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Higher Frequency Designs forIcosahedral Geodesic DomesA frequency-2 large dome is difficult to transport. These designs might solve that problem, but they introduce other problems. |
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Higher Frequency Designs forIcosahedral Geodesic DomesA frequency-2 large dome is difficult to transport. These designs might solve that problem, but they introduce other problems. |
Many geodesic geometries could be used to form a dome of a given diameter. Lower frequency geodesics use fewer, longer poles and simpler geometry. Higher frequency geodesics use more, shorter poles and more complex geometry.
Higher frequency designs allow for shorter and thinner poles, which are easier to transport, cheaper to buy, and less unwieldy. Frequency-4 designs, however, use 2.7 to 3.8 times as many poles as the analogous freq-2 designs, but since the freq-4 poles are shorter, they can be smaller in diameter for the same rigidity. As frequency increases, dihedral angles between faces decrease, so vertices more easily invert, and therefore the structural integrity diminishes.
Frequency-4 icosahedral domes are presented here for both class-1 and triacon varieties of geodesics, along with frequency-3 designs.
Each pole is called an edge and each place where the poles meet is called a vertex. To simplify assembly and reduce the chance of errors, the ends of each pole should be painted according to the color scheme in the schematic diagrams presented here.
For each design, I present a set of "optimized" dimensions. These are optimized in the sense of trying to minimize wasted pole material while maximizing dome size, under certain constraints. The optimized designs assume that the builder wants the largest dome which uses poles around 5 feet long, that these poles will be cut from 120-inch long metal conduit stock, that the drill holes centers are 3/4 inch from the ends of the poles, and that the pole ends are bent 2 inches in from the ends. The angle of the bend should be roughly 9 degrees from a straight line.
For ease of assembly, I recommend using 4-inch long all-thread bolts
for all but the head-height vertices. The vertices at head-height
should use shorter bolts, and those bolts should be capped with plastic
bolt caps, to reduce the risk of impalement. I recommend using bolts
with 3/8 inch diameter shafts. The bolts should not be the cheapest,
lowest grade of metal because on the playa those materials will corrode
and the heads will shear off during assembly or disassembly. Use
high-grade or stainless steel bolts. Coat the bolts, nuts and washers
in lubricant (such as WD-40) before assembly and after disassembly.
This is the bottom view of a frequency-4, class-1 icosahedral geodesic dome.
| Pole lengths for a frequency-4 class-1 icosahedral geodesic dome |
To perform class-1 division (also called "alternate" division), divide each edge of a polyhedron by N where N is called the frequency of the geodesic. Within each original polygon, connect those points with edges. For N greater than 2, those new edges will intersect each other to produce yet more points. Each triangle will then become N^2 triangles. Finally, raise all of the new vertices to the circumscribing sphere.
As the diagram shows, this design has 6 pole lengths, 250 poles, 91
vertices. Euler's formula (V-E+F=2 or F=2+E-V) tells us that there are
161 faces. One of those is the bottom, so there are 160 triangular
faces on the top.
| Pole lengths for a frequency-4 class-1 icosahedral geodesic dome |
To minimize materials waste, use a dome radius of 192.624 inches. This corresponds to a dome with a height of about 16 feet, a diameter of about 32 feet, and which has a ground-floor area of about 810 square feet. This design requires 125 of 120-inch long poles. Note that the lengths 58.234 and 61.766 sum to exactly 120. Those are the length pair which establish the optimized dimensions for this design.
| Pole cuts for a frequency-4 class-1 icosahedral geodesic dome |
The table has ?? in the "number" field because I fucked up the original count and don't have time right now to fix the count.
This "optimized" design, as compared to the analogous optimized design for the frequency-2 class-1 icosahedral design, results in about the same size dome.
Assuming 3/4-inch diameter metal conduit would be used at a price of $2.00 per 120-inch pole, the cost of tubing for this design is $250. The cost of fasteners would probably be around $50.
By comparison, a frequency-2 class-1 icosahedral dome with a nearly
32-foot diameter, using 1-inch diameter metal conduit, would cost $240
for conduit plus about $15 for fasteners. Such a freq-2 design would
be less strong (by maybe 50%) than the analogous freq-4 dome.
Although triacon division was the first used by Buckminster Fuller to
generate geodesic geometries from Platonic polyhedra, this method is
less often seen in hemisphere structure design. Notice that the bottom
perimeter of this design does not touch the ground with a polygon as do
the other designs. That is not a problem, just something to notice.
The bottom vertices can be joined to complete the dome bottom.
Also note that the cartoon renderings show vertical poles extending
through the ground. Those should be ignored. They are shown because
they exist in the complete triacon sphere but they are useless in the
triacon dome. They could be used to halve the length of the poles
around the bottom perimeter, but aside from keeping pole lengths down,
there is not much point in doing that.
This image depicts the bottom view of a frequency-4 triacon icosahedral geodesic dome.
| Pole lengths for a frequency-4 triacon icosahedral geodesic dome |
To perform triacon division, bisect each original angle of a polyhedron. This will create a set of intersections of bisectors. Raise those new points to the circumscribing sphere. Connect those new, raised intersections with edges. A final step to remove the original edges is typically performed.
This design has 5 pole lengths, 175 poles (plus 10 more poles, not
shown, to complete the bottom perimeter), and 66 vertices. The
relative length of the 10 bottom poles is 0.618034. Euler's formula
tell us there are 111 faces. One face is the bottom, so there are 110
triangular faces on top.
| Pole lengths for a frequency-4 triacon icosahedral geodesic dome |
To minimize materials waste, use a dome radius of 168.740 inches. This corresponds to a dome with a height of about 14 feet, a diameter of about 28 feet, and which has a ground-floor area of about 620 square feet. The table presents the resulting pole lengths. This design requires 93 of 120-inch long poles for the skeleton shown, plus 10 poles for the bottom perimeter. Notice that the 60.000 inch poles are cut two to a 120-inch stock pole. This determines the optimized dimensions of this design.
| Pole cuts for a frequency-4 triacon icosahedral geodesic dome |
This "optimized" design, as compared to the analogous optimized design for the frequency-4 class-1 icosahedral design above, results in a 13% smaller dome, with 23% less floor area.
Assuming 3/4-inch diameter metal conduit would be used, at a price of
$2.00 per 120-inch pole, the cost of tubing for this design is $185.
The cost of fasteners would probably be around $40.
| Pole lengths for a frequency-3 icosahedral geodesic 4/9 dome |
Click on an image to see a larger version. Move mouse over oblique view to see 4/9 truncation, or off oblique view to see 5/9 truncation.
The odd-frequency designs can not be made into a hemisphere. They are therefore either extended just past the hemisphere, or just short of it. The images above show 5/9 and 4/9 domes. The bottom view of a 5/9 dome would look unclear because some of the poles and vertices would overlap, so the smaller dome is shown for that view. The oblique view, however, shows the slightly larger 5/9 dome when the mouse is not over the image, and shows the slightly smaller 4/9 dome when the mouse is over the image.
Note that for the 5/9 dome, the bottom tier of triangles is nearly vertical. The peculiarity that odd-frequency domes have no true geodesic polygon, i.e. that they can not make an even hemisphere, ends up also avoiding one of the more annoying aspects of an even frequency dome, which is that even frequency domes have no vertical walls at the bottom.
Note that the vertices of the bottom layer are not exactly at the same level; They vary in height by 0.016 radii, or about 1.6 percent of the dome height. For a 12-foot tall dome, that variation would be about 2 inches (only about twice the size of a typical EMT pole diameter). Depending on the application, that could be considered too small to worry about. The structure could be warped slightly, small blocks could support the vertices which lie above the ground, or the poles on the bottom layer could be adjusted to exactly meet the horizontal ground plane. The main problem with making that length adjustment is that the poles on the bottom layer could no longer interchange with other, nearly identical poles elsewhere in the dome. Keeping track of the slightly different poles would entail difficulties. Note that the oblique view shows a 5/9 dome with an uneven bottom layer, but the height variation of those vertices is not obvious.
Odd-frequency geodesics have no triacon varieties.