Geometry : Structures made with Dodecahedra or Icosahedra
Created on August 26, 2023.
Structures made with Dodecahedra
Following my interest about polycubes, I started to wonder about what kind of shapes one would be able to make by attaching dodecahedra one to another, face to face. The pentagonal symmetry makes for wonderful flower of tree like structures that are quite fun to build. At some point during this exploration, I remembered that amongst the Stewart's solids available in Stella software, there was a ring of dodecahedra, 8 of them in the shape of a rhombus. The next solid in the list was a Triacontahedron made by repeating this rhombus around, a toroidal shape with high genus. Then, I realized that the rhombus was without a doubt a Golden Rhombus which meant that the other Rhombic solids could be built. At this point, if you don't know what rhombic solids are, please have a look at this post of mine. I also have it updated with new shapes recently.
You may have noticed in the rhombus and triacontahedron that some dodecahedra acts as the "vertices" of the base shape and are linked together with other dodecahedra that makes for the "edges". There is only one "edge" dodecahedra in the case of the two Stewart's solids but you can have any odd number of dodecahedra for the edge, it simply make the shape larger. The more you add, the thinner the "edges" will look like with regards to the whole structure and the more it will tend towards a wireframe representation of the corresponding base rhombic solid.
I started making all the common rhombic solids, the acute rhombus, the obtuse rhombus though I had to use five dodecahedra for each edge because the shape would intersect itself with only three. I continued with the rhombic dodecahedron, the rhombic icosahedron and the hexecontahedron and it worked. I have never proved that the angles in the 8 dodecahedra ring are actually the same as for a golden rhombus, I leave that as an exercise to the reader.
From there on, I explored different shapes, extending the hexecontahedron by continuing to replicate the rhombus ring over and over again, adding one shape to another, making DNA helices, viruses, ... The realm of exploration is infinite. This is so much fun !
And then I wondered again, since the icosahedron is the dual of the dodecahedron, can I make golden rhombic solids with icosahedra ?
Structures made with Icosahedra
The answer is no but you can make still make a rhombic dodecahedron, the other one, it is not golden since the ratio of the diagonals of the rhombus is the square root of two. This basic rhombic ring, made of 8 icosahedra, is a Stewart's solid and was in the Stella database (of course !), the rhombic dodecahedron was not. I could also make the corresponding acute and obtuse prism and replicate the fact that this dodecahedron paves space.
And then the day was over so I stopped playing, made the viewers and wrote this article.
THE END
