In 1956, my good friend and excellent sculptor Bill Underhill organized a team of architecture students to design and build an all-aluminum geodesic dome birdcage, which is still extant in Merritt Park in Oakland, CA. It was funded by the Kaiser [Aluminum] Foundation, and Don Richter of Kaiser was our engineer. The following writeup can be found on the localWiki https://localwiki.org/oakland/Geodesic_Bird_Dome
The Geodesic Bird Dome is a part of the Lake Merritt Wildlife Refuge in Oakland, California. Built in  with Kaiser Foundation supplied materials, it was used for years as an exhibit cage for a variety of wild birds, but in later years has been utilized as a cage for sick and injured birds. 1
The birds in here have long been considered unwell and poorly kept. It is not clear if the birds are actually being warehoused or if this is a sanctuary.
A plaque on it says “designed by Buckminster Fuller”, which is incorrect. Fuller did the math behind geodesic domes and did much to popularize them, but was not the inventor of them. This particular dome was designed by William Underhill, Gordon F. Tully, Dick Schubert, Dan Peterson, and Marshall K. Malik, who were architecture students at UC Berkeley.
As I worked out the geometry and saw the project through to the end, I thought it worthwhile to set down the details of how it was designed. I have drawn over the photograph of the dome to help explain its geometry. The meanings of the lines, dots and diamond shapes are explained in the text.
Geodesic domes are based on the icosahedron, a 20-sided figure that is one of the five “Platonic” solids, the only convex solids with equilateral faces. The other Platonic solids are the tetrahedron, with four triangular faces, the cube with six square faces, the octahedron with eight triangular faces and the dodecahedron with 12 pentagonal faces. Bucky Fuller’s riveting lectures on the relationships among the Platonic solids inspired the dome, and the geometry continues to fascinate both Bill and myself (see the Wikipedia entry on Platonic solids at https://en.wikipedia.org/wiki/Platonic_solid )
Five triangles meet at each of the 12 vertices of the icosahedron. By spotting these five-spoked intersections, you can figure out the geometry of any geodesic dome. In the diagram, these intersections are shown with large white dots. In a complete icosahedron, there are 30 edges connecting the 12 vertices. The diagram highlights ten of these edges with yellow lines; our dome has a total of 20 such edges.
The vertices of all Platonic solids lie on the surface of an imaginary circumscribing sphere. The icosahedron is chosen as the basis for geodesic domes because it has the most faces, and they are all structurally rigid triangles. However, an icosahedron creates a crude, pointy structure with long edges. Since the edges of the underlying icosahedron become the straight members that form the dome’s structure, the larger the dome, the longer the edges. So for both practical and aesthetic reasons, it becomes necessary to subdivide the edges of the icosahedron to create shorter members and additional vertices which, when brought out to the surface of the imaginary circumscribing sphere, make the dome more nearly spherical.
The number of times each edge of the underlying icosahedron is subdivided is the “frequency” of the dome. To create reasonably sized members and screen panels for the dome, we chose to make it a “third frequency” structure, with each icosahedral edge subdivided into three segments. You can see in the diagram that each yellow edge is broken into three segments that bend outward. A complete sphere subdivided in this way has 180 triangular faces, 90 edges and 80 vertices.
Even if you subdivide the edges of the icosahedron into three equal lengths, the triangles will not be equal in size (if they were, this would be a Platonic solid with 180 sides, which does not exist). The best you can do in a third frequency design is to have two edge lengths and two sizes of triangles.
As we conceived of the dome to be a flight cage, it made sense to maximize its volume. We did this by designing it as a bubble-shaped three-quarter sphere, incorporating 15 of the 20 icosahedral faces. Each face of the icosahedron is subdivided into nine triangles. If you do the math, you will find that our three-quarter sphere, third-frequency structure has 135 triangular faces, 75 edges and 65 vertices.
Tables were available later on to aid designers in calculating the lengths of the struts. Not having such tables, I did the the complex spherical geometry calculations on a Marchant electro-mechanical calculator owned by the Oakland Park Department, in whose office we did our design work. Marchant calculators were much faster and more sophisticated than any others on the market at the time. (Wikipedia records that the firm was bought by Smith Corona in 1958 and the new firm, unsuccessful in switching to electronics, was gone by 1980).
The ingenious aspect of the design was suggested by our engineer from Kaiser, Don Richter. His idea was to join pairs of triangles to form 65 diamond-shaped panels (plus five infill triangles at the base). The two triangles forming each diamond bend around the dome like the covers of an opened book, and so are not in the same plane. When the screen is stretched across the diamond, it naturally forms a doubly curved surface (a “hyperbolic paraboloid” or “hypar”). The screen panels are cut so that one set of screen wires runs from end to end of the diamond, curving outward, while the perpendicular strands run across the diamond, curving inward. As opposed to a flat screen, which has to deform before it can take pressure, our screens are already bent and can resist pressure immediately from either side. The diagram below shows how it works – naturally the screening is closer woven than is shown in the diagram.
The doubly-curved screening also contributes to the strength of the dome. Richter had constructed such a structure for Kaiser Aluminum out of a 1/16th inch thick corrugated aluminum panel, reinforced at the edges by beams. It was about 20 feet corner to corner. They tested the structure, which supported 18″ of sand before one of the beams buckled (it buckled upward, showing that the failure was due to compression and not bending). You can make a crude model by folding a piece of aluminum foil into corrugations, then flattening and doubling over the corrugations on opposite edges to create the edge beams. The foil pops into a hypar shape – a neat party trick.
There are two sizes of diamond-shaped panels. 20 symmetrical panels cut across the middle of each icosahedral edge, shown in light yellow in the diagram. The other 45 panels form the corners of each icosahedral face, shown in light red. These panels are asymmetrical. The diamond panels meet in the center of each icosahedral face, forming six-pointed vertices.
Across the center of each diamond, we designed a tubular strut to complete the triangular structural grid. If the strut were straight (like the spine of the folded book), it would lie outside the screening. Amadee Sourdry, our supervisor at the Oakland Park Department, vetoed this configuration because the exposed struts would form a perfect jungle gym, which would be an attractive nuisance and therefore a liability to the city. We were forced to put the struts on the inside of the screening, bending them inward to stay inside the curved screening. Each end of the tubular strut was flattened and bent, then bolted to the inside flanges of the C-shaped frame members that form the edges of each diamond.
The screen is clamped between the outside flange of the C-shaped frame member and a neoprene gasket held in place by (as I recall) a 3/8″ x 1″ aluminum plate, all secured by closely-spaced bolts. The outside flanges of the diamond panel frame members is bent inward so that the flanges of two adjoining frame members lie in the same plane, allowing them to be bolted snugly together. At the points of the diamonds, the bottom flanges are likewise bolted together. The complex construction of the screen panels and struts made it necessary to hand-craft all the members and hand-assemble each screen panel. If you built hundreds of these, you could use specialized machinery to do the job. This being a one-off project, we instead relied on a wonderful Polish metalworker. The cross section through a typical screen frame member shows all these parts.
After the foundation was poured, assembly of the dome took one long day, after which we celebrated with a spaghetti dinner. Unfortunately, Bill had been drafted into the army and could not be there. The dome was assembled from the top down, suspended from a crane that raised it up as diamond panels were added and bolted to adjacent panels. The weight was supported only at the top (I don’t recall whether they used a single cable or five separate ones, as the pictures of the dome under construction have disappeared).
As a result of the limited number of supports while the dome was being assembled, bolts began to pop as sections were added near the equator , causing considerable alarm. Luckily, enough survived until the dome was set on its foundation, where it was held up at a sufficient number of points that it became (more) structurally sound. I speculate that any dome which is more than a hemisphere is structurally suspect because it wants to bulge at its equator.
The bulbous configuration of the dome was a result of it being designed as a flight cage, with a small footprint and a large volume. Only well into design and construction, did we learn from Sourdry that it would house waterfowl instead of perching birds. It would have made more sense to widen the base somewhat by eliminating the bottom layer of triangles.
Five triangular infill panels were required at the bottom to complete the enclosure, one of which made a natural entry. The blue triangle in the diagram shows the nearest such panel – the actual entry is on the opposite side. To keep the birds from escaping, an entry lock was needed with a door at each end. Because the panel leans toward the ground, if the panel was used as a door, it would have to open inward and upward.
[The following paragraph was revised in February 2017] Instead, we built an awkward-looking but serviceable tunnel between two rectangular doors, transitioning to fit into the triangular sector as it passes through the dome. My memory here is somewhat fuzzy, but you can check out the actual solution by visiting the dome (which I haven’t done since it was built).
All this is still vividly imprinted on my memory 60 years after its construction in 1956. It was a most rewarding experience, one of the highlights of my career.
PS: I can’t resist inserting a reference to an extraordinary item I found on the web at https://bfi.org/about-fuller/resources/everything-i-know/session-11 , a verbatim transcript of one of Bucky Fuller’s long lectures. He gave several of these lectures while I was at Berkeley, and held his audience from 2 PM to 11 PM, with a break for dinner. In it there is a reference to Don Richter, which I insert below. I also found a patent filed by Richter in 1955 and granted in 1959 for the corrugated hypar roof that he had tested. at https://www.google.com/patents/US2891491
One of my boys at the Institute of Design in Chicago was Don Richter. Don was an extraordinary man and he stayed with me during all the early years of the developing of the geodesic dome, after he graduated from the Institute of Design. He had been a sailor in the Merchant Marine during the war. Please hold the pictures for a minute. Don’t do anymore with them for a second. And Don wanted to really go on. Many architectural students asked me what they ought to do, and I would say, what I think you ought to do is to get production engineering. And the only way you can do that, to really get it first class, would be in the aircraft industry. Don did work for a while with Kaiser Aluminum and he then got a job in Texas with the Republic Aircraft. They were building an enormous bomber and he began he did so well in general engineering that he did get into production engineering, and he lived with the Head of the Production Engineering and developed extraordinary capability.
Don, then, Kaiser Aluminum Company were looking for somebody with design capability and I recommended Don and he went to them, and Don had made his small geodesic dome of aluminum and had it on his desk. He made it at home, and brought it in one day and put it on his desk, and Henry Kaiser, old Henry Kaiser walked by the desk and he thought this was a Kaiser product and he simply said, “I’d like to have one of those built for Hawaii,” and he had just been building a big hotel out there, and so everybody just takes Henry’s orders and so they had to make deals with Don, and there was a great deal of negotiating from there on`. The Kaiser patent attorneys came in to get license from my patent attorney.