In the mid-1960s, Steve Baer was involved with Drop City, an artist's community outside of Trinidad, Colorado. Baer was very interested in dome structures, but was frustrated with some of the features of geodesic domes popularized by Buckminster Fuller. Baer wanted something more adaptable, extensible, and modular, so he explored the geometry of zonohedra, polyhedra with rings of parallel edges.
Through his studies, Baer became intimately familiar with the Platonic and Archimedean solids. At Drop City, Baer and others built a variety of dome buildings "on the cheap", salvaging car tops for the panels in the domes. The most iconic of these was Baer's triple dome, constructed with parts of three rhombicosidodecahedra fused together. For brevity, I'll use "RID" instead of "rhombicosidodecahedron" (or "RIDs" plural) in the rest of this article.
Most of the figures in this article are interactive 3D views. Use your mouse or touch to rotate, pan, and zoom.
To construct his triple dome, Baer had to overcome a slight problem: when he tried to fit the three RIDs around a point, they didn't connect. The construction requires chopping off two caps from each polyhedron, exposing partial decagon faces. The problem is that the angle between those faces is not 2𝜋/3, required if they are to all meet around a point. You can try to build three such modules in Zometool, but they won't join up with normal Zome balls -- you would need two kinds of special connectors at the boundary, and you're still left with an angular gap. Baer had to "fudge" it, using force to close the slight gap the third joint.
Baer's son José recently shared with me an unpublished essay by Steve Baer himself, in which he describes this gap and the dismay that it caused him:
I was so enthralled by these forms, I assumed they were merely clumsy bubbles. Only later did I realize that these polyhedral bubbles did not fit. The fusing angles were not a perfect 120° as with the soap bubbles, but irrational angles of 116.56505°. This was cause for grief, betrayal by nature, by geometry. This was new for me. An uneven student, I was alternately excited and discouraged by science and mathematics, yet I had not encountered such disappointment in private investigations. —Steve Baer, RID -- a love story
The 4D Approach
In four dimensions, where the three RIDs can each be in their own 3D space (hyperplane), they join seamlessly if those three hyperplanes are at the appropriate angles. In fact, there is a uniform 4D polytope that has exactly this configuration. This polytope contains dodecahedral cells (seen here in a cutaway view for clarity) that are separated by prisms and tetrahedra. This effectively wraps the dodecahedra in overlapping RIDs.
We are seeing the usual Zometool projection from 4D to 3D, so any cluster of RIDs we see here can be built with Zometool, or with an equivalent system scaled up for architectural use. To replicate the three-fold symmetry of Baer's triple dome, we can use a triple RID cluster as seen here. Let's discard the rest of the 4D polytope for a clearer view.
In terms of simplicity of construction, this approach looks quite favorable at first glance. There are only three kinds of struts needed, and only seven kinds of panel. However, things get a little more subtle when we try to define a floor plane. The symmetries of our squashed RIDs do not provide a natural place to introduce the floor. Any choice we make will require very non-standard struts and panels where our structure meets the floor.
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