What follows is an explanation very similar to the one about Cubes and Octahedrons. We are now starting to see how two other Platonic solids are tied together: the Dodecahedron with its 12 pentagonal faces and Icosahedron made up by 20 equilateral triangles (I remember Platonic solids are those made up by identical regular polygons, and with the same number of faces meeting at each vertex).

As for Cubes and Octahedrons, also Dodecahedrons and Icosahedrons share a reciprocal "duality" (in geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other).

This is the main reason Dodecahedrons and Icosahedrons are tied together. Well, if you already read the explanation about Cubes and Octahedrons, I guess you are expecting to find also another reason... so I invite you to follow reading!

Let's see some Dodecahedron related polyhedra. Each edge of a Dodecahedron can be divided in three parts (see the edge b-e in the drawing below on the left) so that the extremities b-c and d-e are equal each other, as are the segments a-c, c-d and d-f. Doing the same operation on every face, and cutting out the Dodecahedron vertices along the red lines, we obtain a "Truncated Dodecahedron":

After the truncation, each Dodecahedron face has turned into a regular decagon, and each vertex have turned into an equilateral triangle, where the length of the decagon and triangle sides are the same each other. The Truncated Dodecahedron is composed of 32 regular faces (12 decagons and 20 triangles), 60 edges, and 90 identical vertices.

 
Here is the Geomag Truncated Dodecahedron model, where the green rods are the polyhedron edges and the metal rods are only needed to keep the model rigid (this is valid for all the following models too):

 

There is another way to cut the Dodecahedron vertices: it's enough to divide each edge into two halves and join the obtained central points each other across the cube faces. The result is shown below on the right: it's name is... well, I'll tell you later!

Let's put aside the Dodecahedron for a while, considering now the Icosahedron instead. Each edge of an Icosahedron can be divided into three identical parts: in the drawing below on the left the red lines join these intermediate points. Cutting the vertices of the Icosahedron over these lines, we obtain one regular hexagon for each face, and one regular pentagon for each vertex. The length of the hexagon and pentagon sides are the same each other.

The result is a Truncated Icosahedron, composed of 32faces (12 pentagons and 20 hexagons), 120 edges and 60 vertices:

As for the Dodecahedron, also the Icosahedron edges can be divided into two halves (see below on the left). Cutting the Dodecahedron vertices we obtain what shown on the right:

But... the last drawing is exactly the same as #4! Let's see them side by side:

Actually this polyhedron can be obtained by cutting either the Dodecahedron or  the Icosahedron vertices, and this is the second reason (after their duality) Dodecahedrons and Icosahedron are tied together. Look at the next image:

The blue pentagons are what remain of the Dodecahedron faces after truncating its vertices, while the green triangles are what remain after the truncation of the Icosahedron.

The name of this newer solid name is Icosidodecahedron, with 32 faces (20 triangles and 12 pentagons), 30 vertices and 60 identical edges. Here is the Geomag model:

So there are 5 polyhedra can be converted one another just by truncations (shown in red in the following animation) or additions (shown in green): Dodecahedron, Truncated Dodecahedron, Icosidodecahedron, Truncated Icosahedron and Icosahedron .

The work on Dodecahedrons and Icosahedrons doesn't finish here! Actually Archimedes found other three most interesting solids, so let's start from the first of them!

The pentagonal faces of the Dodecahedron can be expanded so that the distances between the vertices b-c, b-e and c-e be equal to the original Dodecahedron edges (a-b, c-d). Joining the vertices of the pentagons we obtain the model below on the right: the Small Rhombicosidodecahedron, with 62 faces (12 pentagons, 30 squares and 20 triangles), 120 edges and 60 vertices.

The same operation can be done expanding the decagonal faces of the Truncated Dodecahedron, so that the distances between the vertices b-c, d-e and a-f be equal to the decagon edges (a-b, c-d, e-f). Joining the vertices of the decagons we obtain the model below on the right: the Great Rhombicosidodecahedron, which has 62 faces (12 decagons, 20 hexagons and 12 decagons), 120 vertices and 180 edges. This is the biggest of the Archimedean solids (given the same edge length).

 

Here are the two Geomag Rhombicosidodecahedron models (the Small on the left, the Great on the right):

The Great Rhombicosidodecahedron is the most difficult to build with Geomag: my model crashed just after taking two or three shots!

The last Archimedean polyhedron related to Dodecahedrons and Icosahedrons is (in my opinion) absolutely amazing: the Snub Dodecahedron. Let's see how it is obtained.

On each of the expanded decagons shown on drawing #13 we can draw a simple pentagon joining only the odd (or even) vertices, see below on the left. On the right the pentagon vertices are joined as to form some equilateral triangles: this is just the Snub Dodecahedron, which has a total of 92 faces (12 pentagons and 80 triangles), 150 edges and 60vertices (each vertex is formed by one pentagon and four triangles).

One interesting thing is, if we join the even instead of the odd vertices of the decagons (or vice-versa), we obtain the following:

These two polyhedra are both Snub Dodecahedrons, but are not identical: it is as if one was a mirrored version of the other, so they are "Chiral" polyhedra. In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations only.

Here are the two Geomag models:

So here is the list of all the Dodecahedron and Icosahedron related polyhedra:

1 - Dodecahedron
2 - Truncated Dodecahedron
3 - Icosidodecahedron
4 - Truncated Icosahedron
5 - Icosahedron
6 - Small Rhombicosidodecahedron
7 - Great Rhombicosidodecahedron
8 - Snub Dodecahedron, CCW version
9 - Snub Dodecahedron, CW version

Note: All these polyhedra have the same edge length: just one Geomag rod.

Article written by Guest Blogger Aldo Cavini (aka aldoaldoz)