GREAT DODECAHEDRON,

{5, ⁵/₂} or (5)^5/2

  Definition and notation

Our next Kepler-Poinsot polyhedron is the great dodecahedron, illustrated at right. It has intersecting pentagonal faces (indicated by the ‘5’ in the symbol), and each vertex figure is a pentagram {⁵/₂}.

To understand the structure of this solid better, it will help to observe that it is built around a regular dodecahedron, and the final figure is inscribed in a regular icosahedron.

We are now going to determine the number of vertices, edges and faces. In determining the number of faces, again remember that it is the whole pentagons that make up the faces here, not the individual parts. Similarly, the vertices of the solid are where the (five) vertices of the pentagonal faces meet. And the edges are the long (not completely visible) line segments joining the vertices.

• How many faces, edges, vertices does it have? Can you explain the symbols above?
   Complete the table below (# stands for ‘number of’). Do you notice something strange about your last entry?

V	# pentagonal faces, F	E	V – E + F

Check your answers ...

   You should have obtained:

V	# pentagonal faces, F	E	V – E + F
12	12	30	– 6

Since each pentagonal face is centred on the face of the inscribed 12-faced regular dodecahedron, F = 12. And since the vertices coincide with the vertices of a regular icosahedron, the number of vertices of this solid is also 12. For the edges, each of the 12 faces has five edges, giving a count of 5 x 12; since each edge is counted twice in this calculation, we deduce that E = 30. We observe the number –6 which appears, not quite so unexpectedly as before, in the Euler formula.

We can understand the construction of the great dodecahedron better by visualizing how the faces are constructed, starting with an icosahedron.

         green face              yellow face               red face

MathWorld gives without proof some of the basic measurements of this polyhedron.

  Duality

• Compare the values of V, E and F  for the great dodecahedron and the small stellated dodecahedron studied earlier. Compare also the symbols for the solids. What do you notice. Does this indicate duality?

We observe that in this case, V = 12 = F, and E = 30. These figures are exactly the same as those obtained for the small stellated dodecahedron, but the critical fact for establishing duality is that the value of V for the great dodecahedron is the same as the value of F for the small stellated dodecahedron, and conversely. The dual nature is also reflected in the symbols for the solids:

the small stellated dodecahedron  {⁵/₂, 5), and

the great dodecahedron  {5, ⁵/₂}.

  Stellations

We saw previously how the small stellated dodecahedron can be obtained from a dodecahedron by ‘stellation’ – extending the faces until they meet again, so forming the new solid. The great dodecahedron can also be obtained by stellation: starting with the small stellated dodecahedron, we can extend the faces to obtain this new solid. Another way of saying this is to call the great dodecahedron the second stellation of the dodecahedron.

Dodecahedron	Small stellated dodecahedron	Great dodecahedron

  Further properties

As with the Platonic solids, we can use this Java applet to play with each of the Archimedean solids. Perhaps you can find some interesting new property! It is not impossible: new results are being discovered in elementary geometry even today. The only reason they have not been discovered before is that no-one thought to ask the right question.

• Click the ‘all colours’ button to turn off the colours and view the underlying dodecahedron.

• Click off one colour to view a pentagram lying in front of a regular 15-gon. Why can we say that this polygon is regular?

• With just one colour checked, notice how the faces of the polyhedron occur in parallel pairs, and how they are related to the underlying dodecahedron.

• Highlight just three colours to surround a dimple. Show that there is an opposite dimple which can be reached in three easy ways. But does this dimple have the same orientation as the original?

• Do some investigations of your own. Notify the author of any good discoveries you make!

  Star puzzle

There is a version of the Rubik’s Cube in the shape of the great dodecahedron.
It is called Alexander’s Star puzzle, and is pictured at right.

  Model making

The model of the small stellated dodecahedron is easy to make. See this construction page for some details.

  Vertex coordinates

It is a simple matter to derive a set of vertex coordinates for the great dodecahedron. For, as the figure at right shows, these vertices are just the vertices of a regular icosahedron. Hence from our previous work, we can take the 12 vertex coordinate sets to be:

( , 1, 0 ),  (0, , 1 ),  (1, 0, ).

References