The Rhombicuboctahedron

The Rhombicuboctahedron, sometimes called the small rhombicuboctahedron, is a complex and visually striking Archimedean solid. It can be imagined as derived from either a cube or an octahedron through a process called expansion, where faces are moved outwards and new faces are created in the gaps at the original edges and vertices.

It features a total of 26 faces: 8 equilateral triangles and 18 squares (6 oriented like the faces of a cube, and 12 aligned with the cube's edges). It has 24 identical vertices, each meeting one triangle and three squares, and 48 identical edges.

Expansion and Complexity

Metaphysically, the Rhombicuboctahedron represents complexity arising from simple principles through expansion or outward expression. Starting with the basic cube (Earth/stability) or octahedron (Air/intellect), the expansion process reveals a more intricate underlying structure. The resulting solid, with its numerous square faces providing stability and triangular faces adding dynamism, symbolizes the multi-faceted nature of reality.

It suggests the unfolding of potential, the revelation of hidden complexity within simple forms, and the integration of different types of structure (squares) and energy (triangles). Meditating on this shape could aid in understanding complex systems, appreciating different facets of a situation, or expanding one's perspective to encompass greater detail and interconnection.

Order Within Complexity

Despite its apparent complexity with 26 faces, the Rhombicuboctahedron maintains a high degree of symmetry and order. All vertices and edges are identical. The arrangement of faces follows a precise pattern, demonstrating how intricate structures can still possess underlying harmony and balance.

It can symbolize the ability to find order within chaos, to manage complex situations through underlying principles, or the beauty inherent in intricate systems. It represents a sophisticated balance, integrating multiple stable square platforms with dynamic triangular connectors, suggesting a robust and adaptable structure capable of handling diverse inputs.

Geometric Notes

The dual of the Rhombicuboctahedron is the Deltoidal Icositetrahedron. While not space-filling on its own, it appears in various complex structures and tessellations when combined with other shapes. Leonardo da Vinci famously illustrated this polyhedron for Luca Pacioli's book "De Divina Proportione".