GREAT DODECAHEDRON, {5, 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 { 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.
Since each pentagonal face is centred on the face of the inscribed 12-faced regular dodecahedron, We can understand the construction of the great dodecahedron better by visualizing how the faces are constructed, starting with an icosahedron.
Duality
We observe that in this case, 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.
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.
There is a version of the Rubik’s Cube in the shape of the great dodecahedron.
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 MathWorld : http://mathworld.wolfram.com/GreatDodecahedron.html Wikipedia : http://en.wikipedia.org/wiki/Great_dodecahedron Construction : Wenninger, M. J., |