8octahedron

The 8octahedron is a polyhedron that combines properties of both octahedrons and cubes, forming a unique geometric shape within the broader category of Catalan solids. It is the dual of the cuboctahedron, meaning its vertices correspond to the faces of the cuboctahedron and vice versa. The 8octahedron is also known as the *disdyakis triakis tetrahedron* in some geometric classifications, though this term is less commonly used.

This polyhedron consists of 24 congruent isosceles triangles arranged symmetrically around its surface. Each vertex of the 8octahedron is shared by four triangles, and the shape exhibits a high degree of symmetry, belonging to the octahedral point group. Its structure can be visualized as two interlocking octahedrons, though it is not identical to a simple octahedron or a cube.

The 8octahedron is notable in the study of uniform polyhedrons, which are shapes with regular polygonal faces and identical vertex configurations. While it is not a Platonic solid—due to its non-regular faces—it serves as an important example in the exploration of more complex polyhedral forms. Its name, derived from its geometric construction, reflects its relationship to the octahedron, though it differs significantly in structure.

The 8octahedron appears in mathematical and architectural contexts, particularly in the analysis of space-filling tessellations and modular designs. Its properties make it useful in computational geometry and theoretical studies of polyhedral symmetry. Despite its complexity, the shape remains relatively obscure compared to more well-known polyhedrons like the dodecahedron or icosahedron.