vertices each exhibit a degree of n in the polyhedron's , as each connects to n equatorial vertices, while the equatorial vertices each have 3, linking to two neighboring equatorial vertices and one polar vertex. The edge set comprises 4n edges in total, divided into 2n polar edges radiating from the apices to the equatorial band and 2n equatorial edges forming the zig-zag connections among the equatorial vertices. These equatorial edges alternate in length, reflecting the twisted configuration of the two n-gonal rings that constitute the equatorial structure. In the trapezohedral graph, each edge is incident to exactly two faces, ensuring the polyhedron's closed surface topology. The graph maintains a girth of 4, corresponding to the quadrilateral kite faces, and is 3-vertex-connected, characteristic of simple convex polyhedral graphs. As the of an n-gonal , the trapezohedron exhibits a combinatorial structure: its 2n + 2 vertices correspond to the antiprism's 2n + 2 faces (2n triangles and 2 n-gons), its 2n faces correspond to the antiprism's 2n vertices, and its 4n edges align one-to-one with the antiprism's edges. This duality maps the two n-gonal faces of the antiprism to the two polar vertices of the trapezohedron and the triangular faces to the equatorial vertices. The relationship preserves face-transitivity in the trapezohedron, mirroring the vertex-transitivity of the antiprism.

Properties

Symmetry

The symmetry group of an n-gonal trapezohedron is the D_{nd} (also denoted D_{nv}) of $4n, consisting of prismatic symmetries enhanced by a mirror plane perpendicular to the principal axis. This group incorporates n rotations of n around the principal (vertical) axis, n twofold rotations around axes perpendicular to the principal axis, n mirror planes containing the principal axis and bisecting adjacent rotation axes, and one mirror plane. The rotational is the D_n of $2n, comprising the n-fold principal rotations and the n twofold rotations. For the special case of n=3, the trigonal trapezohedron generally possesses D_{3d} symmetry of order 12, but achieves the higher full O_h of order 48 when it is , as in the case of the , which is the of the . The n-gonal trapezohedron is the of the n-gonal , which has the Wythoff | 2 \, 2 \, n and U_{77} for n=5 as a representative example. Trapezohedra are isohedra, meaning the acts transitively on the faces, mapping any face to any other, but the action is generally not transitive on vertices except in special cases like the .

Coordinates and Dimensions

A trapezohedron can be constructed in Cartesian coordinates with its principal axis aligned along the z-axis. The two polar vertices are positioned at (0, 0, \pm h). There is an upper ring of n vertices at height z = d > 0, radius r, with positions (r \cos(2\pi k / n), r \sin(2\pi k / n), d) for k = 0 to n-1, and a lower ring of n vertices at z = -d, radius r, rotated by \pi / n, at (r \cos(2\pi k / n + \pi / n), r \sin(2\pi k / n + \pi / n), -d) for k = 0 to n-1. The parameters r, d, and h are chosen to ensure the kite faces are planar and to achieve the desired proportions, such as in the canonical form derived from the dual antiprism. Canonical dimensions are often normalized such that the short edge length is 1. Under this normalization, the long edge length is e_n = \frac{1}{2 - 2\cos(\pi/n)}, and the height (full, from pole to pole) relates via h_n = \frac{1}{4 \csc^3(\pi/(2n)) \sin(\pi/n)} for the distance from equator to pole (adjusting for full height 2h_n). The volume is V_n = \frac{n \cot(\pi/(2n)) \csc^2(\pi/(2n)) (2\cos(\pi/n) + 1)}{24 \sqrt{2 + 2\cos(\pi/n)}}. The surface area is S_n = \frac{n}{4} \csc^2(\pi/(2n)) \sqrt{4\cos(\pi/n) - 2\cos(2\pi/n) + 1}. These ensure the isohedral properties, with each kite face formed by connecting a pole to two adjacent vertices in one ring and the intervening vertex in the opposite ring. For the specific case of the pentagonal trapezohedron (n=5) under a unit span normalization for the kite diagonal (where the equatorial span across one face is 1), approximate parameters include h \approx 1.538 (half-height) and r \approx 0.8508, yielding a long edge length of \sqrt{r^2 + h^2} \approx 1.758 and short edge length \approx 0.526.

Variants and Special Cases

Convex Forms

The convex forms of trapezohedra are the non-intersecting, isohedral polyhedra in this family, characterized by 2n kite-shaped faces for n ≥ 3, with all faces congruent and the polyhedron dual to an n-gonal antiprism. For the degenerate case of n=2, the digonal trapezohedron reduces to a tetrahedron, where the four triangular faces can be viewed as limiting kites with collinear edges, resulting in 4 faces, 6 edges, and 4 vertices. For n=3, the is equivalent to a , consisting of 6 congruent rhombic faces meeting at two apical vertices, with the rhombi typically featuring acute and obtuse angles such as approximately 60° and 120° in non-cubic cases. A special instance occurs when the rhombi are squares, yielding the as a trigonal trapezohedron. This form highlights the trapezohedron's capacity for high symmetry, serving as the dual to the triangular antiprism. The n=4 case produces the tetragonal trapezohedron, with 8 kite faces that approximate squares but are elongated, dual to the and exhibiting tetragonal symmetry. For higher even n, such as n=6, the hexagonal trapezohedron features 12 faces and is the dual of the hexagonal antiprism, an , illustrating how convex trapezohedra for even n often pair with uniform polyhedra in dual relationships. Odd-n examples include the n=5 , which has 10 congruent faces and is dual to the , another . Across these forms, dihedral angles between adjacent faces are greater than 90° and increase with n.

Star Forms

Star trapezohedra represent non- variants of trapezohedra, constructed analogously to their counterparts but utilizing bases. These polyhedra arise as the duals of uniform , where the antiprism features two parallel regular bases denoted by the {p/q}, rotated relative to one another and connected by isosceles triangular sides. The equatorial cross-section forms a zig-zag with q, and the vertices of this base are connected to two apical poles, yielding 2p -shaped faces that intersect along their edges due to the starring. Each face is a , a with two pairs of adjacent equal sides, extended from the concept but incorporating the . These forms exhibit self-intersections, rendering them non- with a polyhedral equal to q, the of the base , which corresponds to a greater than 1 around the center. They remain isohedral, meaning all faces are congruent and the acts transitively on them, and exhibit the full prismatic group D_{ph} , including reflections, similar to trapezohedra. The intersecting nature arises from the non-planar of the star base, leading to faces that cross without violating the uniform edge lengths. A representative example is the pentagrammic trapezohedron (p=5, q=2), dual to the pentagrammic with {5/2} bases; it possesses 10 intersecting tri-equilateral faces, 20 edges (10 short and 10 long), and 12 vertices (10 of degree 3 and 2 of degree 5), under D_{5h} symmetry. Similarly, heptagrammic trapezohedra exist for q=2 and q=3, with bases {7/2} and {7/3}, respectively, each featuring 14 faces and analogous prismatic symmetry, demonstrating the family for higher p. These examples illustrate how increasing p and varying q produces more complex intersections while maintaining the trapezohedral topology. Star trapezohedra emerged in the broader study of uniform star polyhedra, which extended the classical stellations of solids to include prismatic and antiprismatic forms with star elements, contributing to the enumeration of non-convex regular compounds in the .

Applications

In

In , trapezohedra manifest as closed crystal forms consisting of congruent faces, typically appearing in tetragonal, trigonal, and hexagonal crystal systems. Trapezohedra are closed forms consisting of congruent faces, resulting from 3-, 4-, or 6-fold rotation axes combined with a 2-fold , producing 3, 4, or 6 upper faces offset from an equal number of lower faces. Enantiomorphic pairs of trapezohedra are common in these systems, particularly in classes lacking a center of inversion, where left- and right-handed variants occur due to the absence of mirror . The tetragonal trapezohedron, with eight faces, belongs to the 422 (D_4) and is represented by of the form {hkk}, such as {211}, though less common in minerals like which favor prismatic habits modified by pyramids and pinacoids. In the trigonal system, the six-faced trapezohedron occurs notably in 32 (D_3), exemplified by forms such as {51 \bar{6} 1} in , where the positive trapezohedron dominates the and intersects the c-axis at specific angles defined by . Hexagonal trapezohedra, with twelve faces, appear in 6 2 2 but are less common in natural minerals, often requiring specific growth conditions. Prominent natural occurrences include the 24-faced trapezohedron (trisoctahedron) in , where the {211} form combines with the {110} to produce rounded crystals prevalent in metamorphic rocks. Certain crystals exhibit trapezohedral forms, such as the trisoctahedron {311}, modifying dodecahedral habits in hydrothermal deposits. Trigonal trapezohedra are well-expressed in , frequently combined with pedions or basal pinacoids, aiding in the identification of enantiomorphic varieties through goniometric analysis of . These habits underscore the role of trapezohedra in defining crystal and in mineral assemblages.

In Gaming and Dice

The , featuring 10 kite-shaped faces, forms the basis for the standard ten-sided die (d10) in role-playing games, including . Numbered from 0 to 9, these dice have opposite faces summing to 9, facilitating their use in generating random numbers for mechanics such as damage rolls or ability checks. Introduced amid the rise of tabletop role-playing games in the late , the modern d10 design was patented and commercialized by Gamescience in 1980 as a truncated pentagonal trapezohedron, replacing earlier approximations like modified d20s. The kite-shaped faces promote stable rolling by allowing the die to settle flat on any face, while the isohedral —where all faces are congruent and equivalently positioned relative to the center—ensures fair and uniform probability outcomes. For percentile dice (d100), two d10s are often used together, with one typically in a contrasting color to distinguish tens (00, 10, ..., 90) from units (0-9). Variants of trapezohedral dice include skew designs, such as the asymmetric used for six-sided dice (d6) by The Dice Lab, which maintain fairness despite their irregular appearance through isohedral face arrangements that yield standard probabilities. These dice are manufactured primarily from injection-molded plastic like or for , though premium versions use metal alloys such as for durability and weight. The fairness of trapezohedra stems from their face-transitive geometry, akin to the octahedral d8, enabling equivalent chances for each face to land upward without bias from shape or material distribution.