US 2905113 A
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Sept. 22, 1959 R. B. FULLER 2,905,113
SELF-STRUTTED GEODESIC PLYDOME Filed April 22. 1957 3 Sheets-Sheet 1 INVENTOR. RICHARD BUCKMINSTER FULLER Armen/m. v
Sept. 22, 1959 R. B. FULLER SELF-STRUTTED GEODESIC PLYDOME 3 Sheets-Sheet 2 Filed April 22. 1957 INVEN RICHARD BUCKMINST ER F ER Sept. 22, 1959 R. B. FULLER 2,905,113 Y SELF-STRUTTED GEODESI PLYDOME Filed April 22, 1957 I 5 Sheets-Sheet 5 i AMA@ En U5 n B IN VEN T0 RICHARD BUCKMINSTER FUL ATTORNF sphere.
United States Patent Patented Sept. 22, 1959 iitice SELF-STRUTTED GEODESIC PLYDOME Richard Buckminster Fuller, New York, N.Y.
Application April 22, 1957, Serial No. 654,166
15 Claims. (Cl. 10S-1) The invention relates to geodesic and synergetic construction of ydome-shaped enclosures.
Summary principal characteristics which distinguish it Ifrom the` older architectural forms; so these characteristics will here be reviewed only briey. For a comprehensive review, reference is made to Patent No. 2,682,235, aforesaid.
In geodesic construction, the building framework is` one of generally spherical form in which the longitudinal centerlines of the main structural elements lie substantially in great circle planes whose intersections with a common sphere form grids comprising substantially equilateral spherical triangles. [Great circle planes are defined as planes whose yintersections with a sphere are great circles. Such planes pass through the center of the The earths equator and the meridians of the globe are representative of great circles in the ordinary accepted `meaning of this term.] The grids can, for example, be formed on the faces of a spherical icosahedron. Each of the twenty equal spherical equilateral triangles which `form the faces of the icosa is modularly divided along its edges. Lines connecting these modularly divided edges in a three-way great circle grid provide the out-line for the plan of construction. Each of the smaller triangles formed by the three-way grid is approximately equilateral, i.e. its sides are approximately equal. The extent off variation in length is determined tnigonometrically or by graphic solution of the grids as drawn upon the modularly divided edges of an icosahedron outlined upon the surface of a scale model sphere. It will be found that at each vertex of the icosa ve of the grid triangles form a pentagon, Whereas elsewhere throughout the pattern the grid triangles group themselves into hexagons, this being one of the distinguishing characteristics of .three-way grid construction.
My present invention arises in the discovery that when perfectly flat rectangular sheets are shingled together in a three-way grid pattern and are fastened together Where they overlap in the areas of the geodesic lines of the pattern, a new phenomenon occurs: there are induced t in each flat rectangular sheet, elements of live cylindrical struts defining two triangles of the grid edge to edge in diamond pattern. The effect is to produce a three-Way 2 what we may for simplicity term a self-strutted geodesic ply'clome. The hat sheets become inherent geodesic; they become both roof and beam, both wall and column, and in each case the braces as Well. They become the Weatherbreak and its supporting frame or truss all in one. The inherent three Way grid of cylindrical struts causes ythe structure as la whole to act almost as a membrane in absorbing and distributing loads, and results in a more uniform stressing of all of the sheets. The entire structure is skin stressed, taut and alive. Dead weight is virtually non-existent. Technically, we say that the structure possesses high tensile integrity in a discontinuous compression system.
Description With reference to the accompanying drawings, I shall now describe the "best mode contemplated by me for carrying out my invention.
Fig. l is a perspective view of `a geodesic plydome embodying my inventionin a preferred Iform.
` Fig. 2 is a detail perspective view of a portion o-f the Fig. 1 construction overlaid upon a diagrammatic representation of a three-Way grid `as an imaginary projec- -tion of the induced strutting of the dome. The area comprised is representative of one full face of .the icosa with adjacent one-third sectors vof adjacent faces. Combining the one-third-sectors lying at each side of the respective meeting edges of the 'adjacent faces, We get three arge diamonds; and
Fig. 3 is an enlarged detail view of the sheets which go to make up one of these large diamonds. Here the sheets are shown as they would appear when laid out flat and before they rare fastened together.
` Fig. 4 is a view similar yto Fig. 3. Imagine that this big diamond is now a part of the completely assembled dome, and notice how the structure has inductively produced ve struts in each of the sheets.
Figs. 5 to S, inclusive, show icosa segments of several modied constructions in which pyramidal groupings of the triangular grid faces defined by the induced struts produce in and out convolutions of the spherical surface. In these several constructions the apexes of the pyramids define one sphere and the bases of the pyramids another. Which of the two spheres is the larger depends on Whether the apexes of the pyramids project outwardly or extend inwardly. The sides of the pyramids may ybe regarded as struts connecting elements of the -inner and outer spheres and thus creating a truss.
Fig. 5 follows the `same sheet arrangement as in Figs. 1-4. Because of the convoluted, or involute-evolute construction, we get hexagonal and pentagonal pyramids (pentagons at the vertexes of the icosa), which for simplicity I term a Bhexpent coniiguration. Here the apexes of the hexes 'and pents project outwardly (or upwardly from the plane of the drawing). Notice that a strut is induced along the short axis of each sheet.
Fig. 6 shows a modified hexpent pattern in which the sheets toe in to the apexes of the pyramids.
Fig. 7, like Fig. 5, has the same sheet arrangement as in Figs. l-4. Here the induced geodesic triangles of the sheets form an inverted tetrahedron at the center of each face of the icosa, and one of the induced struts extends the long way of each sheet.
Fig. 8 has a sheet arrangement which may be compared to that of Fig. 6, but with one of the struts extending the long way of the sheet there is formed a pattern of inverted hexpent pyramids.
The construction shown in each of Figs. 5 to S inclusive may be turned inwardly or outwardly. For example if we think of Fig. 5 as representing the outer surface of a dome, we have pyramids projecting outwardly with their apexes in an outer sphere and their bases (or the corners of their bases) in an inner sphere. Or if We Yicosa triangle.
'large diamond 'toward its edges.
different` sheet markings, or types of sheet.
think of Fig. 5 as representing the inner surface of a dome, we have pyramids extending inwardly with their apexes in an inner sphere and their bases (or the corners of their bases) inQan' outer sphere.
ledges of theadjacent' faces. Thuse area OSVT combines afoheethird-fsector-of -icosa triangleRST, namely the sector OST with aone-third sector TSV of the adjacent l call ithe combined `sector areas large diamonds. i It is' helpfulto see the'largefdiainondswhen `analyzing thestructure'as a whole, because, -once the eye becomes practiced at picking them out,both `thepattern of the icosa faces and of the induced three-way strutting is 'more easily discerned. This is especially so -in lthe 'cases of Figs. 1 4, r Fig. 5 and Fig. 7, in each ofwhich all of the sheets are arranged approximately parallel to the major axis of the large diamond. This Vbrings the major axes into focus, outlining the icosa faces. Then the eye finds the center of the icosa face, further identiliableV by the small triangular opening at O, surrounded by a series of kite-shaped openings at the meeting edges of adjacent large diamonds and by square' openings at the edges of the icosa triangle. It is suggested that a brief study of these' characteristic formations with reference to Fig. vl
will be of much help in acquiring a'general grasp ofthe v geodesic alignment of the sheets themselves,and later vof the induced geodesic three-way grid struttingacross -the corners and centers of each sheet. u
Now, if we are." proceeding by the graphic solution method, we first layout the-icosa faces on a scale model sphere, then divide the edges of one of the `faces into the desired number of equal parts, ormodules, whichdetermines what I call the frequency of the 'three-way grid. For example, inFig. 2 I have shown the dot and dash line ST divided into six ymodules numbered l to 6 for identiiication, providing a six-frequency grid. With the three edges of -thelic'osa -face so' divided it is necessary only to 1 join each point of one edge'with every ysecondpoint on pattern of rsubstantially equilateraly triangles. Now we lay out thefsheets' on the grid pattern as shown in Fig. 2, centering `the short axisof each sheetV on alternate grid lines-and working -outwardly from the major axis of a With the frequency of six we get irst a row of three sheets in spaced end-to-end arrangement, a row of two sheets at either'side of this,
and overlapping at the corners, and finally a vsingle sheet coming up to the center Vof veach icosa face, as clearly shown in` Fig. 2.
Notice that the longitudinal centerlines of the sheets (see the representative centerlines a and bin Fig. 2) -lie substantiallyalongV great circles of the-sphere, or lie substantially ingreat circle planes whose intersections with'a common sphere `formgrids `comprising substantially equilateral triangles. The sheets-are nowmarked for interconnection along the klines of the three-way grid previously laid out. These lines of interconnection will be found` to be substantially normal to the aforesaid intersections. Thus the line of'interconnectionmarked onthe three-way grid at a is normal to centerline intersection aand that-marked at bv normal to b, etc. It will be found that the'markings for'interconnection of the sheets will varyf1'rom'oneA sheet to another depending uponits position in the pattern. Thenumber of diiferentsheet markings depends'upon the `frequency of the grid. With the frequency of six shown in Figs. l-4 therevvill be three It is desirable to label,` or color-codethesheets to show how they are to be put together.
351portions yof the shaded'areas. `In"Fig.4l the effecthas been considerably exaggeratedin order to bring out'ithe Vpoint. The fself-'struttingphenomenontakes place during assemblyof 'the sheets according` to their coding'ta'nd fastenings, following the designs laid out as above. Such perforations are shown bythe black dots in a sheet at the lower right of Fig. 3. Notice that additional perforations are provided near the corners of the sheets so that the sheets will be fastened together both in the areas of the grid lines and also at points substantially removed from said lines. This not only buttons down the corners of the sheets, but assists importantly in creating the induced :struts in -the 'completed structure.
Turning nowr to-Fig.V 3, we see the sheets for a large diamond as they would appear when laid out -flat and before they* are fastened together. `The 'shadedareas at the outer overlapping corners show the amount of increased overlap which occurs when'the sheets are brought into position for fastening them together. Once they have been brought into position and fastened, the sinuses 7, 8, etc., between the grid line markings close up and the structure-'assumes its'desired spherical form. -Concurrently, there are induced'in eachlflatv rectangular sheet, elements of ve cylindrical struts dening two triangles of the geodesic grid edge to edge in diamond pattern.
-Asshown'in Fig. 4, four of these struts cross the corners 'ofthe' sheet' andthe ifthy extends the-short way `of the 'sheetto form the base of the two Vgeodesic triangles.
These struts, as may be 1discerned from 'theshadng, iare tentv of overlap of the sheets, thickness vof 'the `sheetsand "possibly other factors. `In some cases the radius of the -bendmay be so4 large that the struttingis not clearly "visible;or is"perhaps only /visiblejto'a practicedV eye. I
'hav'ehad the draftsrnan try to' simulate -vthephotographic appearanceof the particular dome represented in Fig. l,
where the geodesic strutting shows u p in the high-lighted fastening 'them together in the designated areas of the grid lines andat their corners kas markedforfactory- "drilled for the' fastenings. When Fig. 4l is'imagined asa 'part of the completely assembleddomd'a comparisonof Figs. 3 and 4 will helpto give an'idea ofthe inductive struttingaction. Fig.'3 is a lstatic assembly of related parts'which know the three-way -geodesic grid' pattern of 'the ficosa; Fig. 4 a dynamicresolution lof the'pattern "into (a) 'spherical form, (b) with inherent struts express- 'ing Vthe pattern in' terms of gentle bendsk in the'sheets,
each bendcomprising elements of a cylindrical surface. It 'seems remarkable that the bends locate themselves, at
least in part, even in the double thickness' of the overlapping corners of the sheets Where it might have been supposed that the stiffness of the double thickportions would suggest a greater resistance to bending. This resultimplies strongly that the inherent structuring ofthe geodesic grid pattern is so'natural and strong'in'its tendency to produce a perfect self-supporting sphere that it departs from behavior patternspredicted fromordinary -principles of mechanics and strength of materials. 5 Since Ithe behavior of the system as a whole is unpredicted lfrom 'its parts,we say that the resulting structure is synergetic. Such"structures are vastly stronger, `pound"=for "pound, "than any heretofore known.
The curve of the bends in the sheets, variable according'to the vfactors named in 'the preceding paragraph,l
may comprise elements of a circular cylind'eror elements -of a cylinderof varying radius. This is-to say, the radius *ofcurvatureof aparticular strut neednot be uniform. vTosorne degree this 4factor mayv beifinfluenced by the leverage imposed by-the overlapping areas where the vvv`side ofthe geodesic-line, but the strut will in every case remain substantially a true geodesic linein the sensethat its `axis will liel in aiplanewhse intersection with `a sphere is an element` of a greatcirc1e. The strut itself becomes a chord of that sphere. v
To keep the drawings clear and readable, the fastenings have been omitted, except as the holes for them have beendepicted in Fig. 3 and as the geodesic grid lines used in locating them are shown in Figs. l2 'and 3. The fastenings themselves may be of any conventional type, and in some constructions it would befeasible to use adhesive means for holding the sheets together in the same geodesic alignment.
The sheets may be` of anydesired material, such as plywood, aluminum, steel, plastics, plastic-coated wallboard, composites of plywood and aluminum, plywood and aluminum sheet or foil, etc. I have found that marine plywood `in standard ,sheet sizes has ,excellentcharacteristics for induced strutting.
If desired, the openings between the sheets can be closed up, this being merely a function of the selected frequency of the grid in relation to sheet size. The proportions of the sheets also are subject to variation, but I recommend adherence to substantially a three to live ratio between width and length as giving best results for most building purposes. It is even possible to use sheets of other forms than rectangular, but an essential advantage of my construction is that it permits the use of plain rectangular sheets which are so readily available, stack so compactly for shipment and are least expensive. If the openings between the sheets are not closed up by the boards themselves, they may be used as skylights, and I have had good results with the use of thin skins of transparent mylar plastic for covering the openings. In some cases it may be desired to use an overall plastic inner or outer lining to weatherproof the dome; or weatherproofing may be secured by sealing the joints with plastic compounds or tape, and painting. Also, the overlapping of the sheets one upon another can be arranged so that the entire structure is weathershingled to shed water outwardly and downwardly over the surface of the dome. Such shingling of the sheets can also be arranged to cover the openings where they come together, or additional sheets can be slipped in to shingle over the openings.
By laying out the three-way grid pattern so that the radial lines of the polygons are longer than the lines forming the sides of the bases of the polygons, we obtain the hexpent and tetrahedral forms of the multiple-sphere trussed constructions explained Ain the outline description of Figs. 5 to 8, inclusive q.v. Thus in Fig. 5 we have, in a four-frequency grid design, a typical hexagonal pyramid with its apex at the center of the icosa face RST, this pyramid being formed by three sheets, and the two induced triangles of each sheet making two sides of the pyramid. Pentagonal pyramids occur at each vertex of the icosa. The pattern is one comprising hexagonal and pentagonal pyramids the apexes of which define an outer sphere, and the corners of the bases of which define an inner sphere.
In Fig. 6, we again have a pattern of hexagonal and pentagonal pyramids, but here six sheets toe in to the apex of a hexagonal pyramid and five sheets toe in to the apex of a pentagonal pyramid. In both this view and Fig. 5, one of the induced struts extends the short way of the sheet and in this respect there is a similarity to the neutral, or one-sphere, form of Figs. l-4.
In Fig. 7, the induced geodesic triangles of the sheets form an inverted tetrahedron at the center of the icosa face, and one of the induced struts of each sheet extends the long way of the sheet to form the common base of the two induced triangles. Each triangle is two frequency modules wide, one frequency module high.
In Fig. 8, we again have a pattern of hexagonal and pentagonal pyramids, six sheets toeing in to the apex of a hexagonal pyramid and ve sheets toeing in to the apex .of a kpentagonal pyramid, and one of the induced struts extending the long'way of the sheet. 1 v
The terms and expressions which I have employed are used in a descriptive and not a limitingsense, and I have no intention of excluding such equivalents of the invention described, or of portions thereof, as fall with- ,in the scope of the claims.
`spherical triangles, said sheets being initiallyV flat and marked for interconnection' along lines substantially normal to said intersections, said framework 4being characterized by the fact that the sheets are fastened together in the areas of said lines and cylindrical struts are induced in the sheets defining two geodesic triangles in each sheet.
2. A building framework characterized as defined in claim l, in which said interconnected sheets are rectangular in form.
3. A building framework characterized as defined in claim 1 in which said interconnected sheets are fastened together also at points substantially removed from said lines.
4. A building structure of generally spherical form comprising overlapping sheets arranged in a geodesic three-Way grid pattern, said sheets being initially flat and arranged for interconnection along lines normal to the lines of the grid pattern, the sheets being fastened together in the areas of the lines of interconnection and cylindrical struts being induced in the sheets defining two geodesic triangles in each sheet.
5. A building structure of generally spherical form comprising rectangular sheets arranged in a geodesic three-way grid pattern on the faces of a spherical icosahedron with the sheets overlapping at their corners and fastened together at the overlaps and having induced cylindrical struts defining two geodesic triangles forming a diamond in each sheet.
6. A building structure according to claim 5, in which the arrangement of the sheets on the faces of the spherical icosahedron is this: in the diamond formed by geodesic lines joining common vertexes of a common side of adjacent spherical faces with the centers of said adjacent faces, the sheets are arranged in parallel rows aligned with the major axis of said diamond.
7. A building structure according to claim 6, in which the induced geodesic triangles of the sheets form a pattern of hexagonal adn pentagonal pyramids the apexes of which define an outer sphere and the corners of the bases of which define an inner sphere.
8. A building structure according to claim 6, in which the induced geodesic triangles of the sheets form a pattern of inverted hexagonal and pentagonal pyramids the corners of the bases of which define an outer sphere and the apexes of which define an inner sphere.
9. A building structure according to claim 6, in which the induced geodesic triangles of the sheets form an inverted tetrahedron at the center of each face of the spherical icosahedron and one of the induced struts of each rectangular sheet extends the long way of the sheet to form the common base of said two geodesic triangles.
l0. A building structure according to claim 6, in which the induced geodesic triangles of the sheets form a tetrahedron at the center of each face of the spherical icosahedron and one of the induced struts of each rectangular sheet extends the long way of the sheet to form the common base of said two geodesic triangles.
l1. A building structure according to claim 5 in which the geodesic triangles of the sheets form a pattern of hexagonal and pentagonal pyramids, six sheets toeing in *Fuller -AJune 29,11954 UNITED STATES PATENT OFFICE CERTIFICATE OE CORRECTION Patent No.- 2,905,113 September 22, 1959 Richard Buckminster Fuller It is hereby certified that error appears in the printed specification of' the above numbered patent .requiring correction and that the said Letter s -f Patent should readI as corrected below.
Signed and sealed this 22nd day of March 1960.,
KAEL H. AXLTNE ROBERT c. WATSON Attesting OHicer Commissioner of Patents