Building the Omnitruncated 24-Cell
The Event
I visited the London Knowledge Lab on June 22, 2007, to commune with some fellow Zome enthusiasts. As we often do, we had planned a large-ish Zome construction as part of the fun. Often we seek to build something that nobody has done before, typically a 3-dimensional shadow of some 4-dimensional polytope with the symmetry of the "hyper-dodecahedron", or 120-cell. Since LKL is already the site of the largest of those efforts to date, I wanted to look for novelty in a novel direction. The next largest 4D symmetry group is that of the 24-cell, and I had recently added vZome support for polytopes in that family, so I decided to attempt the largest member of that family.
The model was made possible by the generous loan of Zome kits from several participants as well as from Tarquin Publications.
The photos below show a few of the ten or so participants, some later stages of the construction, and many views of the final model, including two that show it in front of the LKL's permanent "biggie", the omnitruncated 120/600-cell.
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The leftmost pair of images in each row below forms a "wall-eyed" stereo pair, while the rightmost pair forms a "cross-eyed" pair. Also, you can click on any thumbnail image to load a larger version of the image.
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Additional photos have been posted on the LKL website by Phillip Kent.
The Polytope
The polytope we constructed is called the omnitruncated 24-cell. In its "uniform" (most regular) form, it has faces that are squares, regular hexagons, and regular octagons. The model we built used non-regular octagons. The "cells" of the 4D polytope are hexagonal prisms and truncated cuboctahedra; when projected to 3D, most of these cells, and some of the faces, are distorted to various degrees.
As above, the pairs of images below form stereo pairs. For these vZome-generated images, you can click on either image to load the model in vZome, using Java Web Start.
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This polytope, like all the members of its family, is "self-dual". Without needing to understand the weighty mathematical definition, you can appreciate it as follows: choose a large cell (truncated cuboctahedron) and the hexagonal prisms that are joined face-on to its hexagonal faces; continue selecting the other large cells so adjoined to those prisms, and so on. When you're done, you'll find that you've selected exactly half of the cells of the model, and another identical (in 4D) set of cells intertwines with the set you selected! It is as if inside the walls of your house were the rooms of another, identical house.
Lessons for Group Model-building
The principal challenges in planning this event were to assemble enough kits (at the last minute) to supply the required parts, and to guarantee that all the loaned Zome kits would be returned intact, with as little mixing of parts as possible. My solution entailed a great deal of labeling (though not enough), careful attention to counting and distributing the required parts in plastic bins, and a modular, "workstation" approach to construction. The model made this possible because it can be naturally subdivided into eight equal parts. I used four tables, two workstations per table, with each workstation building one eighth (octant) of the model.
Each workstation had a set of bins for required parts. Each bin was labeled with the parts it contained and their provenance. Each workstation was labeled with the octant it was contributing, as in "top front left". The labels I should have added but didn't were: for each kit, which workstations had its parts; for each bin, which workstation it belonged to; and octant labels for the model itself.
Overall, I would avoid this problem of merging kits if at all possible. In this case, the problem was exacerbated by a characteristic of this model (see "chirality", below). However, with careful labeling, and perhaps more overseers during construction, the problem is manageable. For less modular objects, planning the construction in the presence of kit merging is probably too time-consuming.
One difficulty with this model is its use of green struts, especially in building regular green hexagons. These hexagons are quite difficult to construct de novo, and even difficult for novices when a sample is available. Finally, we noted some variability in the quality of the green struts and perhaps the balls, so some participants had a very difficult time. If the participants are Zome novices, it is best to avoid green struts entirely.
Because this model has octahedral symmetry, and Zome only supports pyritohedral symmetry, four of the octant modules must be "right-handed", and four "left-handed". The difference is subtle, even for experienced Zome users. I anticipated this problem, but not its scope; we ended up disassembling and reassembling a couple of octants, and rearranging them as well, as a result. This complicated the problem of restoring the parts to the correct kits.
