geodesic domes
Introduction
A geodesic dome or geodesic dome is part of a geodesic sphere, a polyhedron generated from an icosahedron or a dodecahedron, although it can be generated from any of the Platonic solids.
Historical review
Richard Buckminster Fuller is considered the inventor of geodesic domes, as he held his patent in 1954. Fuller developed them in the 1940s, creating one of the best-known geodesic domes in 1967 at the Montreal World's Fair, 76 m in diameter and 41.5 m high.
There are earlier examples of geodesic domes, such as in the Imperial Palace of China (1885) or in the planetarium of the Carl Zeiss workshops (1922).
In the Imperial Palace of China (Forbidden City, Beijing), belonging to the Ming and Qing dynasties, you can see a sphere with a geodesic subdivision of an icosahedron. It is a sphere under the paw of a guardian lion, similar to another at the Summer Palace in China" (near Beijing), which dates from approximately 1885.
As for the planetarium of the Carl Zeiss workshops, it is a 16-frequency geodesic dome created by Walther Bauersfeld, which came to be called "the wonder of Jena." From this, many others were created, until the idea was developed by Fuller.
Geometric description
The faces of a geodesic dome can be triangles, hexagons, or any other polygon. The vertices must all coincide with the surface of a sphere or ellipsoid (if the vertices do not lie on the surface, the dome is no longer geodesic). The number of times that the edges of the icosahedron or dodecahedron are subdivided giving rise to smaller triangles is called the frequency of the geodesic sphere or dome. For the geodesic sphere, Euler's theorem for polyhedra is fulfilled, which indicates that:
Where C is the number of faces (or number of triangles), V the number of vertices (or multiple unions) and A the number of edges (or bars used). For a partial dome that is not a complete sphere, the following is true:
To construct geodesic spheres, the formulas for the radii of the dodecahedron or icosahedron are used. The radii allow the new vertices of the subdivisions to be raised to the surface of the sphere that will pass through the original vertices of the body.