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flat-unit-hollow-truncated-hexahedron-Haűy-construction-01 | by Ardonik
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Over the course of the last three months, dozens of people have asked me what exactly this is. Since I designed this on a whim from a pencil drawing of a few corner cubes, I had no answer for them at first. However, over time it has become clear that this is the Haűy construction for a truncated cube. Interestingly, crystallographers have documented other examples of these phenomena at the molecular level (and indeed this was Haűy's original motivation); one of my favorite online collections to explore for Haűy construction inspiration is this mineralogy page.


This concave, toroidal polyhedron consists of twelve 3-cube columns (each having 3 * 4 + 2 = 14 squares) and eight 3-cube "triads" (made from 6 * 3 = 18 squares.) The total number of cubes is 60, but the total number of square flat units is 312. The project was therefore not very large, but I was only able to work on a handful of units each week, hence the delay.


This polyhedron is only hollow (as opposed to being paneled shut) because it seemed more challenging--and therefore interesting--to make it that way. The work was tedious and slow, and while it did indeed prove interesting, "challenging" is not the word I would use to describe it. Once you've got the pattern, you've got it, and the only real dangers at that point are the coloring mistakes that I encountered in abundance.


One consequence of this early design decision, and one of my major concerns during assembly, was that each "vertical" column (and there are of course 12 such columns altogether; only 4 columns are vertical in this photograph) was being held up by just four thin joining tabs at the base of the column. Because this structure is made entirely of squares, building off of the columns at 90° angles was wobbly work. I was almost certain that one of the columns would tear at the base before I managed to finish the project, and the completed structure remains desperately fragile.


While there were disadvantages to making this thing hollow, there are also benefits. Instead of having my foil origami paper clash with the solid colors on the outside, I reserved all of the gold units for the inner surfaces, where they complement one another to cast a shimmering light from within. The effect is really nice, and I'll try to repeat it with other types of units in the future.


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Taken on November 15, 2011