The hemi‑dodecahedron, also referred to as the projective dodecahedron, is a regular abstract polyhedron that can be realized as a tiling of the real projective plane. It is denoted by the Schläfli symbol ${5,2}$, indicating that each face is a regular pentagon ($5$-gon) and each vertex is incident with two faces.
Geometric definition
The hemi‑dodecahedron can be obtained by taking a conventional Euclidean dodecahedron and identifying each pair of antipodal (diametrically opposite) points, edges, and faces. This quotient operation maps the dodecahedron onto the projective plane, producing a non‑orientable surface in which each pentagonal face appears only half as many times as in the original solid.
Combinatorial structure
| Element | Quantity |
|---|---|
| Faces (pentagons) | 5 |
| Edges | 10 |
| Vertices | 6 |
Each vertex is the meeting point of two pentagonal faces, and each edge is shared by two faces, consistent with the ${5,2}$ configuration.
Symmetry
The symmetry group of the hemi‑dodecahedron corresponds to the projective special linear group $PSL(2,5)$, which is isomorphic to the alternating group $A_5$. This group acts transitively on the faces, edges, and vertices, confirming the polyhedron’s regularity.
Realization
While the hemi‑dodecahedron cannot be embedded as a conventional polyhedron in three‑dimensional Euclidean space without self‑intersection, it can be visualized through models of the projective plane, such as the Boy surface or via stereographic projection of a dodecahedron with antipodal identification.
Historical context
The concept of hemi‑polyhedra, including the hemi‑dodecahedron, arose in the study of regular maps on non‑orientable surfaces in the early 20th century. Notable contributions were made by mathematicians such as H. S. M. Coxeter and W. Magnus, who examined the classification of regular abstract polyhedra.
Mathematical significance
The hemi‑dodecahedron serves as an example of a regular map on a non‑orientable surface and illustrates the relationship between three‑dimensional polyhedral geometry and two‑dimensional projective geometry. It is frequently cited in discussions of abstract polytope theory, combinatorial topology, and the classification of regular maps.