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Design and
Implementation for a
Geodesic Dome
Portable robust shade structure for Geodesika and other hard core temporary outdoor scenarios.
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Design and
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Geodesika is a festival in the mountains at the beginning of summer. Burning Man is a festival in the desert at the end of summer. It gets hot and sunny during the day, so shade structures are crucial. The larger and more portable, the better. They should also be cheap, easy to assemble, and stable enough to withstand strong winds. A geodesic dome fits that description.
Portable geodesic domes and other large tent structures are commercially available. (Some information is provided below for some manufacturers.) However, they tend to be either expensive or flimsy, and definitely less fun and versatile than a home-made structure.
Some materials for the geodesic dome covers are translucent and white,
and could have light projections coming from either outside or inside.
One of the benefits of a geodesic dome is that they are made of triangles. Triangular structures can not deform without deforming the edges, unlike parallelepipeds or other structures based on polygons with more than 3 edges. For example, a triangle can not be squashed, but a rectangle can be squashed into a parallelogram just by changes in angles at the vertices. The idea behind geodesics is to exploit geometry to obtain greater structural integrity. However, geodesic structures have a drawback: In practice, it is difficult to align the vertices, and if there is misalignment, the structural integrity is compromised. The smaller the number of edges, the less problems with alignment, but the longer the edges. Longer edges require stronger materials to maintain their integrity. The design presented here is an attempt to compromise on these issues.
This is the bottom view of a frequency-2, class-I (aka "alternate
division") geodesic dome. Each pole is called an edge and each
place where the poles meet is called a vertex. The thicker
magenta edges are all of the same length,
which is 0.61803399 times the desired radius. The thinner
green edges are all of the same length,
which is 0.54653306 times the desired radius.
(Note that the ratio of the edge lengths is 1.13082636 and that the sum is 1.16456705. This will be useful to remember later.)
For example, for a geodesic dome with a diameter of 30 feet (i.e. a radius of 15 feet), the edges represented by thick magenta poles would be (15*0.61803399)=9.27 feet long, and the edges represented by the thin green poles would be (15*0.54653306)=8.20 feet long.
In this design, there are 35 long edges, and 30 short edges. There are 26 vertices. 6 vertices (1, 12-16) have 5 edges. 10 vertices (2-11) have 6 edges. 10 vertices (17-26) have 4 edges.
I have presented high-4 designs for a larger geodesic dome that allow the edges to be shorter, although the number of edges is about quadruple of this design.
For small geodesic domes (shorter than about 15 feet tall), the triangles at the bottom are too short to walk through so having a larger entrance is desirable. To make an entrance, remove the radial edges of one of the lower triangulated pentagons (such as all edges touching the vertex 12, i.e. edges V12-V2, V12-V7, V12-V17, V12-V26, and V12-V11). After removing the edges, the geodesic dome will partially collapse at that region, so it will need to be reinforced.
To reinforce the entrance, add edges from the middle of the top two edges of the pentagon to the bottom two vertices of the same pentagon (e.g. from vertex 26 to mid-way along the edge between vertices 2 and 11, and from vertex 17 to mid-way between vertices 2 and 7).
The cartesian coordinate of a new vertex midway between vertices 2 and
7 is (0.606961, 0.25, 0.688191). The vector from vertex 17 and that
new 2-7 vertex is (0.344096, 0.059017, 0.688191). The length of that
vector is 0.771681 dome radii.
redundant check:
The cartesian coordinate of a new vertex midway between vertices 6 and
10 is (0.050203, -0.654509, 0.688191). The vector from vertex 24 and
that new 6-10 vertex is (0.050203, 0.345491, 0.688191). The length of
that vector is 0.771681 dome radii. This matches the length computed
above.
The issues involved in implementing a geodesic dome include deciding what materials to use, and how the pieces will be connected together. The solution should be as simple as possible, and use as little material as possible.
The poles could be made from thin steel conduit, also known as electrical metal tubing or EMT. Conduit varieties include variations in wall thickness, diameter, and material. The cheapest feasible possibility is to use non-rigid zinc-coated or galvanized steel conduit. (Steel rusts rapidly. The coating is intended to reduce corrosion, but if the dome will be oft exposed to wetness, apply primer or paint.) The diameter of conduit to use depends on the lengths of each edge: The edges should not bend under the weight of the load they will bear. Longer edges will require larger diameter poles.
If the edge lengths exceed 10 feet each, then a problem occurs: conduit poles are 10 feet long. In order to construct edges which are longer than 10 feet, multiple conduit poles will have to be joined for each edge. One method is to use two different diameters of pole, and insert the smaller into the larger, creating a "telescoping" pole, similar to the legs of a tripod or the barrels of a collapsible hand telescope. Telescoping is a big hassle and reduces the integrity of the structure, but it is easier than using a higher frequency geometry. Still, I have found that using telescoping poles leads to an unsatisfactory structure.
Some people use PVC or ABS (kinds of plastic) pipes for their domes. PVC is light and cheap, but very flexible, even for very large diameters. The hubs are sometimes made of conduit, and the PVC is attached to the hubs using pins or small bolts. In other cases, the hubs are simply made using lashings or tape. Several people have had success with PVC, but I find it too flimsy for my purposes. It is not possible to hang heavy things from PVC domes, not even from the vertices. PVC domes can collapse if they bear much load. Some designs avoid this problem cleverly, but I will not be discussing PVC frames.
Another edge material that has been used is fiberglass. Fiberglass poles, about 4 feet long, are mass produced for use as broom handles, and if you are lucky enough to live near a manufacturer of these poles, you might be able to obtain 65 of them to make a dome. These poles are strong, light, and rigid. To connect them at the hubs, conduit can be used.
Mesh or other very breathable fabrics can be used as a covering. They lack the wind-catching problems that parachutes have. Mesh fabrics can be expensive. Shop around to find inexpensive fabric. Also, it is unlikely to find a single, large, circular sheet of mesh fabric, so if this material is used, a way to drape it over the dome such that material is not wasted will have to be found, such as cutting the material into triangular panels and sewing them together.
Parachutes and mesh coverings have a serious drawback -- they are not opaque and do not sufficiently block sunlight. This makes them look interesting at night when they are lit from inside but during the daytime, the problem is horrible. Dark chutes improve the shade some, but a better solution would be to use an opaque cover, such as a silver tarpaulin, to cover the dome.
Drawbacks of tarpaulins include the fact that the fabric is slightly inflexible and that it makes an annoying sound when blown in wind. To mitigate those problems, custom-fit covering patterns can be cut from the tarps.
Tarps are not entirely opaque but that can be improved by gluing
aluminum foil to the tarp. Use spray adhesive. After the adhesive
dries, the foil can peel off, but not easily. In fact, the
foil coated tarps can be folded, wrinkled, wadded up and abused without
the foil tearing significantly. Plus the shiney look of the foil is
neato.
| Conduit prices from near Boulder, Colorado as of winter 1998. All prices listed are for non-rigid, galvanized steel conduit. |
Shop around for the place that sells conduit for the lowest price -- a lot is required. In Boulder, the prices varied by a factor of two!!! McGuckin's Hardware charges literally more than twice as much for conduit than most other stores in the area. Sullivan's Hardware charged 25 percent more than what could be found at Home Depot or Eagle.
Larger diameter poles will be less flexible than smaller diameter poles.
Poles with thicker walls also give more strength to the structure, and there is a "rigid" variety of conduit, but it is much more expensive and difficult to deal with. Note that there is a difference between "thick walled" conduit and large diameter conduit. "Thick walls" refers to the actual thickness of the metal, where the diameter of the pole refers to the circular cross section size. These two properties are somewhat independent.
Bolts (3/8 inch diameter shaft, 9/16 inch hex head, steel) cost about
US$0.25 each. Cost depends on length. Bolts 2 to 3 inches long are
needed, depending on the number of edges that meet at a vertex. The
cost of corresponding nuts is about US$0.05 each. Washers probably
cost about US$0.02 each. For 26 vertices, the total cost of the
bolt/washer/nut combos is about US$9 including sales tax.
The vertices, or hubs, could be implemented by pressing the pole ends flat and parallel, drilling a hole near the end, bending the ends, and sliding a bolt through all of the pole ends at each vertex.
The way parachutes sizes are advertised is by the diameter of the circle they make when laid flat. This is not the same as the diameter of the hemisphere they cover.
If laid flat, the parachute should be circular, like a disc. When the parachute is draped over a spherical dome, the diameter of the parachute should be equal to half of the circumference of a sphere with the same diameter as the dome. Some simple geometry and math yields this relationship between the diameter of the parachute and the diameter of the dome:
/ 2
E.g., a parachute to cover a 15-foot diameter dome would have a
semi-circle arc that rises over the top of the dome has an arc length
of *(15 feet)/2=23.562 feet.
After the size of the dome is determined, the sizes of the poles is determined by using the factors given above: Multiply the radius (i.e., half the diameter) of the dome by 0.61803399 for the long poles, and by 0.54653306 for the short poles.
There are two principle aspects of constructing the dome pieces that require accuracy and precision: Making sure that the pole ends are flat and parallel, and placing the vertex holes in the right place.
A significant impediment to assembling the dome is if the flattened ends of the edges are not parallel. If the edge ends are not parallel, then when the pole ends are joined, the effective thickness of the vertices will be much larger than necessary, requiring either a longer bolt, or complicated maneuvers with C-clamps to hold the edges together while the bolts are inserted. Much effort will be saved by making sure that the flattened edges are parallel.
What determines the edge length, more so than the cut length of the poles, is the distance between vertex holes in the poles. If the edges are the wrong length with an error of about 1/16 inch each, then after 10 consecutive segments, the cumulative error can be 5/8 of an inch. This is a quite large error, which would make it difficult to connect the edges together. However, machining these edges to an accuracy of better than 1/16 inch is difficult.
In practice, the dome is somewhat flexible, so that if the vertex holes are not drilled precisely, it will be possible to bend the dome to line up the vertex holes, although bending the dome into shape is less elegant than simply driling the holes in the right place.
In order to flatten the pole ends, several methods can be employed.
This is the most elegant method, if you have the means. If you do not have access to a metal shop, then you can probably hire one to press and drill all of your poles for under $100.
Another method of flattening is to use a large, heavy hammer to flatten the poles. Such hammers are cheap, portable, fast, and require less effort than using a vise.
A "spud" bar is the back end of a device used to split concrete. The splitting side has a blade, and the back has a flat base, and the flat base can flatten poles with relatively little effort. This device looks like a pipe with a blade at the end. The pipe has a length of about 4 to 5 feet. The device weighs about 20 pounds, and costs around $20. One benefit of this device is that you can rig a pipe sleeve mounted vertically to guide this hammer so that you only need to place the pole end under the sleeve-guided hammer, then repeatedly lift and drop the hammer a few times to flatten the pole. Using a spud bar is much better than using a sledge hammer.
The major problems with using a hammer include these:
One method is to use a vise to flatten the poles. Using a vise will give precision in flattening the pole. A vise is also useful for many other tasks, such as holding poles while sawing or drilling them. The major problems with using a vise to flatten are that a large 5-inch vise costs about $60 (which is more expensive than a hammer), and that the force (or torque) required to flatten metal poles is significant, so the process takes a long time and a lot of effort.
A heavy duty vise with large jaws (5 inches or more) is desirable. It should be mounted onto a stable, immobile, strong platform. If the vise is simply mounted to a table, even a heavy table, pushing on the vise rod will move the table around. (Bending steel, after all, requires a lot of force.)
The vise could be attached to a 2x12 length of board, then there would be a large, sturdy platform which could then be clamped to a much larger, immobile table. This allows the vise to be semi-portable while still having a sturdy mount.
Also recommended is extending the vise rod by about 12 inches, perhaps by slipping a length of conduit over the existing vise rod. The extra length will increase the torque applied to the vise, which reduces the amount of force required. The amount of force required is inversely proportional to the length of the vise rod (or rather, to the distance between the pivot point and the point where the force is applied); If the length of the rod is doubled, the amount of force needed is halved. Be careful, though; if the vise rod is too long then a modest force can produce enough torque to destroy the threads on the vise screw. More than doubling the vise rod length is not recommended.
Work incrementally. Make a first pass, partially flattening the length, starting at the end and squeezing about 1/2 of the pole at a time, working toward along the length, then make a second pass to fully flatten. After one end is flattened, place the other end into the vise, close the jaws just enough to grab the pole, then use a level to align the flat end so that the two ends will be parallel when they are both flat.
A combination approach of using a hammer and a vise is a good compromise. Using the hammer for only one end of the pole, while using the vise to flatten the other end gives some of the ease and speed of the hammer, while giving precision of the vise for keeping the ends parallel. Yet another possibility is to use the hammer to flatten one end, then to use the vise to partially flatten the other end (just enough to make sure the rest of the flattening will indeed be parallel to the fully flattened end), then finish flattening with the hammer. Still yet another compromise, one that I prefer most, is to use a hammer to flatten the pole ends, then use a vise to refine the work, to make sure the pole ends have no "bubbles".
Placement of the vertex holes requires the most precision of all of the steps in making the dome. To make sure the holes are precisely in the right place, consider subtle length-modifying effects (listed below), measure carefully, drill carefully and systematically, and allow for some play.
| Diagram showing the bending effect length modifier. (The angle depicted in the diagram is exaggerated for clarity of illustration.) |
When the ends of the poles are bent in order to mate up with each other at the vertices, the poles are effectively slightly shortened. This shortening must be accounted for. This would be added to the distance between vertex holes. The angle of the bend will be about 16 to 18 degrees, depending on which pole you are bending.
(Derivation of angle: The sum of exterior angles of a polygon is always 360 degrees. The polygon forming the great circles of this dome is 10-sided. For a 10-sided regular polygon, each exterior angle will be 36 degrees. Each edge will have to bend by half that angle to meet the adjacent edge. This works for the geodesic polygons, but not for other angles, but the other angles are approximately the same. For a more exact computation of angle, have a look at the list of vertex coordinates by clicking on the bottom view schematic image.)
If the bend is made d inches before the target vertex point then the edge would be shortened by a length of
where the factor of 2 comes from the fact that there will be bends at both ends of the conduit.
For d=1 inches, the length change due to bending is 3/32 inch.
For d=2 inches, the length change due to bending is 3/16 inch.
For d=3 inches, the length change due to bending is 9/32 inch.
The edge lengths are the distances between vertices. The actual lengths of the conduit for each edge will have to include extra length to allow for room for the bolts to go through. The diameter of the bolts will be around 1/4 inch or 1/2 inch. Leave extra space for making oversized holes and to have a margin for structural integrity. About 1 or 1.5 inches should be sufficient. This length is outside of the distance between the vertex holes.
Use a heavy-duty, industrial bit which is designed to drill through metal, preferably zirconium-nitride coated, and apply lubrication (such as Tap Oil) to the drilling area to keep the drill bit sharp.
The pole end will be bent at a place a distance "d" (the same distance "d" used in the calculation of the bending effects, above) inside, from the center of the vertex holes. This is the place where the bend will be.
Using the vise, bend the end of each pole. For the frequency-2, class-I dome, the bend angle should be 18 degrees. Use a speed square to measure the angle, locking the rotating arm at the 18 degree location for easy measurement. Do this at both ends of each pole. Make sure the bends are in the same direction for each pole. I.e. the poles should end up mildly C-shaped, not mildly Z-shaped.
