File:Stellation 3 of dodecahedron and convex hull.svg

English: “Dodecahedron” means “polyhedron of twelve faces”. The faces of a regular dodecahedron are twelve regular polygons of same size, which have five edges: convex pentagons or pentagonal stars. Two regular dodecahedra have convex faces: a convex polyhedron also called “Platonic dodecahedron”, as well as a “great dodecahedron”, which is not convex. A “great stellated dodecahedron” is a nonconvex regular dodecahedron of twenty vertices, the faces of which are pentagonal stars. Its vertices are those of its convex hull: a Platonic dodecahedron. On any image of the series, such a dodecahedron hides one of its vertices at the drawing center. The transparent thick lines on certain images reveal the edges of the convex hull. When the image shows a Platonic dodecahedron inside a great stellated dodecahedron, a same colour at the surface of two polyhedra indicates two faces in a same plane. In this case, any edge of a face of a given colour is parallel to one of the five edges of the other face of the same colour.

Any regular dodecahedron has thirty edges. A great stellated dodecahedron can be constructed from a Platonic dodecahedron, by considering thirty diagonals of the convex polyhedron as the thirty edges of a new regular dodecahedron. Another means is a polyhedral stellation of Platonic dodecahedron: each face of the convex dodecahedron is extended, and becomes a face of the new regular dodecahedron, in the same plane as the original face. The twelve polygonal stellations of faces of a Platonic dodecahedron have twelve vertices. These twelve points are the vertices of the first two polyhedral stellations of the dodecahedron: a “small stellated dodecahedron” and a great dodecahedron. Sharing all their vertices, these two polyhedral stellations have a common convex hull: a convex regular “icosahedron”, i.e. a convex regular polyhedron of twenty faces, also called “Platonic icosahedron”. The faces of a great dodecahedron are intertwined convex regular pentagons, of which the twelve polygonal stellations are the faces of a great stellated dodecahedron, the topic of these images: the third polyhedral stellation of Platonic dodecahedron.

The symmetry center of any drawing of the series represents several remarkable points of the three-dimensional space: two opposite vertices, and the center of one or two dodecahedra of the image. The present image represents three concentric regular dodecahedra, and each dodecahedron is invariant under the symmetry with respect to the common center of the polyhedra. The projection of each dodecahedron onto the drawing plane distorts all its faces, but it does not shrink certain edges. For example, the outline of the convex hull of the great stellated dodecahedron consists of twelve segments, and six of the twelve segments are not shrinked. They have the same length. Other example, there is an equilateral triangle at the center of the image, of which each edge is parallel to a segment of the outline of the convex hull, which is not shrinked. The projection does not distort this equilateral triangle, of which each edge is a part of an edge of the great stellated dodecahedron, that the projection does not shrink.