Truncated cuboctahedron
The truncated cuboctahedron, also known as the great rhombicuboctahedron or rhombitruncated cuboctahedron, is an Archimedean solid consisting of 12 regular squares, 8 regular hexagons, and 6 regular octagons as its faces, meeting in the vertex configuration (4.6.8).[1][2][3] It features 48 vertices and 72 edges, with each vertex surrounded by one square, one hexagon, and one octagon in cyclic order.[1][2] This polyhedron exhibits full octahedral symmetry (Oh), the highest symmetry group for polyhedra derived from the cube or octahedron, with a symmetry order of 48.[1][2] It is one of the 13 Archimedean solids, characterized by regular polygonal faces of more than one type and identical vertex figures.[1][3] The truncated cuboctahedron can be constructed as the cantellation (rhombtruncation) of a cuboctahedron or as the Minkowski sum of three cubes oriented along perpendicular axes.[1] Geometrically, for an edge length of 1, its volume is $22 + 14\sqrt{2} \approx 41.799, its surface area is $12(2 + \sqrt{2} + \sqrt{3}) \approx 61.752, and its circumradius is \frac{\sqrt{13 + 6\sqrt{2}}}{2} \approx 2.318.[1][2] The dual polyhedron is the disdyakis dodecahedron, a Catalan solid with 48 triangular faces.[1][2] Notably, it is an equilateral zonohedron with Dehn invariant 0, meaning it can be dissected into polyhedra of equal volume using only translations and rotations.[1]Overview
Definition and construction
The truncated cuboctahedron is an Archimedean solid, which is a convex uniform polyhedron composed of regular polygons of more than one type meeting with identical configuration at each vertex.[4] It is a semiregular polyhedron characterized by the vertex configuration (4.6.8), where a square, hexagon, and octagon alternate around each vertex.[5] This arrangement ensures that all faces are regular polygons and all edges are of equal length, confirming its uniformity.[6] The polyhedron consists of 26 faces—12 regular squares, 8 regular hexagons, and 6 regular octagons—72 edges, and 48 vertices.[6] It can be constructed through the truncation of a cuboctahedron, a quasiregular Archimedean solid with 8 triangular and 6 square faces. In this process, the original triangular faces are replaced by regular hexagons (as each edge cut introduces three new sides to the truncated triangle), the square faces become regular octagons (with four new sides added), and new square faces emerge at the positions of the original vertices, where two triangles and two squares previously met. The truncation depth is set such that vertices are placed one-third along each edge from the original vertices, ensuring the resulting faces are regular.[7] Additionally, the truncated cuboctahedron arises in the Wythoff construction of uniform polyhedra, generated from the Wythoff symbol 2 3 4 | based on the octahedral reflection group.[8] It belongs to the cuboctahedral series of uniform polyhedra sharing the full octahedral symmetry of the cube and octahedron.[9]Basic properties
The truncated cuboctahedron is a uniform Archimedean solid, characterized by being vertex-transitive with regular polygonal faces of multiple types meeting in identical configuration at each vertex.[4] This uniformity ensures that the arrangement around every vertex consists of one square, one hexagon, and one octagon in cyclic order (vertex figure (4.6.8)).[10] The polyhedron exhibits the full octahedral symmetry group O_h of order 48, which acts transitively on the vertices and separately on each class of faces (squares, hexagons, and octagons).[10] It possesses 48 vertices, 72 edges of equal length, and 26 faces comprising 12 regular squares, 8 regular hexagons, and 6 regular octagons.[5] These counts satisfy the Euler characteristic \chi = V - E + F = 48 - 72 + 26 = 2, confirming that the truncated cuboctahedron is topologically equivalent to a sphere and has genus 0, as expected for a convex polyhedron. The Schläfli symbol for the truncated cuboctahedron is t_{0,3}\{4,3\} (or equivalently rr\{4,3\}), reflecting its construction as the rhombitruncation (omnitruncation) of the cube or its dual octahedron under octahedral symmetry.[1]Nomenclature and history
Names
The name "truncated cuboctahedron" was coined by Johannes Kepler in his 1619 treatise Harmonices Mundi, where he described the polyhedron as arising from truncating the cuboctahedron, one of the 13 Archimedean solids he systematically enumerated.[10] In Latin, Kepler referred to it as a form of truncation of the cuboctahedron. However, the designation is somewhat misleading, as a genuine truncation of the cuboctahedron—cutting off vertices until edges disappear—would yield rectangular faces in place of the observed squares.[11] Other designations include the great rhombicuboctahedron, which differentiates it from the related rhombicuboctahedron by incorporating "great" to reflect its expanded structure, and the rhombitruncated cuboctahedron, emphasizing the rhombi (squares) introduced via rhombification followed by truncation.[1]Historical context
The truncated cuboctahedron was first described by Johannes Kepler in his seminal 1619 work Harmonices Mundi, where he classified it among the semi-regular polyhedra alongside other uniform convex solids derived from truncations and expansions of Platonic forms.[12] Kepler's inclusion of this polyhedron stemmed from his broader exploration of geometric harmony in the universe, drawing on earlier traditions while extending the catalog of known regular-faced solids beyond the classical Platonic bodies. In the late 19th century, the polyhedron received further systematic attention through Edmund Hess's 1876 study on regular polyhedral compounds and divisions of the sphere, where it was formally cataloged as one of the 13 Archimedean solids, emphasizing its uniform vertex configuration and role in spherical tessellations.[13] Hess's work built on Kepler's foundations by providing rigorous enumerations and constructions, solidifying the truncated cuboctahedron's place in the canon of convex uniform polyhedra.[14] The 20th century saw continued recognition through physical modeling and enumeration efforts, notably in Magnus Wenninger's 1974 book Polyhedron Models, which featured detailed construction instructions for the truncated cuboctahedron among Archimedean forms, and in George W. Hart's digital explorations of polyhedral geometry. Modern cataloging traces to Norman W. Johnson's 1966 paper on convex polyhedra with regular faces, where the truncated cuboctahedron was assigned index U15 in the complete list of 75 uniform polyhedra, facilitating its integration into computational geometry and symmetry studies.[15]Geometry
Cartesian coordinates
The vertices of the truncated cuboctahedron, when centered at the origin and scaled to have edge length 2, are located at all permutations of the coordinates\left( \pm 1, \ \pm (1 + \sqrt{2}), \ \pm (1 + 2\sqrt{2}) \right).
This set generates the 48 distinct vertices required for the polyhedron.[10][16] These coordinates arise from the cantitruncation of a cube (or dual octahedron), where vertices are positioned by expanding the faces outward and truncating the original vertices and edges to introduce the square, hexagonal, and octagonal faces while maintaining uniformity. Alternatively, they can be derived by truncating a cuboctahedron—whose vertices are at all permutations of (\pm 1, \pm 1, 0)—such that the cut depth reduces the triangular and square faces to regular hexagons and octagons, respectively, with the scaling adjusted to equalize all edge lengths to 2. (from "Gems of Geometry" by Webb, referencing standard derivations in polyhedral geometry) For normalization to unit edge length, divide all coordinates by the scaling factor 2, yielding all permutations of
\left( \pm \frac{1}{2}, \ \pm \frac{1 + \sqrt{2}}{2}, \ \pm \frac{1 + 2\sqrt{2}}{2} \right). [10]