Did I mention we need 60 of those?
The next step was to assemble some of the pentagonal faces of the excavated dodecahedron. Since with the Polydrons you can’t reach inside of structures once they’re built, it seemed like it might be best to make these rings of octahedra connected by triangles — the red triangles would eventually become part of the face-central icosahedra. Checking back on the statistics, we’d need 12 of those.
Next it was time to start connecting those rings. Three of them come together at each vertex of the overall dodecahedral structure.
For a while things seemed to be going pretty well with just connecting three of the rings at each eventual junction, like so.
But shortly after the above picture, the whole structure became hopelessly floppy and started to disconnect at numerous junctions as it wobbled apart. So for a second try, I filled in more of each icosahedron before proceeding with its neighbors, leaving off just a spot for an eventual “cap” of five triangles.
This led to a much more rigid structure and stable build. And then when I had all of the connections made to a given icosahedron,
I could add on that cap to complete it.
At this point it was stable enough to start adding a second tier of the five-way icosahedra.
Things now proceeded pretty steadily. Here is the model with three icosahedra in the upper tier, and more of the ones in the first tier capped off.
And here we are just before and after adding the final five-way icosahedron at the top of the structure.
All that remained was to cap off that last central icosahedron, and finish off the five green icoshedra surrounding it. Unfortunately, while performing what by that point seemed like routine connections within the first of those five icosahedron, some connection in the lower tiers gave way under the accumulated weight of the model. That rip set off a chain reaction of self-disassembly, leading to various strips of Polydrons on the floor and a seriously disrupted structure.
So, despite coming close, it seemed as though this medium for building the model wasn’t going to work, either. I’d need to find some other means of building it… leading to the subject of next week’s column.
As a footnote, though, I do want to say that it’s possible that someone aware of the potential collapse near the end and working with sufficient delicacy could get the full excavated dodecahedron to hang together with Polydron. Moreover, Polydron does also offer framework pieces like these, which are significantly lighter, and I don’t doubt that with a sufficient supply of framework pieces it would be possible to assemble this entire structure.