Related Polyhedral Figures
Frustums
A frustum is the portion of a pyramid between two parallel planes cutting all lateral faces, featuring two non-congruent polygonal bases. It is formed by truncating a pyramid parallel to its base, thereby removing a smaller pyramid cap.[41]
The lateral faces of a frustum consist of trapezoids connecting the two bases. The volume VVV of a frustum is calculated using the formula
where hhh is the perpendicular height between the bases, and B1B_1B1 and B2B_2B2 are the areas of the two polygonal bases.[42]
A representative example is the frustum of a square pyramid, characterized by trapezoidal lateral faces with non-parallel slanting edges. These shapes find practical application in engineering, such as in the construction of buckets and hoppers for material handling.[43][44]
In distinction from full prisms, the lateral edges of a frustum are non-parallel and converge toward one another.[45]
Truncated Prisms
A truncated prism is a uniform polyhedron obtained by performing a uniform truncation on a uniform prism, where the vertices are cut off until the original edges are reduced to points, resulting in all edges being of equal length and all faces being regular polygons. This process preserves the convexity and uniformity of the original prism while introducing new faces at the truncated vertices.[15][46]
In the structure of a truncated n-prism, the two original n-gonal bases are transformed into regular 2n-gons, as each original edge of the base is replaced by two new edges separated by the truncation cuts. The n original quadrilateral lateral faces (squares in the uniform case) become regular octagons, with each of the four original edges replaced by two, and new edges added from the truncations at the four vertices. Additionally, each of the 2n original vertices, which are 3-valent, is replaced by a new regular triangular face. Thus, the truncated n-prism has 2 regular 2n-gons, n regular octagons, and 2n regular triangles as its faces. The number of vertices is 6n, and the number of edges is 9n, satisfying Euler's formula for polyhedra.[47][48]
Representative examples include the cases for small n. The truncated triangular prism (n=3) consists of two regular hexagons, three regular octagons, and six regular triangles; it has 18 vertices and 27 edges. The truncated square prism (n=4) features two regular octagons from the bases, four regular octagons from the lateral faces (indistinguishable from the base-derived ones in type), and eight regular triangles. The truncated pentagonal prism (n=5) has two regular decagons, five regular octagons, and ten regular triangles. These examples illustrate the progression in complexity while maintaining uniform edge lengths.[47]
The symmetry of a truncated n-prism is the same as that of the original uniform prism, belonging to the dihedral group D_{nh}, which includes rotations around the principal axis by multiples of 360°/n, reflections through planes containing the axis, and horizontal reflections perpendicular to the axis. This prismatic symmetry ensures vertex-transitivity and the regular nature of all faces.[16]
Truncated prisms for n=3,4,5 are among the convex uniform polyhedra with prismatic symmetry, extending the class of Archimedean solids in broader enumerations of semi-regular polyhedra beyond the 13 with full tetrahedral, octahedral, or icosahedral symmetry.[46][49]
Star and Crossed Prisms
A star prism is a type of non-convex uniform polyhedron formed by two parallel regular star polygon bases connected by rectangular lateral faces, typically squares in the uniform case, where the bases have a density greater than 1.[50] These bases are star polygons, denoted in Schläfli symbol as {n/k} with k > 1, such as the pentagram {5/2}, leading to indentations and a star-like appearance in the overall structure.[51] Unlike convex prisms with simple polygonal bases, star prisms are vertex-transitive but exhibit reduced symmetry, belonging to the dihedral group D_{nh}, and their non-convexity arises from the intersecting edges of the base polygons without the full polyhedron surface self-intersecting in the standard construction.[52]
Crossed prisms represent a further non-convex variant where the two star polygon bases are rotated relative to each other—often by an angle like 36° for pentagonal cases—causing the lateral faces to intersect and producing self-intersecting polyhedra with higher density.[53] In the pentagrammic crossed prism, for instance, two {5/2} pentagrams are linked by triangular lateral faces, resulting in a density of 2 and a structure where vertices lie on a hyperboloid surface, enabling the crossing while maintaining geometric regularity.[54] These figures may form compounds or exhibit compound-like symmetries, with properties including self-intersection of faces and potential embedding on ruled surfaces, distinguishing them from non-crossed star prisms by their intersecting lateral elements.[53]
Examples of star prisms include the pentagrammic prism (uniform polyhedron U78), with 10 vertices, 15 edges, and 7 faces (two pentagrams and five squares), and the stellated triangular prism, a variant using a stellated base configuration to achieve non-convexity through extended facets.[55] Crossed variants, such as the octagrammic crossed prism, demonstrate how rotation introduces hyperbolic geometry influences, with vertices on equilateral hyperboloids.[53] Some star prisms relate to the Kepler–Poinsot regular star polyhedra, like the great stellated dodecahedron {5/2, 3}, through shared use of star polygons and uniform vertex figures, though prisms maintain prismatic symmetry rather than full icosahedral.[51]
Higher-Dimensional Prisms
In higher dimensions, an n-prism is defined as the Cartesian product of an (n-1)-dimensional polytope with a line segment, resulting in an n-dimensional polytope where the two parallel (n-1)-dimensional facets correspond to the copies of the base polytope at the endpoints of the segment.[56] This construction generalizes the familiar 3D prism, with the lateral (n-1)-facets being the products of the (n-2)-facets of the base with the line segment.[56]
Uniform n-prisms extend this idea by requiring the two parallel (n-1)-cells to be regular polytopes and the connecting prism cells to be cubic in their lateral structure, ensuring vertex-transitivity and equal edge lengths throughout.[57] Notable examples include the 4-dimensional tesseract, which can be constructed as a prism over a square base extruded through two dimensions (equivalent to the product of a square and two line segments), featuring eight cubic cells.[58] Similarly, the 5-dimensional penteract (or 5-cube) arises as a prism over a cubic base, with ten tesseractic 4-cells.[59]
The n-dimensional volume of an n-prism is given by the product of the (n-1)-dimensional volume of the base polytope and the length hhh of the line segment, V=Vn−1×hV = V_{n-1} \times hV=Vn−1×h.[56] The symmetry group of an n-prism is the direct product of the symmetry group of the base polytope and the dihedral group D2hD_{2h}D2h acting on the extrusion direction, allowing reflections and rotations that preserve the parallel bases.[60]
Infinite families of uniform prismatic polytopes exist in higher dimensions, including prismatic honeycombs that tile Euclidean n-space, such as the product of an (n-1)-dimensional cubic honeycomb with a line segment, forming space-filling tessellations with uniform cells.[61]