The hemi-icosahedron is the regular map on the real projective plane obtained by treating opposite points on the boundary
of a regular icosahedron as a single point.
Its 6 vertices, 15 edges,
and 10 triangular faces
are the opposite pairs of vertices, edges,
and faces of the icosahedron. Its type is 
, where the subscript is the length of a Petrie
polygon (Leemans and Schulte 2007).
The schematic disk in the figure contains five triangles meeting at the central vertex and five more attached along the outer edges of those triangles. Identify opposite boundary edges with the same letter by matching their arrow directions. The ten triangles then tile the real projective plane.
The skeleton is the complete graph 
(McMullen and Schulte 2002, p. 137). Each vertex
of the regular icosahedron is joined to one
vertex in each of the five other opposite pairs.
The dual map is the hemi-dodecahedron.
The hemi-icosahedron is the shape of each facet of the
11-cell (Leemans and Schulte 2007).
Acting on Polytopes." Adv. Geom. 7,
529-539, 2007.