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Square antiprism

A square antiprism is a uniform polyhedron consisting of two parallel regular squares rotated by 45 degrees relative to one another and connected by an alternating band of eight equilateral triangles.^[1] It features 10 faces (two squares and eight triangles), 16 edges, and 8 vertices, with each vertex incident to one square and three triangles.^[2]^[3] As one of the 75 uniform polyhedra, the square antiprism exhibits full symmetry under the dihedral group D_{4d}, which has 16 elements and includes rotations and reflections preserving its structure.^[4] Its dual polyhedron is the tetragonal trapezohedron, a catalan solid with kite-shaped faces.^[2] For an edge length of 1, the surface area is $2(1 + \sqrt{3}) and the volume is \frac{1}{3} \sqrt{4 + 3\sqrt{2}}, reflecting its compact, space-filling potential in geometric constructions.^[2] The square antiprism can be extended to form other uniform polyhedra, such as the gyroelongated square pyramid (Johnson solid J10) by capping one base with a square pyramid, or the gyroelongated square bipyramid (J17) with caps on both bases.^[3] In applications, it appears in molecular geometry for eight-coordinate complexes and in architectural designs requiring high symmetry and efficiency.^[5] Coordinates for its vertices, centered at the origin with edge length 2, include points like

(0, \pm \sqrt{2}, 1/\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{2})

and

(\pm 1, \pm 1, -1/\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{2})

, enabling precise modeling in computational geometry.^[3]

Definition and Properties

Definition

A square antiprism is the second member in the infinite family of uniform antiprisms, constructed by placing two parallel regular squares in rotationally offset parallel planes, rotated by 45 degrees relative to each other, and connecting corresponding vertices with 8 equilateral triangular faces.^[1] This configuration results in a polyhedron with 10 regular faces: 2 squares and 8 triangles, all of equal edge length in the uniform case.^[6] It is also known by the alternative names anticube and squap.^[7]^[6] The Schläfli symbol for the square antiprism is s\{2,4\}, reflecting its structure as a rectified digonal prismatoid, while an alternative notation is \mathrm{sr}\{2,4\}. The vertex configuration is (3.3.3.4), indicating that three equilateral triangles and one square meet at each of the 8 vertices.^[8] The square antiprism is classified as a uniform polyhedron, meaning it has regular polygonal faces and is vertex-transitive, and as a semiregular polyhedron due to its regular faces meeting in a consistent arrangement at each vertex, though the vertices themselves do not form regular polygons.^[7]^[9] Antiprisms, including the square variant, originated in studies of uniform polyhedra, with early recognition by Johannes Kepler in 1619, who grouped them with prisms as semiregular solids, and further elaboration by Louis Poinsot in 1810; the square antiprism was specifically examined in 19th- and 20th-century works on prismatic and antiprismatic polyhedra.^[10]^[11]

Basic Properties

The square antiprism is a polyhedron with 10 faces, consisting of 8 equilateral triangular faces and 2 square faces.^[12]^[2] It has 16 edges, with 8 forming the boundaries of the square bases and 8 lateral edges connecting the bases in a twisted configuration.^[12] The structure includes 8 vertices, each incident to one square and three triangles.^[12] The Euler characteristic of the square antiprism is calculated as V - E + F = 8 - 16 + 10 = 2, confirming its topology as a genus-0 surface homeomorphic to a sphere.^[12] As a closed polyhedral surface without self-intersections, it is orientable.^[6] The dual polyhedron of the square antiprism is the tetragonal trapezohedron, also known as the square trapezohedron or square antitegum, which features 10 quadrilateral faces.^[2]^[12]^[6] The vertex figure at each vertex is an isosceles trapezoid with edge lengths 1, 1, 1, and \sqrt{2} (assuming unit edge length for the polyhedron).^[6] As a uniform polyhedron, the square antiprism belongs to the family of prismatic polyhedra, where all vertices are equivalent and faces are regular polygons.^[6]

Geometric Measures

Cartesian Coordinates

The Cartesian coordinates of a square antiprism with edge length 1 can be specified relative to its center at the origin. The four vertices of the upper square lie in the plane z = 8^{1/4}/4 \approx 0.42045, positioned at \left(\pm \frac{1}{2}, \pm \frac{1}{2}, \frac{8^{1/4}}{4}\right) (all sign combinations). The four vertices of the lower square lie in the plane z = -8^{1/4}/4 \approx -0.42045, rotated by 45° relative to the upper square, positioned at \left(0, \pm \frac{\sqrt{2}}{2}, -\frac{8^{1/4}}{4}\right) and \left(\pm \frac{\sqrt{2}}{2}, 0, -\frac{8^{1/4}}{4}\right).^[6] These coordinates place the polyhedron symmetrically about the origin, with the upper and lower bases as regular squares of side length 1, since the horizontal distances between adjacent vertices in each base equal 1 (e.g., distance between (0.5, 0.5, z) and (0.5, -0.5, z) is 1). The 45° rotation of the lower base relative to the upper ensures that each vertex of one base connects to two non-adjacent vertices of the other via edges of length 1, forming eight equilateral triangular faces; the specific value of z = 8^{1/4}/4 is chosen precisely to satisfy this edge length condition, as the horizontal distance between a vertex of the upper base and a connected vertex of the lower base is \frac{\sqrt{4 - 2\sqrt{2}}}{2} \approx 0.5412 and the vertical separation is $8^{1/4}/2 \approx 0.84090, yielding \sqrt{ \left( \frac{\sqrt{4 - 2\sqrt{2}}}{2} \right)^2 + \left(8^{1/4}/2\right)^2 } = [1](/page/1).^[6] The height h between the parallel bases is $8^{1/4}/2 = 2^{-1/4} \approx 0.84090.^[6] The circumradius R, or distance from the center to any vertex, is \sqrt{2(4 + \sqrt{2})}/4 = \sqrt{(4 + \sqrt{2})/8} \approx 0.82267.^[12] This follows directly from the coordinates, e.g., \sqrt{ (1/2)^2 + (1/2)^2 + (8^{1/4}/4)^2 } = \sqrt{1/2 + (8^{1/4}/4)^2}.^[6] The midradius \rho, the radius of the midsphere tangent to the edges at their midpoints, is \sqrt{2(2 + \sqrt{2})}/4 \approx 0.65328. For example, the midpoint of a base edge such as between (0.5, 0.5, z) and (0.5, -0.5, z) is at (0.5, 0, z), with distance \sqrt{ (1/2)^2 + (8^{1/4}/4)^2 } \approx 0.65328 from the center.^[12] The inradius r, the radius of the inscribed sphere tangent to all faces, is \sqrt{4 + 3\sqrt{2}} / [2(1 + \sqrt{3})] \approx 0.52573 for edge length 1. It is computed as r = 3V / S, where V = \sqrt{4 + 3\sqrt{2}}/3 \approx 0.95700 is the volume and S = 2 + 2\sqrt{3} \approx 5.46410 is the surface area (two squares of area 1 each plus eight equilateral triangles of area \sqrt{3}/4 each). To arrive at this: first, S = 2 \cdot 1 + 8 \cdot (\sqrt{3}/4 \cdot 1^2) = 2 + 2\sqrt{3}; the volume formula for a uniform n-gonal antiprism specializes to the given expression for n=4; thus r = \sqrt{4 + 3\sqrt{2}} / [2(1 + \sqrt{3})].^[12]^[1]

Volume and Surface Area

The surface area of a square antiprism with regular faces and unit edge length consists of two squares, each with area 1, and eight equilateral triangles, each with area \sqrt{3}/4, for a total of S = 2 + 2\sqrt{3} \approx 5.464.^[2] This value arises directly from the constituent face areas, confirming the polyhedron's uniformity. The volume is given by V = \frac{1}{3} \sqrt{4 + 3\sqrt{2}} \approx 0.957, or equivalently V = \frac{(\sqrt{2} + 1) 2^{1/4}}{3}.^[2]^[12] To derive this, the polyhedron can be treated as a prismatoid and decomposed using the formula V = \frac{h}{6} (B_1 + B_2 + 4M), where h \approx 0.841 is the height, B_1 = B_2 = 1 are the square base areas, and M is the area of the intermediate regular octagonal cross-section; alternatively, pyramid decomposition from the centroid to each face, based on the Cartesian coordinates, yields the same result through tetrahedral volume summation.^[12] The coordinates, detailed in the Cartesian Coordinates subsection, position the vertices to ensure unit edges and enable such computations. For comparison, a right square prism with the same unit edge length and height adjusted to match the antiprism's (a non-uniform prism) has volume $1 \times 0.841 \approx 0.841, but the uniform square prism (cube) with all edges 1 has volume 1, larger than the antiprism's despite the twist reducing the effective height.^[12] The surface area density, S/V \approx 5.71, is lower than the unit cube's 6, reflecting the antiprism's more compact form. The isoperimetric quotient Q = 36 \pi V^2 / S^3 \approx 0.635 exceeds the cube's \pi/6 \approx 0.524, assessing greater spherical-like uniformity.^[12]

Symmetry

Symmetry Group

The full symmetry group of the square antiprism is the point group D_{4d}, a dihedral group of order 16 that incorporates 8 proper rotations and 8 improper isometries (reflections and rotoinversions).^[13]^[14] This group arises from the uniform structure of the polyhedron, where the two square bases are rotated by 45° relative to each other, preserving the overall antiprismatic arrangement.^[15] The rotational subgroup is D_4, of order 8, generated by rotations around symmetry axes aligned with the polyhedron's features. It includes 4-fold rotations by 90°, 180°, and 270° around the principal axis passing through the centers of the two square bases, two 2-fold (180°) rotations around axes through the midpoints of opposite lateral edges, and two 2-fold (180°) rotations around axes passing through pairs of opposite vertices.^[13] These operations ensure that the equilateral triangular faces and square bases are mapped onto themselves or equivalent elements. The D_{4d} group acts transitively on the 8 vertices of the square antiprism, resulting in a single orbit under its operations, which underscores the uniformity of the polyhedron.^[6] In certain 2D projections, particularly orthogonal views along the principal axis, the symmetry elements project to those of the wallpaper group p4mm, reflecting the underlying 4-fold rotational invariance.^[14]

Dihedral Angles

The uniform square antiprism features two distinct types of dihedral angles due to its composition of triangular and square faces. The dihedral angle between two adjacent triangular faces (3-3) is given by \arccos\left(\frac{1 - 2\sqrt{2}}{3}\right) \approx 127.55160^\circ.^[6] The dihedral angle between a square face and an adjacent triangular face (4-3) is \arccos\left(\frac{\sqrt{3} - \sqrt{6}}{3}\right) \approx 103.83616^\circ.^[6] These angles can be derived by calculating the angle between the normal vectors to the planes of adjacent faces, obtained from the Cartesian coordinates of the polyhedron's vertices.^[16] Alternatively, spherical trigonometry on the unit sphere centered at a vertex can yield the same results by considering the great circle arcs corresponding to the face angles.^[16] Due to the D_{4d} symmetry group, all dihedral angles of each type are equal across the polyhedron.^[6] In comparison, the corresponding square prism has dihedral angles of exactly $90^\circ between all adjacent faces. The dihedral angles in the square antiprism contribute to its enhanced rigidity relative to the prism, as the triangular lateral faces provide additional bracing against deformation. Since both types of dihedral angles are less than $180^\circ, the uniform square antiprism is convex.^[2]

Prisms and Antiprisms

An antiprism is a polyhedron consisting of two parallel regular n-gonal bases rotated relative to each other by an angle of \pi / n radians (or 180°/n), connected by a band of 2n equilateral triangles that alternate in orientation.^[1] For the square antiprism, where n=4, the bases are squares rotated by 45° with respect to one another, resulting in eight triangular lateral faces.^[2] In comparison to a prism, which features aligned bases connected by n rectangular lateral faces, the antiprism's rotation produces exclusively triangular sides, creating a more compact and twisted structure.^[1] For the square antiprism to be uniform—all edges equal and all faces regular—the height between bases must satisfy h = a \cdot 2^{-1/4}, where a is the edge length, ensuring the lateral triangles are equilateral.^[1] The square antiprism belongs to an infinite family of uniform antiprisms indexed by n ≥ 3, including the triangular antiprism (a regular octahedron), pentagonal antiprism, and higher members.^[1] In Conway polyhedron notation, it is represented as A4, where "A" denotes an antiprism seed and 4 specifies the square bases.^[17]

Derived Polyhedra

A gyroelongated form is obtained by attaching two square pyramids to the square bases of the square antiprism, with each pyramid rotated by 45 degrees relative to its base; this construction yields the gyroelongated square bipyramid, also known as Johnson solid J17, consisting of 16 equilateral triangular faces and possessing D_{4d} symmetry.^[18] The snub square antiprism (Johnson solid J85) represents a snub derivation, where the square antiprism undergoes a chiral snubbing operation that introduces additional triangular faces while preserving the overall antiprismatic topology, resulting in 24 equilateral triangles and 2 squares.^[19] The uniform truncated square antiprism is obtained by truncating the vertices of the square antiprism, resulting in 2 regular octagons (from the bases), 8 regular hexagons (from the triangles), and 8 regular squares (from the vertices). The dual of the square antiprism is the tetragonal trapezohedron, an isohedral polyhedron with 8 identical kite-shaped faces; operations on this dual, such as rectification or truncation, generate related polyhedra like truncated trapezohedra, which correspond to duals of modified antiprisms in the trapezohedral family.^[20]

Topological Variants

Twisted Prism

The twisted prism represents a topologically equivalent but geometrically deformed variant of the square antiprism, featuring a continuous twist deformation that introduces infinitesimal flexibility while preserving the overall combinatorial structure. It consists of 10 faces (two squares and eight triangles), 16 edges, and 8 vertices, identical to the square antiprism.^[21] This structure serves as a classic example in the study of flexible polyhedra, where infinitesimal motions allow small deformations without changing edge lengths to first order.^[22] Construction of the twisted prism begins with a square prism, where the four lateral rectangular faces are each bisected by a diagonal to form eight triangular faces, establishing the 10-face topology. A uniform twist is then applied along the height of the prism, rotating the top square base relative to the bottom by a variable angle θ; as θ approaches 45°, the structure deforms into the uniform square antiprism configuration.^[23] This deformation enables an infinitesimal isometry, permitting small changes in the twist angle that preserve edge lengths to first order while altering dihedral angles, a property central to models of flexible polyhedra.^[22] Such flexibility distinguishes the general twisted form from the rigid uniform square antiprism, with the twist angle θ parameterizing a family of realizations sharing the same topology. In Conway's classification of polyhedra, the rectangular twisted prism exemplifies this family, maintaining the 10-face structure analogous to the square antiprism when bases are square.^[24]

Crossed Antiprism

The crossed square antiprism is a non-convex, self-intersecting polyhedron topologically identical to the convex square antiprism, sharing the same vertex arrangement of eight vertices. It is constructed by placing two parallel squares in a rotated orientation of 45 degrees relative to each other, with lateral edges connecting corresponding vertices in a manner that causes the edges to cross, resulting in a star polyhedron configuration. This crossing produces intersecting lateral faces, distinguishing it from the non-intersecting convex form. The faces comprise two regular squares and eight triangular faces that form intersecting "dents" due to the self-intersections, yielding a total of ten faces, sixteen edges, and eight vertices, consistent with the Euler characteristic of 2 for a spherical topology. The underlying topology is orientable and genus 0. This structure relates to the square retroprism, an isohedral variant where the triangular faces are crossed, maintaining isogonal symmetry but emphasizing face-transitive properties over vertex-transitivity. The crossed square antiprism, a non-uniform star polyhedron, was discussed in the context of crossed antiprisms in the enumeration by Coxeter et al..^[25]^[26]

Applications

Molecular Geometry

The square antiprismatic geometry is commonly observed in eight-coordinate metal complexes, where the ligands arrange around the central atom to form two nearly parallel squares rotated by 45° relative to each other, adopting D_{4d} point group symmetry. This arrangement is exemplified by the [XeF_8]^{2-} anion, in which the xenon(VI) center is surrounded by eight fluoride ligands in a distorted square antiprism, as determined from its crystal structure in the salt (NO)_2[XeF_8]. Similarly, the [Zr(ox)_4]^{4-} complex, where ox denotes oxalate, features a zirconium(IV) ion coordinated by eight oxygen atoms from four bidentate oxalate ligands in a square antiprismatic configuration. The cyclic allotrope S_8 of elemental sulfur adopts a crown-shaped structure with D_{4d} symmetry, which corresponds to the geometry of a square antiprism, positioning the eight sulfur atoms at the vertices of the polyhedron. This molecular form is the stable orthorhombic polymorph of sulfur under standard conditions. The D_{4d} symmetry of the square antiprism minimizes ligand-ligand repulsion energies compared to other eight-coordinate geometries, such as the cube or square prism, by maximizing interligand distances and providing a low-energy pathway for ligand arrangement around the central metal. This stability contributes to its prevalence in coordination compounds, including those of divalent first-row transition metals like Fe(II), where square-antiprismatic complexes exhibit a high-spin S=2 ground state due to small d-orbital splitting. Lanthanide ions also favor this geometry, as seen in Gd(DOTA)^-, where the gadolinium(III) center is nine-coordinated—with eight donors from the macrocyclic DOTA ligand (1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetate) and one water molecule—in a twisted square antiprismatic (TSAP) arrangement, enhancing the complex's rigidity and utility in MRI contrast agents.^[27] Capped square antiprismatic variants extend this geometry to nine-coordination by adding a ligand to one triangular face of the square antiprism, resulting in a distorted structure often approximated as C_{4v} symmetry. Examples include certain lanthanide and actinide complexes, such as those of praseodymium(III) with multidentate ligands, where the additional capping ligand occupies an axial position.^[28]

Architectural Uses

The One World Trade Center in New York City exemplifies the architectural application of a square antiprism form, where the building's square base tapers upward through chamfered edges into a twisted square top rotated by 45 degrees, creating an elongated square antiprism composed of eight isosceles triangular faces.^[29]^[30]^[31] This twisted configuration enhances both structural stability—through a diagrid system that distributes wind loads efficiently—and aesthetic appeal, providing a dynamic, sculptural profile that symbolizes resilience in high-rise design.^[32] Beyond skyscrapers, square antiprism geometries appear in pavilion designs. Geodesic approximations of antiprisms also feature in expo structures, deriving from prismatic or antiprismatic subdivisions to form expansive, efficient lattice shells.^[33] In contemporary parametric architecture, tools like Grasshopper within Rhino enable the generation of facades incorporating antiprismatic twists, allowing architects to optimize surface patterns for light diffusion and ventilation in modular building envelopes.^[34] Physical scale models of square antiprisms are realized through polyhedral architecture kits, such as straw-based constructions or HyperTiles systems, facilitating educational exploration of their geometric properties in design prototyping.^[35]^[36]

References

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