Hexagonisia

Hexagonisia is a hypothetical geometric concept representing a three-dimensional solid characterized by its faces being exclusively regular hexagons. Unlike Platonic solids, which have a fixed number of faces and edges meeting at each vertex, hexagonisia could theoretically exist in various forms depending on the arrangement and number of hexagonal faces.

In theoretical geometry, the existence of a convex polyhedron composed solely of regular hexagons presents a challenge. This is primarily due to the interior angle of a regular hexagon, which is 120 degrees. For a polyhedron to form a closed surface, the sum of the angles around any vertex must be less than 360 degrees. If three or more hexagons meet at a vertex, the sum of their angles would be 360 degrees (3 x 120 = 360) or more, preventing them from forming a convex vertex.

However, the concept of hexagonisia might be explored in non-Euclidean geometries or as a basis for tessellations in higher dimensions. It could also be a conceptual stepping stone in understanding the properties of polyhedra with regular faces, even if a simple convex realization in three dimensions is not possible under standard Euclidean geometry. The term itself is not widely established in mainstream geometric literature, suggesting it is likely a less common or perhaps a more specialized theoretical construct.