DOME MAGAZINE: Summer 1995, Vol. 7 | No. 4
The language of geodesics is sometimes difficult to understand by the layman, so by default, this article will attempt to explain some of the common terms in dome description.
Frequency
This is a common term used to describe a property of a geodesic dome. For instance, a dome with a fixed radius, say ten feet, could consist of a small number of large struts, (low frequency), or a large number of short struts (high frequency).
When selecting a frequency for a dome of a given diameter, one tries to avoid large, bulky, expensive struts and panels (low frequency) as well as numerous flimsy, small struts (high frequency). Most domes have struts that are less than eight or nine feet in length.
We begin with a one-frequency dome. This is simply an icosahedron, a 20-sided geometric solid. Each face is an equilateral triangle between faces and on faces and all edges are equal (figure 1).
A two-frequency dome is a structure that is made by taking each icosahedron strut and dividing it into two parts and then inserting another triangle within the basic icosahedron triangle. This must be done in such a manner that each hub that connects that strut is touching the shell of a sphere that passes through all of the vertices of the original icosahedron (Figure 2 ).
A three-frequency dome is a structure formed by dividing the original icosahedron struts into three parts. The icosahedron triangles change (Figure 3, page 16). Each vertex, or hub, is still in the sphere generated by the basic icosahedron. By utilizing special cord lengths on our three-frequency domes, they have the ability to lie flat when truncated by the 4/9 and the 5/9 levels, without having to alter the base triangles.
A four-frequency dome is formed by dividing the icosahedron struts into four parts. The icosahedron triangles are shown. As with the other domes mentioned, each vertex or hub, is in the sphere generated by the basic icosahedron (Figures 3 & 4).
Truncation
We use different truncation numbers from those of some of the dome manufacturers. The numbers more accurately explain the lines where a sphere is cut off and set on the ground in building. The sphere is made up of bands or rows (horizontally) of triangles from the north pole to the south pole.
In a two-frequency sphere, there are six rows. In the drawing, three rows are shown; therefore, it is truncated at the 3/6 line, or halfway point (Figure 5).
The three-frequency sphere has nine bands of triangles from top to bottom. The example shows 5 bands. Therefore, it is a 5/9 dome. The 4/9 line is also flat and could truncate at the 4/9 level. A quick way to find out how many bands are in a dome is to multiply the frequency of the dome by three (Figure 6).
In order to maximize the advantages of a geodesic, it is necessary to select the proper frequency for the intended use, size and building materials. It should also be noted that the higher the frequency, the
more spherical the dome becomes. The frequency subdivisions also produce different truncations (cut-off points) on the sphere. Domes usually sit on one of the straight bands which make up the original framework (see Figure 7).
Truncation is usually displayed in fractions, which indicate what portion of the total sphere you are getting: e.g., 5/9 – more that half; 4/9 – less than half. Only even numbered frequencies truncate at the hemisphere (1/2). Determining the correct truncation for a specific application is important, both economically and spatially. A 4/9, 32-inch diameter dome has approximately 800 square feet of floor area and is 13 1/2 feet high, ideal for a small cabin or studio where limited loft space is planned. But if you need extra bedroom or office space, a 5/9, 3 frequency, 32-foot diameter dome will give you a 19 1/2-foot ceiling and enough room for a 400-600 square-foot second floor (see Figure 8).
The 2-Frequency Dome
The above description covers the geometry of domes produced by certain specific manufacturers. There are different dome designs that may differ from the above structures’ geometry, but the specific terms generally apply.
One of the problems of today’s society is the lack of adequate housing throughout the world. The two-frequency dome in the 26-foot and 29-foot diameter sizes represent an economical structure for low-cost housing. The basic dome kit requires 65 struts and 40 panels. These two-frequency domes have only two colors, two different lengths of struts and two different sizes of triangles, making them easy to build and assemble.
The 29-foot diameter dome on a permanent wood foundation, providing approximately 1,500 square feet of living space, can be built for a finished cost of less than $35 per square foot. These costs include the double-walled, energy-efficient design, as well as all necessary components to finish the home.
There are no patents on this design and it can be built from a basic hardware kit or from a basic kit that includes all of the lumber with the hardware attached, ready for installation.
Fuller’s Dream
Buckminster Fuller’s dream was to have housing for all the peoples of the world. We know that his dream is far from fulfilled. We feel strongly that the dome industry, with its collective experience and creativity, can bring about designs of efficiency and economics that will make that dream possible. This may not happen in the current political climate. Yet great changes have occurred in the world in the past few years that make this dream a possibility. If we, as dome advocates, continue with our focus and dedication to that dream, the change to make the dome the housing of choice may occur sooner than we think.