The Pseudorhombicuboctahedron is a captivating polyhedron, a geometric marvel resembling the rhombicuboctahedron but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the elongated square gyrobicupola (J37), is classified as a Johnson Solid because its faces are regular polygons, but it is not a uniform polyhedron or an Archimedean Solid due to its non-standard vertex configuration. Unlike uniform polyhedra, not all of its vertices are congruent; this means the arrangement of faces around each vertex is not identical throughout the solid, a property known as a lack of vertex transitivity. This distinction is crucial for its classification as a Johnson Solid, which permits regular polygonal faces but does not require uniform vertices or edges.
The image illustrates the construction of the pseudorhombicuboctahedron. From left to right: the elongated square gyrobicupola, its decomposition into two square cupolas and an octagonal prism, and the rhombicuboctahedron.
Structurally, the pseudorhombicuboctahedron, or elongated square gyrobicupola, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central octagonal prism and attaching two square cupolas to its opposite octagonal faces. Crucially, one of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation). This rotation directly breaks the higher symmetry of the standard rhombicuboctahedron, where both cupolas would be perfectly aligned, leading to a much more symmetrical structure. The 45-degree rotational displacement of one cupola relative to the other is the defining characteristic that alters the arrangement of its faces and vertices, making it unique and distinguishing it from its uniform counterpart. This "gyro" operation results in a noticeable twist that is central to its pseudo-uniform nature.
Like the rhombicuboctahedron, the pseudorhombicuboctahedron has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent. Specifically, it has two types of vertices: 16 vertices where three squares and one triangle meet (often denoted as (3.4.4.4)), and 8 vertices where two squares and two triangles meet (denoted as (3.4.3.4)). This fundamental lack of vertex transitivity is precisely why it falls outside the category of uniform polyhedra and Archimedean solids. Its symmetry group is D4d, representing dihedral symmetry of order 4. This implies a single 4-fold rotation axis passing through the centers of the two octagonal faces, along with perpendicular 2-fold rotation axes and dihedral mirror planes. This is a significantly lower level of symmetry compared to the full octahedral symmetry (Oh) of its uniform counterpart, further highlighting its distinct, lower symmetry arrangement.
This unique construction highlights the diversity within convex polyhedra and serves as an excellent example of how slight modifications, such as a rotational twist, can lead to entirely new geometric classifications. These "pseudo" forms challenge traditional definitions of regularity by demonstrating shapes that visually resemble their uniform counterparts but fundamentally differ in their underlying symmetry properties. The concept of "pseudo" polyhedra extends to other shapes as well, such as the Pseudo Great Rhombicuboctahedron, which similarly introduces a break in uniform symmetry within a related family of polyhedra, showcasing a broader principle in geometric classification.
