English: Up to similarity, ie considering all objects similar to a given object as one only object, there are five regular polyhedra which are convex.  They are called the five Platonic solids.  The face centers of such a regular polyhedron are the vertices of a second Platonic solid.  And then, the face centers again of the second solid are the vertices of a third polyhedron, similar to the initial one.  Thus, each of the first two solids is called the dual of the other, with the same number of edges.  One of them has as many faces as the other has vertices.  One of the five Platonic solids is itself its dual:  the regular tetrahedron, with six edges, four faces, four vertices.

On this page, a Platonic dodecahedron is seen from above:  a convex regular polyhedron with thirty edges, twelve faces and twenty vertices.  The thirty straight lines that extend all its edges intersect outside the dodecahedron at twelve points, which are the twelve vertices of a Platonic icosahedron:  a concentric specimen of the dual of the dodecahedron.  Each of the twelve points, like  T,  is the intersection of six lines:  five of the thirty lines and the axis of two opposite faces of the dodecahedron.  For example, point N  is a vertex of the dodecahedron, and the line passing through  N  and  T  contains a slanted edge, depicted by a dotted segment.

Each Platonic solid is symmetric with respect to point Ω,  its center shared with its dual.  Each solid is also transformed into itself under certain rotations, around one of its sixteen axis.  Each axis passes through center  Ω,  two opposite vertices of one solid, and the centers of two opposite faces of the other solid:  its dual.  For example, the slanted line that passes through  T  and  Q  is an axis of two opposite faces of the dodecahedron, like the vertical passing through  Ω.  An elevation view also surrounds with a half-disk in orange each of points  K,  L,  M  and  Q,  to inform of the coplanarity of these points in a horizontal plane.

This top view is a projection of the three-dimensional figure.  Under an orthogonal projection onto a horizontal plane, the vertical axis of the polyhedra is represented as any of its points, for example Ω  has the same image as two of the twelve vertices of the icosahedron.  Considering these twelve vertices and the thirty lines that contain the edges of the dodecahedron, each line passes through two vertices, for example  L  and  H,  or  H  and  Q,  or  Q  and the higher vertex, lying on the vertical axis.

The stellated pentagon with vertices depicted by black crosses is built by extending the five upper edges of the dodecahedron.  In other words, this regular star is a stellation of the upper face of the dodecahedron.  The thirty edges of the icosahedron are also the edges of a great dodecahedron:  a stellation of the Platonic dodecahedron as shown in another image, where the accent is on other properties of the figure.  For example, here we cannot see that the axis of two opposite faces of the icosahedron passes through  Ω  and two opposite vertices of its dual, like the axis of triangle  HMT  passing through  Ω  and  P.  Here, all the faces of the icosahedron are distorted, although certain lengths are preserved, for example between  H  and  T  the edge length, or the distance between two opposite edges like between  M  and  Q,  because the two points are the endpoints of a horizontal segment.

The dodecahedron is placed on a horizontal jigsaw puzzle.  Sixty puzzle pieces are visible.  The two last paragraphs below deal with the two kinds of puzzle pieces, and the idea of "gnomon".

Another image exhibits an analogous tiling, that partly covers a face of a Platonic dodecahedron.  This other tessellation shows two sides of two hexagonal sections of the solid:  two equal segments which here represent two extensions of edges of the dodecahedron.  The image of the ten extensions is a regular decagonal star, more visible in an image where the dodecahedron is removed, so it no longer hides thirty puzzle pieces in the decagonal center.

Here the top view is an image of the figure through a vertical orthogonal projection.  One only point may be the image of several objects through this projection, for example these five objects: the center of the pentagonal puzzle, two of the twelve points, the midpoint of the segment joining them, which is the center of the two solids denoted by  Ω,  and the common vertical axis of the Platonic solids.  The image of  Ω  is also the center of two convex regular decagons, the sides of which form the outlines of the two solids.

Here the edges of the icosahedron are not visible, except the horizontal edge that joins  K and  L,  and the ten slanted edges depicted on its outline.  The boundary of the puzzle contains five sides of this outline, four of them are thick blue lines.  By dividing up the fifth side into two segments in red and green, the image exhibits the unit length:  the larger side length of a puzzle piece is chosen as unit length.

Given a vertex of the icosahedron depicted on its outline, either it lies on the horizontal plane containing the puzzle and the bottom face of the dodecahedron, in this case the vertex is depicted like  L  or  M as the center of a small half-disk in orange, or else the vertex like  H  or  T  lies on the opposite plane, depicted by a black cross.  Such a point which lies on five extensions of edges is also denoted by  T  in two views of another image, where point  T  lies on the bottom plane, while here it lies on the upper plane.  The puzzle plane contains five horizontal edges of the icosahedron, like the one joining  L  and  M.  The puzzle plane divides up the icosahedron into two parts.  The part below the puzzle is a regular pentagonal pyramid, which shares its vertical axis with the Platonic solids.

We see a sketch of three unfolded faces of the dodecahedron, of which vertex  L   is in the bottom left image.  Within another image with the same partial net of the dodecahedron, the accent is on two geometric sequences of five lengths and five areas, with common ratio φ: the golden ratio.

Behind the puzzle pieces, there is the idea of "gnomon":  a good way to remember a few geometric shapes and some formulas about the golden ratio.  The concept of gnomon is not mathematic:  in order to get a similar figure by adding another figure to a given one, the most interesting figure to add is chosen, called the gnomon of the initial figure.  The two shapes of puzzle pieces are called "golden triangles", each shape is the gnomon of the other shape.

All the puzzle pieces are isosceles triangles of two different kinds.  Two pieces of the same kind are "isometric".  In other words, given two triangles of a given kind, we can superpose them in such a way they coincide exactly.  Denoting by "type 1" the shape of an acute puzzle piece of which the equal sides are its larger sides, by "type 2" the other triangular shape, one piece of type 1 and two pieces of type 2 together form the small pentagon with green sides, shown in the upper right image.  This small pentagon is regular and convex.  Calling "golden triangle" any triangle which is similar to a puzzle piece, two equal angles of a golden triangle measure either 36 degrees or 72 degrees.  Given a golden triangle of a given type, its gnomon is a golden triangle of the other type.  In a golden triangle, the golden ratio is the ratio of the larger side length to the smaller one:
φ = 2 cos 36° = ${\displaystyle {\tfrac {1}{\,2\,\cos \,72^{\circ }\,}}.}$