Projecting stars, triangles and concrete. The Early History of Geodesic Domes, from Walter Bauersfeld to Richard Buckminster Fuller

Martino  Peña Fernandez-Serrano; José Calvo-López

22 architectura Band 46 / 2016 1. Walther Bauersfeld, hemispherical dome using a system of metal bars in Jena, 1922

23 architectura Band 46 / 2016 Martino Peña Fernández-Serrano and José Calvo López Projecting stars, triangles and concrete The Early History of Geodesics Domes, from Walter Bauersfeld to Richard Buckminster Fuller Sometimes artefacts that had been designed with a sci- entifc purpose turn into cultural icons that infuence art and architecture, mainly because the viewer’s rela- tionship with the work and the very act of receiving it is quite different. In this way, the receiver extracts same of the properties of the original artefact and uses them in other prototype that appears in a completely different way to our eyes. This is the case with the frst hemispherical dome using a system of metal bars connected by pin-joints (fg. 1) made by the Zeiss Company headquartered in Jena, Germany, designed by its chief engineer Walter Bauersfeld, and its recep- tion some years later by the artistic and architectural avant-garde. The Wonder of Jena In 1913 the company Zeiss was entrusted by the Deutsches Museum in Munich with the construction of a projector that would reproduce the movements of the stars and planets on a hemispherical dome. From the very beginning, the nature of the assignment was going to change it into a cultural artefact. So far this kind of building was known as Sternentheater or Sternenschau [›Star Theatre‹ or ›Star Show‹]. They could accommodate few people, mainly due to the concept of the show. The planetarium, as known so far, was based on a fxed projector; the celestial dome was rotated to simulate both the movement of the 2. Walther Bauersfeld, Projector in the Zeiss Planetarium, 1926

24 architectura Band 46 / 2016 stars as the earth. Such technique was applied in the so-called Atwood Celestial Sphere, designed by Wal- lace A. Atwood, director of the Museum of Science in Chicago. He designed in 1911 a spherical planetarium with a diameter of 4.57 meters, illuminated from the outside. 1 This type of planetarium could not accom- modate a large audience. Since the projector is fxed, it involves a quite complex mechanism in order to rotate the dome to simulate the movements of stars, planets and the earth itself. For the Munich dome, Bauersfeld flipped the concept, casting images from a mobile projector on a static surface. He designed a machine (fg. 2) that could rotate around its axis, casting images from 32 small projectors that reproduced the motions of the stars. The problem he faced was to divide the area of the projecting head into 32 fat surfaces for the individual projectors. The regular polygon with most faces is the icosahedron, with 20 triangular faces and 12 vertices. Bauersfeld truncated the vertices of the icosahedrons, getting a solid with 32 fat faces, in particular, 20 hexa- gons stemming from the original icosahedron and 12 pentagons resulting from the cutaway vertices. Also, a circumference of the same diameter can be inscribed in both the pentagonal and hexagonal faces. This prin- ciple simplifes the production and the placement of lenses, so it has been subsequently used by Zeiss and other companies. Some documents in the Archiv Carl Zeiss called Kugelunterteilung show how Bauersfeld divided the sphere into 20 faces and later on, into 32 sections. This document, dated from 1 st April 1919, has eight pages dealing with the division of the sphere, labelled as Ku1 through Ku8. In Ku1 (fg. 3) a note states: »Geomet- rical bodies consist of pentagons and hexagons. Every pentagon is surrounded by fve hexagons. The fgure hat 32 faces (20 hexagons and 12 pentagons), 12 × 5 + 20 × 3/2 = 90 edges and 12 × 5 = 2 × 90/3 = 60 vertices.« 2 In this way Bauersfeld invented the star projector. 3. Walther Bauersfeld, manuscript, Ku1 in Kugelunterteilung, 01.04.1919 1 Cf. Krausse 2006. 2 »Körper aus regelmäßigen 5 Ecken und 6 Ecken bestehen. Jeder 5º Eck ist von 5 6º Ecken umgeben. Der Körper hat 32 Flächen (20 Sechsecke, 12 Fünfecke) 12 × 5 + 20 × 3/3 = 90 Kanten und 12 × 5 = 2 × 90/3 = 60 Ecken.« Bauersfeld 1919, 1.

25 architectura Band 46 / 2016 The next problem that Bauersfeld had to solve was the construction of a hemispherical projecting surface where he could test the new projector. He needed a structure approximately 16 meters in diameter; it should resist inclement weather conditions for a cou- ple of months without being neither too heavy nor too expensive. They decided to build it on the roof of the building number 11 of the Zeiss factory complex in Jena (fg. 4). At frst, they had considered solving the problem with a cloth surface, such as the ones used in circus covers. They gave up this idea soon, since canvas, like all textile materials in the period, was very expensive due to high infation. By contrast, steel, a German product, was relatively cheap; thus, they con- sidered the possibility of constructing the hemisphere as a grid of metal rods. 3 However, this construction system did not guar- antee by itself a smooth surface in the inside nor the desired protection from the weather conditions. Tech- nicians of the company Dyckerhoff and Widmann (DYWIDAG), which had already constructed other buildings in the complex of the Zeiss factory in Jena, suggested that the best solution was to project a thin concrete shell onto the hemispherical grid of metal rods, acting thus as a kind of formwork or centring. In order to materialise the geometric form of the hemispherical metal bar frame, Bauersfeld used the same subdivision scheme he had devised for the sur- face of the projecting head, as two series of drawings known as Ku1 and Ku2 in the Zeiss archive shows (fg. 5). Reusing these ideas, Bauersfeld »calculated a spatial network construction in which the iron fat bars with a cross-section of 8 × 20 mm and a length of approx. 60 cm were held together at the intersections by specially developed node connections. The almost 4,000 bars with 50 different lengths [...] The complete network weighed only approx. 9 kg/qm including the projection surface.« 4 However, Bauersfeld needed to divide the sphere into smaller parts in order to use more manageable bars in the construction of the hemispherical dome, rationalising the process in order to get smaller units. Thus, he began to project the icosahedron into the sphere, but he had 20 triangles that were still too big. He also knew that each of the 20 triangles of the spherical icosahedron could be divided into six equal triangles so that the geometrical fgure would have a total number of 120 equal triangles. This implies that he already had a model where he could fnd out the lengths of the bars so that they will be equal in the 120 triangles. This triangle was named by Bauersfeld as Charakteristisches Dreieck [characteristic triangle]. 3 Cf. Breidbach 2011, 57. 4 »Er berechnete eine räumliche Netzwerkkonstruktion, bei welcher die eisernen Flachstäbe mit 8 × 20 mm Querschnitt und ca. 60 cm Länge an den Kreuzungs- stellen durch speziell entwickelte Knotenverbindungen zusammengehalten wurden. Die fast 4.000 Stäbe mit 50 verschiedenen Längen, bei Zeiss gefertigt, besaßen eine Genauigkeit von 1/20 mm ihrer Länge. Das komplette Netzwerk wog einschließlich Projektionsfäche nur ca. 9 kg/qm.« Kurze 2011, 65. 4. Walther Bauersfeld, Planetarium in the Carl Zeiss factory in Jena, 1924 5. Walther Bauersfeld, hemispherical dome using a system of metal bars in Jena, 1922

29 architectura Band 46 / 2016 through an icosahedron inscribed on it. The icosahe- dron should be divided again in order to get smaller triangles reducing the dimensions of the bars required to build the dome. The next step is to fnd out the optimal relation- ship between the length of the bars and the factor K, that is, the number of parts in which each edge of a spherical triangle should be divided. In a sheet with the title K.f.P 38 it is written: Relationship between the longitude l and the number of triangles N in average angles. 6 The text hints that Bauersfeld was seeking relationships between these factors that he could use in order to change the weight of the construction or the numbers of bars. So he arrived at a mathematical formula with three terms, which he could change at will. L is related to the longitude of the bars, R is the radius of the sphere and K, as it is already known, is the total number of sections in which the icosahedrons edge’s triangle is divided. In this way, he fxed the ra- dius (R) and gave it a dimension of 8 meters. With this radius he got different results, from which he could choose different solutions. However, he went forward and decided to give to K the value 16 (fg. 7), so the results were those: With R = 16 and K = 16, L will be equal to 0,6 in meters Using the other mathematical formula N = K 2 20, 16 2 × 20 = 5120 triangles Number of edges = 3/2 N = 7680 bars. Anyhow, Bauersfeld was trying to build a hemisphe- rical dome; thus he only had needed half this number of bars, that is 7680/2 = 3840. This is the same number mentioned by Kurze in the article mentioned earlier, where he defned the longitude of the bars as 60 cm what is 0,60 m. Finally in the documents called Rechnung der Stab- langen für Kugelteilung [Bar length calculations for the division of the sphere], 7 Bauersfeld computed the dimensions of the bars which were going to be used for the dome; as we had seen, the bars actually were used as a formwork for the concrete that was going to be projected on it. The document includes 31 pages where Bauersfeld performs the same geometrical cal- culations and trigonometrical relationships between the different parts in which the characteristic triangle is being divided. Let us remember that the character- istic triangle is one of the six equal triangles in which the spherical triangle could be divided and there are a total of 120 of them in the spherical icosahedron. Bauersfeld knew this fact and he also was aware of it and what is more important, having the length of the bars of one of these characteristic triangles could have the total dimensions of the bars of the hemispherical dome. In the frst 24 pages Bauersfeld divided the edge of the characteristic triangle into four parts and computed the distances in degrees; later on, he translated these 6 Bauersfeld 1922a, Kuppelkonstruktion für das Projekti- onsplanetarium, K.f.P. 38. 7 Bauersfeld 1922b. 8. Walther Bauersfeld, manuscript, Stablängen für den Kuppel Neubau (8 m Radius), in: Kugelunterteilung- Rechnung der Stablängen für Kugelteilung, 1922

32 architectura Band 46 / 2016 Eisenbeton [Method for making domes and similar curved surfaces of reinforced concrete]. The memory stated, that »the new method consists in constructing a spatial network of iron bars in the roof cladding, supporting itself and part of the total intrinsic weight, and using lightweight formwork directly attached to the ironworks.« 9 This formwork should be positioned without introducing any tension that might deform the metal framework of bars. Also, no deformation was appreciated caused by bending stress, during the execution of the reticular frame. As mentioned above, the structural metal frame was seen as an auxiliary system to project the thin concrete shell and would be therefore hidden by it. There are no other data in the patent about how to build the reticular dome, or the way bars are joined to each other. It should be remarked again that the novelty of the method is that it is self-supporting and does not need any other aux- iliary element to build the hemispherical dome, in fact, »the new method avoids the need for costly underpin- ning, which is replaced by the formwork mentioned above, and also virtually eliminates the stresses of the equipment, which would otherwise have to be taken into account when dimensioning the thickness of the individual parts.« 10 Another patent granted to Zeiss and connected with the Jena dome deals with constructive aspects from a hemispherical reticular dome, specifcally the knots joining the metal bars (fg. 13). With the name Knotenpunktverbindung für eiserne Netzwerke, 11 the constructive knot is patented, in the same date, i.e. 9 th November 1922. The patent, which can be translated as ›Knots for a metal frame grid‹, describes a generic knot to join several bars; it is not specifed how many bars meet at one knot. The joints are made with two metal round plates with grooves where the bars are inserted as Bauersfeld had already drawn in his notes (fg 14). Once the bars are put into position, the knot would be closed by a threaded bolt, and the bars would be tightly joined with each other, since the node must be fully capable of transmitting forces arising from each bar. Bars with circular section have a slotted end that allows them to be inserted perfectly into the metal plates. Thus, the design of the system allows joining different numbers of rods at each knot, while the an- gles between bars and the metal plate may be different. The claims supporting the patent fling are those: »1. Node connection for iron networks, in which the rods are held together by two lateral plates inter- spersed with screw bolts and plates and rods engage in one another with a groove and pin, characterized in that the grooves of the plates run round in a circular manner all around. 2. A node connection according to claim 1, characterized in that the wedge pins of the rods do not fll the wedge grooves of the plates to the bottom.« 12 9 »Das neue Verfahren besteht darin, dass ein sich selbst und ein Teil des Gesamteigengewichts tragendes, in der Dachhaut liegendes räumliches Netzwerk aus Eisenstäben aufgebaut und unter Verwendung leichter, unmittelbar an das Eisenwerk angehängter Schalungen, beispielsweise durch das Spritzverfahren, mit dem zur Erreichung der vollen Tragfähigkeit erforderlichen Betonmantel umhüllt wird.« Firma Carl Zeiss 1922a, 1. 10 »Bei dem neuen Verfahren wird eine kostspielige Un- terrüstung vermieden, an deren Stelle die erwähnten Schalungen treten, und es fallen auch die Ausrüstungs- spannungen so gut wie vollständig weg, denen sonst bei der Bemessung der Stärke der Einzelteile Rechnung getragen werden muss«. Firma Carl Zeiss 1922a, 1. 11 Firma Carl Zeiss 1922b, 1f. 12 »1. Knotenpunktverbindung für eiserne Netzwerke, bei der die Stäbe durch zwei seitliche, von Schraubenbolzen durchsetzte Platten zusammengehalten werden und Platten und Stäbe mit Nut und Zapfen ineinandergreifen, dadurch gekennzeichnet, dass die Nuten der Platten kreis- förmig ringsum laufen. 2. Knotenpunktverbindung nach Anspruch 1, dadurch gekennzeichnet, dass die Keilzapfen der Stäbe die Keilnuten der Platten nicht bis zum Grunde ausfüllen.« Firma Carl Zeiss 1922b, 2. 13. Carl Zeiss Company, detail in patent nº 420823, Knotenpunktverbindung für eiserne Netzwerke, 1922

35 architectura Band 46 / 2016 As we have said, what turns a scientifc artefact into an architectural icon is the infuence of the position of the observer and how he or she is going to receive or perceive it. The students of the Bauhaus, who had visited the Jena Planetarium during construction ac- companied by their teachers, including Mogoly-Nagy, Adolf Meyer and Walter Gropius, were utterly fasci- nated by the artefact, despite it was, in fact, a form- work made of metal bars that was going to be hidden after the concrete was projected on it, just the skeleton of a building. This concept had an important impact on architecture, like Krausse argued: »The architects have observed natural structures as the physicians were fnding new patterns using the new x-ray technique. Mies van der Rohe has reduced to skin and skeletons his work in the Friedrichstrasse. Behind the costume and the dress, the true and the construction courage is being sought. All of this is converged when using the x-rays as a theory in an ordinary conversation.« 15 As X-Rays allowed to discover the inner structure of nature, students at Jena were dazzled by a building in skeletal form before concrete was projected. These rays had permitted scientists to observe how nature make its constructions and structures. Long before, a scientist who had worked in Jena, Ernst Haekel, had discovered the geometry of the radiolarian, a micro- scopic marine organism, which is about the same geometry that Bauersfeld used to shape the reticular dome using metal bars. The photograph immortalised by Mogoly-Nagy (fg. 15) shows the construction of the planetarium Zeiss in Berlin, it was built by Bauersfeld although he used a different system. Anyhow, Mogoly-Nagy commented about it: »a new way of occupying the space: a group of people above a foating and transparent network, like aeroplane squadrons in the sky.« 16 What it is really impacting the viewer is the fact that he could watch the fgures through the structures when he was staying at a lower level. There were no divisions between them, strengthening the illusion that the occupants are foating in space, as Krausse noticed too, when he said: »The transparent foating network allows people to fll the space freely, without contact with the ground, without any structural support. They are like tightrope walkers or fyers without any weight emerging in the space.« 17 That is how Bauersfeld’s artefact impacted strongly on the mind of Moholy-Nagy. He was not an architect, but he was deeply in touch with the profession; in his book Von Material zu Architektur he introduced con- cepts as light, energy, mass, volume and space through his experiments in photography and cinematography. These notions were actually extracted from different mathematical and physical felds. In these years many artists were infuenced by Albert Einstein’s theories around the fourth dimension. In Krausse’s words »[…] some of the avant-garde artist as Naum Gabo, El Lissitzky, Adolf Meyer, Siegfried Ebeling, László Mo- holy-Nagy and lately mainly Buckminster Fuller had strongly believed in the relationships, similarities and performances between the new scientifc revolution (through Planck, Einstein, Bohr etc.) and the change perceiving life and mainly the new way of understand- ing the profession in art and design felds.« 18 For Moholy-Nagy, the new scientifc discoveries were going to change our perception of the space, shift- ing the focus of architecture from static structures to dynamic ones. This involves movement relationships between the different spaces of a building, relating the outside closely with the inside, downstairs with upstairs, as well as connections between forces that are 15 »Die Architekten werfen Blicke auf Baukonstruktionen wie die moderne Kristallphysiker, die in den Interferenz- mustern der Röntgenspektrographien Raumgitter ent- decken. Mies van der Rohe reduziert den Bau mit seinem Hochhausproject an der Friedrichstrasse auf Haut-und- Knochen-Architektur. Hinter den Ver- und Umkleidun- gen wird die Wahrheit und Kühnheit der Konstruktion gesucht. All das fokussiert sich im umgangssprachlichen Ausdruck des Röntgenblicks.« Krausse 2006, 67. 16 »eine neue fase der Besitznahme von raum: eine men- schenstaffel in schwebend durchsichtigem netz, wie eine fugzeugstaffel im äter.« Moholy-Nagy 1929, 235. 17 »Das schwebend durchsichtige Netz vermag die Menschen frei im Raum zu halten, ohne Bodenkontakt und ohne irgendeine Stütze. Wie Flieger oder Artisten scheinen die Menschen fast schwerelos im Raum zu agieren.« Krausse 2006, 71. 18 »[So …] glaubten einige der avantgardistischen Künstler und Gestalter, wie Naum Gabo, El Lissitzky, Adolf Meyer, Siegfried Ebeling, László Moholy-Nagy und später vor allem Buckminster Fuller an tieferliegende Beziehungen, Entsprechungen und Analogien zwischen der Revolution des wissenschaftlichen Weltbildes (durch Planck, Einstein, Bohr usw.) und der dramatischen Ver- änderung der sinnlich erfahrbaren Lebensumstände und besonders der auf Wahrnehmbarkeit gerichteten Praxis in Kunst und Gestaltung.« Krausse 2006, 73.

36 architectura Band 46 / 2016 appearing as interactions of bodies. Plasticity among space, »since, in architecture not sculptural patterns, but spatial relations are the building elements, the inside of the building must be interconnected, and then connected with the outside by spatial divisions. The task is not completed with a single structure. The next stage will be space creation in all directions, space creation in a continuum. Boundaries become fuid, space is conceived as fowing-a countless succession of relationships.« 19 The reception of the dome in the Avant-garde in the sixties through Richard Buckminster Fuller Bauersfeld and Zeiss were not aware of the importance of their invention. As Tony Rothman said in Science a la Mode. Physical Fashions and Fictions their patents are not very explicit nor too extensive. 20 By contrast, the artefact had a major reception in the avant-garde architecture of the mid 20 th century, through Richard Buckminster Fuller. Zeiss did not protect the triangulation system itself, but rather the concrete-projection system and the bar-joining knot. In the patents, the reticular metal structure had a sec- ondary function, and it was referred only as a form- work helping as stress frame. The invention may be considered a form of serendipity, that is, the develop- ment of events by chance in a happy way. It was Fuller 19 »[…] da bei der Architektur nicht plastische, bewegte Figurationen, sondern die räumlichen Lagerungen das Bauelement sind. So wird das Innere des Baus durch seine räumliche Gliederung in sich und mit dem Außen ver- bunden. Die Grenzen werden füssig, der Raum wird im Fluge gefasst: gewaltige Zahl von Beziehungen.« Moholy- Nagy 1929, 222, cf. Moholy-Nagy 1947, 63. 20 »The language of the patent is not very precise but is clearly made out to the Zeiss Company of Jena and clearly refers to the technique used to build the Jena dome. To a logician it follows that the patent is for a geodesic dome. But law is not logic and I am not a lawyer; therefore, I will comment no further on the matter. The term ›geo- desic‹ was applied to domes by Buckminster Fuller, who received a U.S. patent in 1954.« Rothman 1989, 59. 16. Richard Buckminster Fuller, Building Construction, Patent nº 2.682.235, 1954

37 architectura Band 46 / 2016 who took up the idea, making it one of the architec- tural paradigms of the 20 th century, turning a tech- nical artefact into an architectural icon. In Krausse’s words: »in some artefacts like the Planetarium, we can see a convergence of different independent felds like exacts sciences, technical inventions and artistic achievements, all of them fnally coming together in a single theoretical discourse.« 21 In 1948 Fuller began to teach at the Institute of De- sign in Chicago; that summer he was invited to give seminars at Black Mountain College in North Caro- lina, an innovative educational institution where John Cage and Merce Cunnigham taught seminars while Anni and Josef Albers brought the Bauhaus tradition to America. In summer 1948 he tried to build a dome made from aluminium strips with his students. This frst attempt was unable to stand, so it was known as ›Supine Dome‹. Next year Fuller went back to Black Mountain. With the help of an engineering team, he tackled the construction of a geodesic dome using al- uminium bars and vinyl sheets; it stayed erected until it collapsed in September. Years later, Richard Buckminster Fuller patented an invention that had some points in common with the one from the Zeiss Company. Fuller used almost the same dimensions and the same system as Bauersfeld had employed to build the frst planetarium on the top of the building 11 in the Jena factory. Rothman talked about the patent from Fuller too, which is called Building Constructions (fg. 16) and deals with ›Geodesic Domes‹, a term introduced by Fuller. He described the invention as a framework for enclosing space; materialised by a spherical form where the longitudinal centre lines of the structural elements lay in diametral or great circle planes. Several geometrical fgures may be projected into the sphere and Fuller discussed this issue. He talked about the spherical tet- rahedron, the spherical octahedron and his preferred solution, the regular icosahedron (fg. 17). All three share the same properties: »As we have learned, there are only three prime structural systems of Universe: tetrahedron, octahedron and icosahedrons. When they are projected onto the sphere, they produce the spherical tetrahedron, the spherical octahedron and the spherical icosahedron, all of those angles corners are much larger than their chordal, fat-faceted, poly- hedral counterpart corners […] they are projections outwardly onto a sphere of the original tetrahedron, octahedron, or icosahedrons, which as planar surfaces could be subdivided into high-frequency triangles without losing any of their fundamental similarities and symmetry.« 22 21 »In bestimmten Artefakten, wie das Planetarium eines ist, sehen wir eine Konvergenz des Denkens, das die unabhängigen Bereiche von exakten Wissenschaften, technischen Erfndungen und ästhetischen Innovationen einschließlich ihrer theoretischen Diskurse zusammen- führt.« Krausse 2006, 78. 22 Fuller 1975, 664. 17. Richard Buckminster Fuller, spherical icosahedrons, in: Building Construction, Patent nº 2.682.235, 1954 18. Richard Buckminster Fuller, LCD, in: Synergetics. Explorations in the Geometry of Thinking, New York 1975

38 architectura Band 46 / 2016 Fuller also knew too that each of the twenty equal spherical triangles might be modularly divided along its edges to generate individual bars; the numbers of them for each edge is dubbed ›Frequency‹ by Fuller. Also he called the process ›geodesic sphere triangulation‹ and defned it as »the high-frequency subdivision of the surface of the sphere beyond the icosahedron.« 23 He also was aware that each of these 20 triangles could be divided into six equal triangles; in fact, Bauersfeld had used a similar concept, giving it the name of »characteristic triangle«. Fuller defned it as spherical triangle LCD (fg. 18) which stands for Lower Common Denominator. Fuller projects the 20 triangles into the sphere in order to measure distances; also »the edges of each spherical triangle are modularly divided and are interconnected by the three-way great circle grids previously mentioned. These grids are formed of a series of struts each of which constitutes one side of one of the substantially equilateral trian- gles defned by the lines of the grid.« 24 The language used by Fuller is not very specifc; the patent uses term as ›approximate‹, ›substantially‹ or ›not precisely‹. This seems to arise from the fact that when the icosahedrons edge is projected into the sphere, and then the spherical icosahedron edges are divided into equal parts, three great circle lines will not concur in one point but two of them. It seems that Fuller used the same method as Bauersfeld did. First the icosahedron edge is divided in equal parts and then is projected into the sphere (fg. 19), giving as a result different bar lengths, as stated by Fuller, »in all of the form of framework I have described, the lengths of the individual struts are substantially equal, but not precisely so.« 25 Notwithstanding that, in Fuller’s prototype bars follow the great circle lines, that is, the geodesic lines of the sphere. Thus, the shorter path between two given points of the sphere is an arc of a great circle; in other words, the grid follows the most economical lines for energy to travel on the surface of the sphere. Thus load transmission is entirely optimised, so there is no waste of material or energy. In this way, nature is creating its own structures. With the projection of the icosahedron onto the sphere, Fuller achieved the geo- metrisation of the prototype. Frequency is the number of modules; when it reaches infnity, the modules will be points, and the spherical icosahedron will be a per- fect sphere. Although in mathematics geodesics means the shortest distance between two points, the term comes actually from the Greeks words ›geo‹ meaning earth and ›dasia‹ standing for division. In this sense, geodesics is the science of the division or measure of the earth; we may surmise that Fuller wanted to build a prototype having the earth as a model. This fts well into his conceptions, considering the big enclosures to cover big cities that he designed or the metaphor having the earth as a spaceship in his publication Op- erating manual for Spaceship Earth. 26 Such an interpretation of the earth as a model or prototype of the geodesic dome also connects Fuller with Bauersfeld, since the German engineer wanted to reproduce the sky, the celestial sphere in order to project the stars and planets on it. As Rothman said, Bauersfeld used the truncated icosahedron to approx- imate the geometric object to the sphere, while Fuller projected the icosahedron into the sphere, so it is referred to a two dimensional surface embedded into a three dimensional space: »The surface of a sphere is a two-dimensional manifold which lies in our or- 23 Fuller 1975, 664. 24 Fuller 1954, 2. 25 Fuller 1954, 4. 26 Fuller 1969. 19. Calvo, José. Projecting a triangle into a sphere.

39 architectura Band 46 / 2016 dinary three dimensional space. A geodesic dome is also a two-dimensional surface which is ›embedded‹ in three-dimensional space. The difference between the sphere and the dome is that the sphere’s surface is curved but the dome is composed of Euclidean trian- gles. The approximation to the curvature comes at the joints between the triangles. Regge calculus extends this idea to spacetime. Spacetime may be viewed as our three-dimensional space moving through time. In oth- ers words, a three-dimensional manifold embedded in a four dimensional space.« 27 Fuller achieved international recognition as the inventor of the geodesic dome, the prototype he pat- ented with the name Building construction in 1954. In its summary he wrote »I have discovered how to do the job at around 0,78 lb per sq. ft. by constructing a frame of generally spherical form in which the main structural elements are interconnected in a geodesic pattern of approximate great circle arcs intersecting to form a three-way grid, and covering or lining this frame with a skin of plastic material.« 28 Actually, this construction system is quite similar to the one devised by Bauersfeld. Both project the icosahedrons edge with their divisions into the sphere where the spherical triangles are placed as shown in fgure 19. As stated above, the frequency is the number of modules in which this edge is being divided. In the Fuller’s patent he decided to set the frequency at 16, »the number of modules into which each edge is di- vided is largely a matter of choice. In the framework of Figs 1,2,3 and 6, the number is 16.« 29 Thus, Fuller used the same divisions that Bauersfeld had employed to shape the prototype on the top of the building 11 of the complex Zeiss in Jena and in the projector that he invented too. Besides, the diameter the Jena dome was 16 meter (fg. 20), while Fuller stated in his patent, »the framework construction illustrated in fgs. 1 to 9 inclusive is representative of the best mode devised by me of carrying out my invention particularly as utilized in structures up to approximately 50 feet in diameter.« 30 50 feet equal 15,24 meters, slightly shorter than the diameter that Bauersfeld used. Fuller’s technical innovation was to lay the bars along the great circles of the sphere where the distances between two points are shorter, and energy travels in the more economical way, hence the name ›geodesic vaults‹. We may conclude that the geo- desic dome that Fuller patented in 1954 (fg. 21) is the same one that Bauersfeld built 30 years before (fg. 22), using the same geometrical process and almost the same dimensions. 27 Rothman 1989, 72. 28 Fuller 1954, 1. 29 Fuller 1954, 3. 30 Fuller 1954, 2. 20. Walther Bauersfeld, section of the dome with a radius of 8 meters, Carl Zeiss Archiv 21. Sketch showing the division of the Geodesic Dome from Fuller’s patent called ›Building Construction‹

Projecting stars, triangles and concrete. The Early History of Geodesic Domes, from Walter Bauersfeld to Richard Buckminster Fuller

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