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Trigonal trapezohedron

A trigonal trapezohedron, also known as a trigonal deltohedron or 3-trapezohedron, is a convex polyhedron consisting of six congruent rhombic faces, eight vertices, and twelve edges.^[1] It belongs to the family of trapezohedra, which are isohedral polyhedra where all faces are identical kites or rhombi, and in this case, it exhibits trigonal symmetry characterized by a threefold rotation axis perpendicular to three twofold axes.^[2] This polyhedron is notable for its role in crystallography as the general form {hkil} in the trigonal point group 32 (D_3), where it appears as two interpenetrating trigonal pyramids rotated relative to each other by an arbitrary angle θ, transforming into a rhombohedron when θ = 60° or a trigonal dipyramid when θ = 0°.^[2] The full symmetry group is D_{3d} of order 12, including reflections and inversion, making it a centrosymmetric figure suitable for applications requiring uniform face accessibility, such as fair six-sided dice.^[3] Special cases include the cube (with square rhombi and higher octahedral symmetry O_h) and the acute and obtuse golden rhombohedra, which have rhombic faces with angles related to the golden ratio.^[1] In geometry and polyhedral combinatorics, the trigonal trapezohedron is the dual of the trigonal antiprism and can be constructed by elongating a triangular antiprism or as a limiting case of more general deltoidal polyhedra.^[4] Its isohedral nature ensures that every face is equivalent under the symmetry operations, a property shared with rhombohedra, but the trigonal trapezohedron's specific configuration ensures full D_{3d} symmetry.^[5]

Definition and Basic Structure

Definition

A trigonal trapezohedron is a polyhedron composed of exactly six congruent rhombic faces, arranged such that three faces meet at each of two opposite vertices to form polar caps, with the remaining faces creating an equatorial belt around the structure.^[1]^[4] This configuration distinguishes it as a specific type of trapezohedron, or deltohedron in the case of rhombic faces, characterized by trigonal (threefold) symmetry along a principal axis.^[1] Historically, the trigonal trapezohedron has been termed a trigonal deltohedron or 3-trapezohedron, reflecting its geometric lineage within polyhedral classification systems.^[1] The name "trapezohedron" originates from "trapezoid," an older term denoting a kite-shaped quadrilateral without parallel sides, while "trigonal" refers to the threefold symmetry axis that defines its rotational properties.^[4]^[6] As a rhombohedron, all faces are identical congruent rhombi.^[5]

Face, Edge, and Vertex Counts

The trigonal trapezohedron possesses 6 faces, 12 edges, and 8 vertices.^[4] These topological invariants satisfy Euler's polyhedral formula for a convex polyhedron of genus 0, V - E + F = 8 - 12 + 6 = 2.^[7]^[4] The polyhedron is 3-valent, with each of its 8 vertices incident to exactly 3 edges and 3 faces.^[8] Each of the 6 quadrilateral faces is adjacent to 4 others, sharing one edge with each, as determined by the total of 12 edges connecting pairs of faces ($6 \times 4 / 2 = 12).^[4]

Geometric Properties

Face Configuration

The trigonal trapezohedron consists of six congruent rhombic faces arranged in three pairs of opposite faces that exhibit a characteristic twist around the central threefold symmetry axis. This arrangement divides the polyhedron into polar regions near the apical vertices and an equatorial belt, with each face spanning from the polar zones to connect with the equatorial structure, forming a closed, convex surface.^[4]^[1] In terms of adjacency, each rhombic face shares its four edges with four neighboring faces, ensuring that exactly two faces meet along any given edge, which maintains the polyhedron's topological integrity without triple meetings at edges. This configuration results in a seamless interleaving of the faces, where the offset rotation between upper and lower triangular layers—by 60 degrees—creates the twisted, staggered appearance characteristic of the form.^[4] As a zonohedron, the trigonal trapezohedron possesses the property of being the Minkowski sum of three line segments oriented in distinct directions, generating three zones of parallel faces: each zone comprises a pair of opposite rhombi whose edges align with two of the generating directions. This zonal structure underscores its equilateral nature and ability to tile space in certain cases, such as the rhombohedral honeycomb.^[9]^[10] For visualization, a conceptual net of the trigonal trapezohedron unfolds into a pattern of six rhombi organized into three parallel bands, each band linking two opposite faces to illustrate the zonal belts and facilitate understanding of how the twisted assembly closes into the three-dimensional form.^[4]

Coordinates and Metrics

A trigonal trapezohedron, as a rhombohedron with six congruent rhombic faces, has its vertices defined by eight points generated from three basis vectors of equal length forming equal angles α between them. The vertices are located at all possible sums of these vectors taken zero, one, two, or three at a time. To ensure trigonal symmetry, the basis vectors u, v, w (each of length a, the edge length) can be positioned such that u lies along the x-axis, v lies in the xy-plane at angle α to u, and w is determined to make angle α with both u and v, with its z-component ensuring unit length (or scaled to a). Explicit components require solving w_x = a cos α, w_y = (a cos α - a cos² α)/ (a sin α) from the dot product conditions, and w_z = a \sqrt{1 - (w_x/a)^2 - (w_y/a)^2}, confirming the volume non-zero for physical α (typically 60° < α < 109.47° for convex).^[11]^[1] In the rhombic case where all faces are congruent rhombi, all 12 edges have equal length a, corresponding to the length of the basis vectors. The dihedral angle δ between adjacent rhombic faces is calculated using the dot product of their outward normals: \cos \delta = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{|\mathbf{n_1}| |\mathbf{n_2}|}, where \mathbf{n_1} and \mathbf{n_2} are the normals to the two faces sharing an edge, such as \mathbf{n_1} = \mathbf{u} \times \mathbf{v}. Due to the D_{3d} symmetry, there is a single distinct dihedral angle; for the rhombohedron, it simplifies to \cos \delta = -\cos \alpha, with specific values depending on α—for example, in the cube special case (α = 90°), δ = 90°. The normals can be computed as cross products of edge vectors from the coordinate basis.^[12]^[11] The volume V of a trigonal trapezohedron in its rhombic variant is given by V = a^3 \sqrt{1 - 3 \cos^2 \alpha + 2 \cos^3 \alpha}, confirming the metric consistency with the coordinate construction.^[13]

Symmetry

Point Group D_{3d}

The trigonal trapezohedron belongs to the crystallographic point group D_{3d}, denoted in Hermann-Mauguin notation as \bar{3}m, which classifies its full symmetry in the trigonal crystal system. This point group has order 12 and consists of the dihedral group D_3 (encompassing proper rotations) combined with an inversion operation, yielding elements that include the identity, rotations, reflections, and improper rotations (rotary inversions). Specifically, the group incorporates one identity element, two non-trivial 3-fold rotations, three 2-fold rotations, one inversion, two 6-fold rotary inversions, and three reflections.^[3]^[14] The symmetry axes of D_{3d} for the trigonal trapezohedron align with its geometric structure: a principal 3-fold rotation axis (C_3) passes through opposite vertices, three 2-fold rotation axes (C_2) lie in the plane bisecting the C_3 axis and pass through midpoints of opposite edges, and a center of inversion (i) is located at the polyhedron's centroid. Complementing these are three dihedral mirror planes (\sigma_d) that contain the C_3 axis and bisect the angles between adjacent C_2 axes. These elements collectively ensure that the eight vertices and twelve edges of the trigonal trapezohedron are transitively permuted under the group action.^[3]^[14] A key consequence of D_{3d} symmetry is the isohedral nature of the trigonal trapezohedron, where all six kite-shaped faces are equivalent and can be mapped onto one another by the group's operations; in crystallography, this polyhedron represents the general form for \bar{3}m, with faces corresponding to the {hkl} form. For achiral variants, the full symmetry holds, but chiral versions of the trapezohedron—lacking the inversion center and mirror planes—reduce to the lower-symmetry rotation subgroup D_3 (order 6), which includes only the proper rotations.^[3]^[14]

Symmetry Operations

The symmetry operations of the trigonal trapezohedron are the elements of the point group D_{3d}, consisting of proper rotations, reflections, improper rotations, and inversion. These operations leave the polyhedron invariant and map its structural elements—faces, edges, and vertices—onto themselves or equivalent parts.^[14] The proper rotations include two 3-fold rotations by 120° and 240° around the principal axis, which cycles three faces meeting near one pole of the polyhedron while simultaneously cycling the three faces meeting near the opposite pole. There are also three 2-fold rotations by 180° around axes perpendicular to the principal axis; each such rotation swaps a pair of opposite faces.^[14]^[3] The reflections comprise three vertical mirror planes (\sigma_d), each containing the principal 3-fold axis and bisecting a pair of opposite faces.^[14] The improper rotations consist of two S_6 operations along the principal axis, where each is a 60° rotation combined with a reflection through the plane perpendicular to the axis, and the central inversion (i) through the polyhedron's centroid, which maps each element to its antipodal counterpart.^[14] Under the action of these operations, the eight vertices form two orbits: one of two polar vertices along the principal axis and the other of six equatorial vertices in the girdle perpendicular to it.^[3]

Special Cases

The Cube

The trigonal trapezohedron assumes the form of a cube when each of its six rhombic faces degenerates into a square, yielding a regular hexahedron with all edges of equal length and adjacent faces mutually perpendicular. In this special case, the polyhedron's structure aligns perfectly with the Platonic solid known as the cube, where the trigonal orientation corresponds to viewing it along a body diagonal. This degeneration preserves the six faces but equalizes the angles of the rhombi to 90 degrees, eliminating the obliqueness characteristic of general trigonal trapezohedra.^[1]^[4] This configuration markedly enhances the symmetry beyond the general trigonal trapezohedron's point group D_{3d} (order 12), elevating it to the full octahedral group O_h (order 48), which includes additional 4-fold rotations and reflections aligned with the square faces. The D_{3d} subgroup persists when the cube is oriented such that its body diagonal coincides with the principal 3-fold axis, demonstrating how the cube embeds as a symmetric endpoint within the trigonal trapezohedron family. This symmetry upgrade underscores the cube's role as the most symmetric member, where the equal edge lengths and right angles enable the full set of octahedral operations.^[8]^[15] Geometrically, the cube exhibits uniform dihedral angles of exactly $90^\circ across all edges, a direct consequence of the perpendicular faces. Its volume is simply V = a^3, where a denotes the common edge length, providing a straightforward metric for this regular case. These properties highlight the cube's departure from the variable angles and volumes of non-square trigonal trapezohedra. Historically, the polyhedron in Albrecht Dürer's 1514 engraving Melencolia I—known as Dürer's solid and identified as a truncated triangular trapezohedron—has been interpreted by some as a distorted variant within the trapezohedron family, evoking a "crazy cube" due to its apparent cubic inspiration; however, the untruncated cube remains the canonical regular example of the trigonal trapezohedron.^[16]^[17]

Golden Rhombohedra

The golden rhombohedra represent special irrational instances of the trigonal trapezohedron, characterized by faces that are golden rhombi with diagonals in the golden ratio \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618. There are two distinct forms: the acute golden rhombohedron, in which three acute face angles meet at each polar vertex, and the obtuse golden rhombohedron, in which three obtuse face angles meet at each polar vertex. Both share the same face configuration but differ in their vertex figure orientations, preserving the D_{3d} symmetry of the trigonal trapezohedron family.^[18]^[19]^[20] The faces of both golden rhombohedra are congruent golden rhombi, with interior angles of approximately $63.43^\circ (acute) and $116.57^\circ (obtuse), derived from the geometry where the diagonals d_\text{long} and d_\text{short} satisfy d_\text{long}/d_\text{short} = \phi. For an edge length of 1, the diagonals are approximately d_\text{long} \approx 1.701 and d_\text{short} \approx 1.051. These rhombi tile the surfaces of more complex golden isozonohedra, including the rhombic triacontahedron (with 30 such faces) and the Bilinski dodecahedron (with 12 such faces), enabling constructions that embed icosahedral symmetry in quasicrystalline models. The acute form emphasizes the smaller angles at polar vertices, while the obtuse form highlights the larger ones, facilitating distinct assembly roles in polyhedral dissections.^[21]^[22] Copies of the acute golden rhombohedron tile three-dimensional space to form a space-filling honeycomb via rhombohedral lattice translations, as do copies of the obtuse form independently. In contrast, the regular rhombic dodecahedron—a space-filling polyhedron with \sqrt{2} diagonal ratio rhombi—can be dissected into four oblate rhombohedra, illustrating a rational counterpart to the irrational golden assemblies. For edge length 1, the body diagonals of the golden rhombohedra exhibit ratios involving \phi, with the tip-to-tip height (shortest body diagonal for the acute form) given by

\sqrt{3 + 6 \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{5}} \approx 2.058

. The dihedral angle \beta between adjacent faces is uniform and computed as \beta = \arccos\left( -\frac{\cos \alpha}{1 + \cos \alpha} \right), where \alpha \approx 63.43^\circ is the acute face angle, yielding \beta \approx 108^\circ and reflecting the adjustment of symmetric base values like \arccos(\pm 1/\sqrt{3}) through the golden ratio geometry.^[23]^[20]^[24]^[19]

Dual Polyhedron

The dual polyhedron of the trigonal trapezohedron is the triangular antiprism.^[4] The triangular antiprism consists of 6 vertices, 12 edges, and 8 triangular faces.^[4] This topological duality follows Euler's formula, with the 6 vertices of the dual corresponding to the 6 quadrilateral faces of the trigonal trapezohedron, and the 8 triangular faces of the dual corresponding to the 8 vertices of the original polyhedron.^[4] In the dual construction, each vertex of the triangular antiprism is positioned at the centroid of a quadrilateral face of the trigonal trapezohedron.^[4] Conversely, each triangular face of the dual connects the centroids of the three quadrilateral faces adjacent to an original vertex, reflecting the degree-3 valence at each vertex of the trigonal trapezohedron.^[4] Both polyhedra share the D_{3d} point group symmetry, preserving the threefold rotational and inversion symmetries of the original.^[4] The trigonal trapezohedron is isohedral with congruent rhombic faces. For instance, when the trigonal trapezohedron specializes to a cube (with square faces aligned along a space diagonal), its dual is the regular octahedron.^[1]

Trapezohedron Family

The trapezohedron family encompasses the n-gonal trapezohedra, a sequence of polyhedra where each member possesses 2n congruent kite-shaped faces arranged in a staggered manner around two polar vertices.^[4] The trigonal trapezohedron occupies the initial position in this family for n=3, resulting in 6 rhombic faces that meet symmetrically.^[4] These polyhedra are also known as antidipyramids or deltohedra, distinguishing them from related forms through their quadrilateral face configuration. The trigonal trapezohedron is equivalent to the rhombohedron, which has six congruent rhombic faces and all edges of equal length.^[25] As a zonohedron, it arises from the Minkowski sum of three line segments (generating zones) in distinct directions, producing three belts of parallel rhombic faces that ensure central symmetry.^[9] This zonal construction underscores its membership among parallelohedra, polyhedra capable of filling three-dimensional space without gaps or overlaps via translational copies, as exemplified in rhombohedral lattices.^[26] In contrast to the trapezohedron family, dipyramids or bipyramids feature 2n triangular faces rather than kites, arising as duals to prisms instead of antiprisms.^[4] The family progresses with increasing n; for instance, the n=4 tetragonal trapezohedron has 8 kite faces, extending the staggered kite arrangement seen in the trigonal case while maintaining the overall antidipyramidal topology.^[4]

Applications

Crystallography

In the trigonal crystal system, the trigonal trapezohedron represents the general crystal form {hkl} in point groups 32 (D_3) and \bar{3}m (D_{3d}), characterized by a three-fold rotation axis and three two-fold axes perpendicular to it.^[27] This form consists of six quadrilateral faces reflecting the enantiomorphic nature in the chiral point group 32 (D_3), which lacks mirror planes, while in \bar{3}m (D_{3d}) it includes an inversion center.^[28] Quartz (SiO_2), belonging to point group 32, commonly exhibits these trapezohedral faces, distinguishing left- and right-handed enantiomorphs through the orientation of the faces relative to the prism.^[29] In hexagonal coordinates used for trigonal systems, trapezohedral faces are indexed as {hkil}, where the indices account for intercepts on the a_1, a_2, a_3, and c axes, with the third index i = -(h + k) ensuring lattice compatibility.^[30] The rhombohedral Bravais lattice underpins space-filling arrangements in the trigonal system, enabling periodic structures that accommodate trapezohedral forms.^[31] Special cases, such as the golden rhombohedron with angles of approximately 63.43° and 116.57°, can tile space to form honeycombs approximating higher symmetries, relevant to certain mineral lattices and quasicrystalline approximations in trigonal contexts.^[32]

Dice and Models

The asymmetric trigonal trapezohedron serves as a fair six-sided die, offering an elongated alternative to the traditional cube while maintaining equal probability for each face due to its D_{3d} point group symmetry.^[33] This shape, known as a skew die, features six congruent kite-shaped faces that are transitively equivalent under the polyhedron's symmetry operations, ensuring that random orientations land equally on any face.^[34] The fairness of these dice is geometrically verified by their isohedral (face-transitive) property, where the symmetry group acts to map any face to any other, guaranteeing uniform outcomes regardless of the die's skewed appearance.^[33] Commercial examples, such as those produced by Tessellations, demonstrate this in practice, with the dice rolling as reliably as cubic d6s for gaming purposes.^[35] Physical models of the trigonal trapezohedron are commonly constructed via 3D printing for educational visualization of its symmetry and as components in polyhedral art installations.^[36] Open-source designs, such as those available on platforms like Printables, allow assembly from parametric files, facilitating hands-on exploration in geometry classrooms or hobbyist modeling.^[37] These models highlight the polyhedron's utility beyond gaming, serving as tangible aids for understanding rotational symmetries in three dimensions.^[38] In role-playing games, the trigonal trapezohedron appears in custom dice sets as a novel d6 variant, providing aesthetic variety without compromising randomness, as seen in specialty products from manufacturers like The Dice Lab.^[33] Historically, such non-cubic polyhedral dice emerged in the late 20th century alongside broader polyhedral sets, with skew designs gaining niche popularity in the 2010s for their mathematical intrigue.^[39]

References

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Trigonal trapezohedron. (hkil). (ihkl). (kihl) y, x, z x + y, y, z x, x y, z ... D3d hexagonal axes. 12 d. 1 x, y, z y, x y, z x + y, x, z y, x, z x + y, y ...
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The dihedral angle is the angle theta between two planes. The dihedral angle between the planes a_1x+b_1y+c_1z+d_1 = 0 (1) a_2x+b_2y+c_2z+d_2 = 0 (2) which ...Missing: rhombohedron | Show results with:rhombohedron
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Aug 20, 2018 · The asymmetric version of a Trigonal Trapezohedron is supposed to be a fair die just like a cube, meaning I can start with one face and rotate ...Coordinates of a rhombohedron - Math Stack Exchangewhat are the symmetries and flags of tetrahedron?More results from math.stackexchange.com
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A parallelohedron is a space-filling polyhedron that fills space using an infinite number of similarly situated copies.
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28-day returnsThis d6 dice in the form of an asymmetric trigonal trapezohedron. These wacky-looking dice are actually just as fair as regular six-sided dice. Unlike other ...
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In stockThese Skew dice are a d6 in the form of an asymmetric trigonal trapezohedron. These wacky-looking dice are actually just as fair as regular six-sided dice.