Figure 4. Concentric HeHagonal Joists ouer Triangular Girders Fi_gure 5. Trian_gular Joists within Triangular Girders Figure 6. Triangles within Triongles within Triangles November/December 1984 RAIN Page 21 ing their history. Each of these was then covered with a piece of the plywood that served as flooring. The obvious way to follow suit in designing the hexagon's floor was to join together six triangular sections; the geometry of the hexagon dictated that the triangles should be equilateral. Such a pattern was suggested by my one source of information about the design of a "hex shed," a sketch in Lloyd Kahn's Shelter. But the sketch showed only a pattern of boards for the flooring, as depicted in Figure 2, and did not show how the floor was to be framed. Since I had decided to build a cabin 16 feet in diameter, each side of a triangular section would measure 8 feet in length (see Figure 3). Clearly, I would need some additional framing members to support the floor; I had, in fact, read that floor joists should be no farther apart than 2 feet. I could have laid joists over the triangular girder sections in a regular pattern, such as the concentric hexagons shown in Figure 4, but it seemed more elegant to continue working with triangles. Besides, I had read that triangles were strong. So I sketched joists that would hang between the girders, connecting adjacent midpoints on the sides of the latter, as shown in Figure 5. The result was pleasing to the eye, for it created within each girder section four triangles of equal size. But each of these triangles had sides of 4 feet, and that was still twice the recommended (maximum) span. Connecting the midpoints on the sides of these smaller triangles would, however, produce yet smaller triangles, 2 feet on a side. And so the pattern for framing the cabin's floor fell into place, as shown in Figure 6. From there, it was only a short step to imagine the floor itself, made up of triangular plywood "tiles" two feet on a side. These could be stained different colors to produce a mandala design. As the casual sketches that accompanied my flights of imagination gave way to the more exacting drawings required for building, I began to discover some of the Imitations of mathematical elegance. The sides of my imaginary triangles had intersected in points. But the pieces of lumber that would form the triangles supporting the floor were wider than a pencil line. I found myself drawing diagrams like that of Figure 7, checking on the width of the framing lumber, and filling up the pages of a notebook with geometrical computations. The mathematics that had been my imagination's playmate became instead a laborious taskmaster; v'3 became my constant companion. After a lot of head scratching and angle cutting, my lesson on the distinction between smart and wise drew to a close. Each of the girder triangles filled up with smaller triangles; the six girder triangles were then assembled over the supports; and the 96 plywood "tiles" were cut, stained, and nailed into place. Now the cabin had a floor, the likes of which I shall never build again. For all my pains, the floor did have an attractive design, as shown in Figure 8. After nearly a decade of use, however, the design has nearly faded. But the memory of the work it took has remained vivid. A sensible solution to the problem of framing a hexagonal floor did not present itself to me for several more years, long after I had stopped looking for one. It was a
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