of the polyhedra images. {| class="wikitable" width=400 |- align=center ! Polyhedron |Medial rhombic triacontahedron |Dodecadodecahedron |Medial triambic icosahedron |Ditrigonal dodecadodecahedron |Excavated dodecahedron |- align=center !Type ||Dual {5,4}6 ||{5,4}6 ||Dual of {5,6}4 ||{5,6}4 || {6,6}6 |- align=center !(v,e,f) |(24,60,30) ||(30,60,24) ||(24,60,20) ||(20,60,24) ||(20,60,20) |- align=center !Vertex figure |{5}, {5/2} |(5.5/2)2 |{5}, {5/2} |(5.5/3)3 | |- align=center valign=top !Faces |30 rhombi |12 pentagons12 pentagrams |20 hexagons |12 pentagons12 pentagrams |20 hexagrams |- align=center ! Tiling |{4, 5} |{5, 4} |{6, 5} |{5, 6} |{6, 6} |- align=center ! χ | −6 | −6 | −16 | −16 | −20 |} Petrie dual The Petrie dual of a regular polyhedron is a regular map whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of skew Petrie polygons. Spherical polyhedra The usual five regular polyhedra can also be represented as spherical tilings (tilings of the sphere): Regular polyhedra that can only exist as spherical polyhedra For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces may be found by: The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3. The restriction m ≥ 3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the regular polyhedron {2, n} is represented as n abutting lunes, with interior angles of 2/n. All these lunes share two common vertices. A regular dihedron, {n, 2} (2-hedron) in three-dimensional Euclidean space can be considered a degenerate prism consisting of two (planar) n-sided polygons connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two line segments. However, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. It is regular if the vertices are equally spaced. The hosohedron {2,n} is dual to the dihedron {n,2}. Note that when n = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic