Octagonal prism

An octagonal prism is a three-dimensional polyhedron consisting of two parallel regular octagonal bases connected by eight rectangular lateral faces.^[1] It possesses 10 faces (two octagons and eight rectangles), 24 edges, and 16 vertices.^[2]^[1] Octagonal prisms can be classified as right prisms, with lateral edges perpendicular to the bases, or oblique prisms, where the lateral edges are slanted.^[1] In real-world applications, octagonal prisms appear in architectural elements such as decorative columns and pillars for structural stability, as well as in everyday objects like birdhouses and certain tin boxes.^[3]^[4]^[1]

Definition and Basic Geometry

Definition

An octagonal prism is a polyhedron consisting of two parallel regular octagonal bases connected by eight parallelogram lateral faces.^[5] In general, a prism is formed by translating (or extruding) a polygonal base along a straight line perpendicular or oblique to the plane of the base, yielding two congruent bases and parallelogram-shaped lateral faces.^[6] When the lateral edges are perpendicular to the bases, the figure is a right octagonal prism, with rectangular lateral faces; in contrast, an oblique octagonal prism has lateral edges slanted relative to the bases, resulting in parallelogram lateral faces.^[7] The concept of prisms originated in ancient Greek geometry, where Euclid defined them in his Elements (Book XI, Definition 13) as solid figures bounded by two equal, similar, and parallel opposite planes with the remaining faces as parallelograms.^[6] Prisms, including the octagonal variant, are part of the infinite families of uniform polyhedra, which also include the finite set of 13 Archimedean solids; they feature regular polygonal bases and equivalent vertices.^[8]

Components

An octagonal prism is composed of two parallel regular octagonal bases and eight parallelogram lateral faces that connect the corresponding sides of the bases, resulting in a total of 10 faces.^[5] The bases are congruent regular octagons, each with eight equal sides and angles, while the lateral faces are parallelograms whose dimensions depend on the side length of the octagon and the height of the prism.^[5] The prism has 24 edges in total: 16 edges from the two octagonal bases (8 per base) and 8 lateral edges that connect the vertices of the bases.^[5] It also features 16 vertices, with 8 vertices on each base corresponding to the corners of the regular octagons.^[5] As a convex polyhedron, the octagonal prism satisfies Euler's formula, where the Euler characteristic \chi = V - E + F = 16 - 24 + 10 = 2.^[9] In a right octagonal prism, where the lateral edges are perpendicular to the bases, the dihedral angle between a base and an adjacent lateral face is 90 degrees.^[7] The dihedral angle between two adjacent lateral faces is 135 degrees, determined by the geometry of the regular octagonal base.^[10]

Dimensions and Measures

Vertex Count and Coordinates

An octagonal prism possesses 16 vertices in total, consisting of 8 vertices on each of the two parallel octagonal bases.^[1] For a right regular octagonal prism centered at the origin, with bases parallel to the xy-plane, unit side length on the bases (a = 1), and height h, the Cartesian coordinates of the vertices are given by two sets of 8 points each. The top base lies at z = h/2, and the bottom base at z = -h/2. The x and y coordinates for the bases are

\left( R \cos\left(\frac{2\pi k}{8}\right), R \sin\left(\frac{2\pi k}{8}\right) \right), \quad k = 0, 1, \dots, 7,

where the circumradius R of the regular octagon is

R = \frac{1}{2} \sqrt{4 + 2\sqrt{2}}.

^[11] This positioning ensures that the 8 edges on each base have length 1, while the 8 lateral edges connecting corresponding vertices between the bases have length h.^[12]

Surface Area and Volume

The surface area and volume of an octagonal prism are calculated assuming a convex right prism with regular octagonal bases, where s denotes the side length of the base and h the height of the prism.^[1]^[13] The area of one regular octagonal base is derived from the general formula for regular polygons, yielding A_b = 2(1 + \sqrt{2})s^2.^[14] This formula arises by dividing the octagon into eight isosceles triangles from the center, each with two sides equal to the apothem and base s, and summing their areas using trigonometric relations, simplified to the closed form above.^[14] The total surface area consists of the areas of the two bases plus the lateral surface area. The lateral surface area is the product of the base perimeter and height; the perimeter of the regular octagon is P = 8s, so the lateral area is $8sh.^[13] Thus, the total surface area is:

SA = 2A_b + 8sh = 2 \left[2(1 + \sqrt{2})s^2\right] + 8sh = 4(1 + \sqrt{2})s^2 + 8sh.

^[1]^[14] The volume is the product of the base area and height:

V = A_b \cdot h = 2(1 + \sqrt{2})s^2 h.

This follows directly from the Cavalieri's principle for prisms, where cross-sections parallel to the base are identical.^[13]^[1]

Symmetry and Variants

Symmetry Group

The symmetry group of a regular octagonal prism is the dihedral prismatic group D_{8h}, which has an order of 32 and describes all isometries that map the prism onto itself.^[15] This group arises from combining the D_8 symmetries of the regular octagonal bases with additional operations along the prism's height, including a horizontal reflection plane./02%3A_Symmetry_and_Group_Theory/2.02%3A_Point_Groups) The group comprises 16 orientation-preserving isometries, consisting solely of rotations, and 16 orientation-reversing isometries, which include reflections and roto-reflections.^[16] The rotational symmetries feature an 8-fold rotation axis aligned with the prism's vertical direction, allowing rotations by multiples of $45^\circ, along with eight 2-fold rotation axes perpendicular to this principal axis and passing through the midpoints of opposite lateral faces or edges./02%3A_Symmetry_and_Group_Theory/2.02%3A_Point_Groups) Reflection symmetries are realized through nine mirror planes: one horizontal plane passing through the midpoint of the prism's height, perpendicular to the principal axis; four vertical planes containing the principal axis and passing through opposite vertices of the bases; and four additional vertical planes containing the principal axis and bisecting opposite edges of the bases.^[16] These elements collectively ensure the full D_{8h} symmetry, preserving the regularity of the octagonal bases and the uniformity of the rectangular lateral faces.^[15]

Uniform and Non-Uniform Variants

The uniform octagonal prism is a prismatic uniform polyhedron composed of two parallel regular octagonal bases connected by eight square lateral faces, achieved when the height equals the side length of the octagons to ensure all faces are regular polygons.^[10] This configuration makes it vertex-transitive, with each vertex incident to one octagon and two squares.^[5] The symmetry group of the uniform octagonal prism is the dihedral group D_{8h}, of order 32.^[17] Although featuring regular faces of multiple types, the uniform octagonal prism is not an Archimedean solid, as Archimedean solids comprise the thirteen specific convex uniform polyhedra excluding infinite families of prisms and antiprisms; prisms like this are instead classified as uniform within the prismatic category.^[18] The dual polyhedron of the uniform octagonal prism is the octagonal dipyramid (also known as an octagonal tegum), which consists of sixteen equilateral triangular faces corresponding to the original's vertices.^[17]^[19] Non-uniform variants of the octagonal prism include oblique forms, in which the lateral edges are not perpendicular to the bases, resulting in parallelogram-shaped lateral faces rather than rectangles or squares.^[20] Right octagonal prisms with regular bases and rectangular lateral faces (where height differs from the base side length) are also non-uniform, as the rectangles are not regular polygons, though they retain the full D_{8h} symmetry. Another non-uniform type features irregular octagonal bases, where the bases deviate from regularity while remaining congruent and parallel, or lateral faces that are non-rectangular parallelograms due to the obliqueness of the lateral edges.^[21] These variants preserve the prismatic structure but lack the regularity and vertex-transitivity of the uniform case.^[7]

Applications and Uses

Practical Uses

Octagonal prism shapes have been employed in architecture for their structural integrity and aesthetic appeal, particularly in towers and lanterns that require balanced load distribution and enhanced light penetration. For instance, in Gothic architecture, the octagonal lantern tower of Ely Cathedral, constructed in the 14th century, exemplifies this use, where the eight-sided form provides exceptional stability while allowing natural illumination through its design.^[22]^[23] For example, Coutances Cathedral in Normandy features an octagonal lantern tower, a typical element in Norman Gothic architecture that provides stability and allows for light penetration.^[24] In optical engineering, octagonal prisms serve as light guides and components in laser systems due to their uniform cross-section, which facilitates total internal reflection to homogenize non-uniform light sources. These prisms are manufactured with precision polishing on all faces to minimize light loss, making them suitable for applications in beam steering and illumination devices.^[25]^[26] Additionally, octagonal prism-based flexible mechanisms are integrated into high-bandwidth tilt-tip mirrors for adaptive optics, leveraging the shape's symmetry for precise angular control and reduced mechanical stress.^[27] Modern applications include 3D printing, where octagonal prism models are fabricated.^[28] The octagonal symmetry in kaleidoscope patterns is produced by mirrors angled at 45 degrees, as in the original 19th-century design by Sir David Brewster, enhancing visual multiplicity.^[29] The high symmetry of octagonal prisms offers advantages in balanced weight distribution, providing greater structural stability compared to square or rectangular forms by enclosing space more efficiently—approximately 20% better than a square of equal perimeter—while distributing loads evenly across eight sides.^[30]^[31] This property is particularly beneficial in engineering contexts like scaffolding, where the octagonal base enhances resistance to tipping under uneven loads.^[32]

In Polyhedral Compounds and Tiling

The octagonal prism appears in certain uniform polyhedral compounds, notably as a component in the stellated octagonal prism, which is equivalent to the compound of two cubes. This compound interlocks two cubes such that their vertices align to form a structure with two stellated octagons as bases and eight squares as lateral faces, maintaining uniform edge lengths throughout.^[33] In tiling applications, the octagonal prism integrates into three-dimensional space-filling arrangements derived from two-dimensional tilings featuring octagons and squares, such as the truncated square tiling where one square and two regular octagons meet at each vertex. Extruding this tiling along the third dimension produces the truncated square prismatic honeycomb, a uniform Euclidean honeycomb composed of infinite cubes and octagonal prisms in a 1:2 ratio, with two cubes and four octagonal prisms meeting at each vertex to achieve complete space-filling without gaps or overlaps.^[34] This construction ensures a packing density of 1 in Euclidean space, leveraging the compatibility of the octagonal prism's uniform variant with square-based prisms.^[10] Such prismatic tilings exemplify how octagonal prisms contribute to layered assemblies, where cross-sections replicate the octagon-square pattern of the base tiling, enabling scalable finite or infinite extensions in practical geometric designs.^[35]

Advanced Contexts

In Uniform Honeycombs

Octagonal prisms appear as cells in several uniform honeycombs that tile Euclidean three-dimensional space. A key example is the truncated square prismatic honeycomb, denoted as tassiph, which features regular cubes and uniform octagonal prisms as its cells. In this structure, two cubes and four octagonal prisms meet at each vertex, following the vertex configuration 4.4.8.8.8.8, where the squares arise from the cubes and the octagons from the prisms.^[34]^[36] This honeycomb arises from the prismation of the truncated square tiling (4.8.8), or equivalently the Cartesian product of the truncated square tiling and an apeirogon, integrating prismatic elements over a semi-regular base tiling. Its Wythoff symbol is given by [4,4]:(011)^[37] or [4,4]:111^[37], reflecting its construction within the Coxeter group framework. The arrangement ensures complete space-filling without overlaps or gaps, as the rectangular lateral faces of the prisms align to form a uniform tessellation, combining cubic and octagonal prismatic components in a manner analogous to other prismatic honeycombs derived from uniform plane tilings.^[38]^[34] Other uniform honeycombs incorporating octagonal prisms include the great prismated cubic honeycomb (gippich), where two great rhombicuboctahedra and two octagonal prisms join at each vertex, and the prismatorhombated cubic honeycomb, which mixes small rhombicuboctahedra, truncated cubes, cubes, and octagonal prisms. These structures highlight how octagonal prisms integrate with Archimedean solids in more complex uniform tilings.^[39]^[40] The enumeration and classification of such prismatic honeycombs build on H.S.M. Coxeter's extensions of uniform polyhedra to infinite tessellations, as detailed in his works from the 1930s onward, including the identification of 28 convex uniform honeycombs in Euclidean space. Coxeter's Wythoff construction method systematically generates these from reflection groups, emphasizing their regularity and symmetry.^[38]

In 4-Polytopes and Higher Dimensions

In four-dimensional geometry, the octagonal prism appears as a cell in uniform prismatic 4-polytopes constructed via the Cartesian product of lower-dimensional uniform polytopes and a line segment. The octagonal prismatic 4-polytope, specifically the product of a uniform octagonal prism and a line segment, is a convex uniform 4-polytope whose cells consist of two octagonal prisms and eight cubes.^[41] This structure maintains vertex-transitivity, with the octagonal prisms serving as cells corresponding to the bases of the base prism.^[41] The octagonal prism also features prominently in duoprisms, another class of uniform 4-polytopes formed by the Cartesian product of two regular octagons. The resulting octagonal duoprism contains 16 octagonal prism cells, arranged such that four meet at each vertex, and exhibits the product symmetry of the dihedral group D_8 \times D_8. Its Coxeter notation is

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, reflecting the disconnected diagram of two separate octagonal branches.^[41] In higher dimensions, the octagonal prism represents a 3-dimensional instance of generalized prismatic polytopes, where an n-gonal prism in d dimensions arises recursively as the product of an n-gon and a (d-2)-dimensional prism, or more broadly as the Cartesian product of an (d-1)-polytope and a line segment. These higher-dimensional analogs preserve uniformity when the base components are uniform, extending the prismatic construction to embed octagonal prism-like cells within n-polytopes for n > 4. For instance, in 5 dimensions, an octagonal prismatic 5-polytope would include multiple octagonal prism cells alongside higher-dimensional cubic and prismatic elements derived from the base.^[41] Uniformity in 4D contexts further manifests in operations like rectification and truncation of regular 4-polytopes, where octagonal prisms emerge as cells in rectified or runcinated forms involving hypercubic symmetry. The Coxeter notation for such prismatic 4-polytopes generally features a disconnected diagram, combining the branch for the octagonal prism's symmetry group [8,2] with an isolated node for the prismatic direction, ensuring the overall vertex-transitive structure.^[41]

Prismatic Relatives

The octagonal prism belongs to the broader family of n-gonal prisms, where the base is an n-sided regular polygon and the lateral faces are rectangles connecting two parallel bases.^[7] For any such prism, the total number of faces is n + 2 (two n-gonal bases and n rectangular sides), the number of edges is 3n (n per base plus n vertical edges), and the number of vertices is 2n (n per base).^[7] These formulas highlight the scalable structure of prisms as n varies, with the octagonal case corresponding to n = 8, yielding 10 faces, 24 edges, and 16 vertices.^[7] Among specific relatives, the hexagonal prism (n = 6) is particularly notable for its occurrence in natural formations, such as the columnar jointing in basaltic lava flows, where cooling contraction produces elongated hexagonal prisms, as seen at sites like Devils Postpile National Monument.^[42] In contrast, the square prism (n = 4), also known as a cuboid or rectangular box, serves as a foundational example in geometry and everyday applications, differing from the octagonal prism primarily in its simpler base and resulting lower face count of 6.^[7] As n increases from the triangular prism (n = 3) to higher members like the dodecagonal prism (n = 12), the number of faces grows linearly while the prismatic uniformity—characterized by regular bases and rectangular sides—remains consistent across the family.^[7] This progression illustrates the evolutionary nature of prisms within polyhedral families, where complexity arises from base sidedness rather than altering the extrusion principle.^[7] Volume scales with the base area and height, expanding as n grows due to larger polygonal bases.^[7]

Antiprisms and Other Extrusions

An antiprism is a polyhedron formed by two parallel regular polygonal bases that are rotated relative to each other, connected by a continuous band of equilateral triangles, in contrast to a prism where the bases are aligned and connected by rectangles. For an n-gonal base, the uniform antiprism rotates the bases by 180°/n and features 2n triangular lateral faces.^[43] This twisted configuration distinguishes antiprisms from straight extrusions like prisms, providing a more compact structure with alternating upward- and downward-pointing triangles along the equator.^[44] The octagonal antiprism (n=8) specifically consists of two regular octagons rotated by 22.5° relative to each other, joined by 16 equilateral triangles, resulting in 18 faces, 32 edges, and 16 vertices.^[43] It is a uniform polyhedron (Uniform #77 in some enumerations) with dihedral antiprismatic symmetry of order 32, denoted D_{8d}, which includes reflections and rotary inversions but lacks the horizontal mirror plane present in the octagonal prism's D_{8h} symmetry.^[45] The height of a uniform octagonal antiprism with side length a is given by h_8 = a \sqrt{ \sqrt{5 + \frac{7}{2\sqrt{2}}} - 1 - \sqrt{2} }, ensuring all faces are regular.^[43] Its surface area is S_8 = 4(1 + \sqrt{2} + \sqrt{3}) a^2, reflecting the areas of the two octagons and 16 triangles.^[43] Beyond antiprisms, other extrusions of the octagon include oblique prisms, where the lateral edges are not perpendicular to the bases but slanted at an angle, producing parallelogram lateral faces while preserving parallelism between bases.^[46] These non-right variants maintain the octagonal bases but alter the overall geometry for applications requiring shear, such as in architectural modeling, though they are generally non-uniform unless the slant aligns to form regular faces.^[18] Trapezoidal or irregular extrusions further generalize this by allowing non-parallel lateral edges, forming prismatoids that encompass prisms and antiprisms as special cases.