About Structures that Approximate the Sphere.

The regularity we take as granted regarding the tessellation of a planar surface is a recent acquisition when discretizing the sphere. The paradigm for this new approach to spherical structures, the Geodesic Dome -a mesh of bars and nodes laid out as the edges and vertices of a sphere’s inscribed polyhedron- is undoubtedly the most characteristic architectural constructive typology  of the 20th century’s second half.

Geodesic Meshes: Fuller’s Model.

Few are aware that the first structure of this type was conceived in a date so far away as 1922, not by Buckminster Fuller but by Dr. Karl Bauersfeld, Chief Engineer for Carl Zeiss’ factory in Jena (Figure 1, left). Its purpose was to shelter his first planetarium prototype. The celestial sphere was reproduced through the projection of a mosaic of images projected from the spherical dome’s center. Equivalent areas of the dome’s inner surface should correspond to each image. To achieve this, the celestial sphere is transferred to the faces of the polyhedron obtained through the truncation of the vertices of an icosahedron. This same line of thought lead Bauersfeld to the conception of his supporting structure, based on the triangulated subdivision of the faces of an icosahedron (Figure 1, right). This structure, once covered with a thin concrete layer, resulted in the first reinforced concrete shell structure known. Curiously enough, Bauersfeld can thus claim the authorship of both the constructive icons of the 20th century: Reinforced Concrete Thin Shell Structures for the first half of the century, Geodesic Domes for the second half.
Figure 1. 1922. Zeiss Planetarium Dome before being covered with concrete and Bauersfeld’s sketches showing his proposed tessellation for the celestial sphere’s image [12].

Fuller arrives to essentially the same solution thirty years later following a path analogous to that of Bauersfeld. Since the late thirties he is interested in alternative representations of the earth globe. The projection of the globe on a flat surface entails problems similar to those which Bauersfeld had to solve: transferring a spherical (at least in appearance) surface to a mosaic of planar surfaces, in this case the faces of a Cuboctahedron (Figure 2, left).In this map, that Fuller named “Dymaxion”, a correspondence is established between the location of a point on the sphere and its position on the plane by referring it to a grid of greater circle arcs whose gnomonic projection on the corresponding planar face of the polyhedron is a straight line. If we compare the triangular grid included in the Dymaxion Map [6] patent documentation (Figure 2, right) and the Jena planetarium structure (Figure 1, left), the similarity is surprising.The Bauersfeld dome had been patented, but only as a shell structure, with no mention to the reinforcement’s geometry. This allowed Fuller to assume in 1951 the paternity of “a building framework 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 defining substantially equilateral triangles”. By this time Fuller had abandoned the Cuboctahedron as the base polyhedron, using, as Bauersfeld had done, the Icosahedron as his structure’s starting point.

Figure 2. Dymaxion Map, in its original version using the Cuboctahedron as base polyhedron, and the tessellation proposed by Fuller for the projection on one of the triangular faces of the Cuboctahedron.

Geodesic Structures were conceived by Fuller during his stay as a visiting professor at Black Mountain College [2] during the summers of 1948 – 1949. Fuller describes in U.S. Patent 2,682,235 [7] a way to calculate the edge lengths which is adequate enough for his art school students:

“One way of determining the strut lengths is to construct a paperboard hemisphere to a scale of, say, 1 inch to 1 foot, and lay out the vertexes of one of the faces of a spherical icosahedron on its surface. These vertexes are next connected by drawing great circle lines (spherical straight lines) therebetween. The edges of the triangle defined by these lines are next divided equally into the number of units represented by the selected grid frequency. The division points are then connected by drawing great circle lines in the manner clearly shown by Figs. 1 and 2. (Note that the points along one edge are connected to every second point on another edge.)
We have now a completed three-way grid pattern. Finally the length of the chordal struts is measured directly with the use of ordinary draftsman’s dividers, allowance being made for the strut fastenings.”

Such a procedure, which obviously satisfies their needs, lacks the precision demanded by real-world building techniques. This is where Computer Science comes in as a new instrument in the Architect and Engineer’s toolbox. The initial effort towards the systematization of the procedures for the design of geodesic tessellations of the sphere was undertaken under the leadership of Joseph D. Clinton in the Southern Illinois University (Carbondale Campus). These studies, that took place between 1964 and 1971, were sponsored by NASA on the belief that this type of structures had possible applications in future space missions. As a result, Clinton published [3] eight methods (with their corresponding computer programs) that could be applied to two different topologies for which he proposed the names of Class I and Class II. The methods described by Clinton can also be applied to Regular Polyhedra with triangular faces other than the Icosahedron.

All of these methods start with a subdivision of the edges of one of the base polyhedron’s faces (or a part of this face) to obtain a tridimensional mesh the vertices of which are transferred by means of a Gnomonic Projection to the surface of the sphere. These methods differ in:

• The topology, that may be of Class I (mesh directions parallel to the face edges) or Class II (directions perpendicular to the edges).
• The tessellation being calculated upon the whole face or upon its Schwartz Triangle (a sixth part of the face, 1/120 of the sphere in the case of the Icosahedron.
• The division of the edge into equal lengths or into unequal lengths derived from equal central angles in the polyhedron (obtaining this way greater circle arcs equal in length).

Due to the symmetry conditions characteristic of these regular polyhedra, calculating one single face is all that is required for the total definition of the structure’s geometric properties.The classification scheme proposed by Clinton was redefined by H. M. Wenninger [15] who added a Class III taking into account the skewed topologies revealed by Caspar and Klug’s [9] investigations about the geometric properties of virus capsules.

Other Polyhedral Structures:The Wester and Yacoe Models.

Professor Ture Wester [17] arrives to the structural model he defines as Plate Structures applying the transformation described by Wenninger under the name of polar reciprocation in his book Dual Models [16]. The Wester Model is derived from the Geodesic bar and node model by applying the principle of duality (Figure 5, left and center). Dual polyhedra possess the same number of edges, but the number of vertices and faces are interchanged. These faces and these vertices fulfill the requisite that for each face of n edges in a certain polyhedron, its dual polyhedron will present a corresponding vertex where n edges meet.

The Geotangent Structures proposed by Yacoe and Davies in 1986 [18] can have two types of faces that are regular polygons on the equator and on the north pole of the sphere), and at least half of the other faces being non equilateral Pentagons or Hexagons. The sphere that is approximated by this kind of polyhedron is tangent to each edge in only one point, its section with the face’s plane being a circle inscribed in that face (Figure 5, right). Each one of these circles is tangent to those sphere section circles inscribed in adjacent polygonal faces. The procedure used for the calculation of the form an dimensions of the faces that is described in the patent document is very complex, reaching a solution through a lengthy iterative process.

Inversion of a Plane on the Sphere.

Reexamining Coxeter’s “Introduction to Geometry” [4], we find a figure (see Figure 6) that can be revealing of the differences between the approach we are dealing with in our research and the classic method for the design of spherical meshes. Two paradigms are here compared: Stereographic versus Gnomonic projections. The figure on the left represents plane σ and the sphere σ’. This sphere results from the inversion of this plane with respect to an inversion sphere (not shown in the drawing) of radius NS centered in N. For each point in plane σ there is a corresponding a point on the sphere σ’ that is the inverse of plane σ with respect to the sphere of radius NS. The inverse of point N is the point at the infinite. In this case we are considering a Stereographic Projection.
In the Gnomonic Projection represented on the right side of Figure 6, instead of projecting from N, the projection is done from the center O of the sphere. In this case each point on the plane is represented by two points on the sphere, P1 and its antipodal P2. The circles on the sphere are inverted in the Stereographic Projection as circles on the plane and viceversa. In a Gnomonic Projection greater circles are projected on the plane as straight lines.

Figure 7 represents the same Stereographic Projection shown in Coxeter’s illustration, but this time upside down. The sphere C_ai with respect to which the inversion is made (omitted in Figure 6) is now a sphere of radius 1, centered in the coordinate system’s origin. The z=1 plane (and consequently points P₁, P₂, P₃, … P_n) are inverted on the sphere C of radius ½, tangent to plane z=1 on its north pole, that has its south pole in the coordinate system’s origin.

The alternative procedure we are presenting assumes a set of points in plane z=1 that are transferred to the radius = ½ sphere through a Stereographic Projection. In this case, one single plane is capable of representing the sphere as a whole, the South Pole being the point at infinity. The mathematical treatment can be found in references [10], [11], [14].

An Approximation to the Three Space Mesh Models.

Classic studies on polyhedra resort to the way this solid relates to the sphere. In this sense, we can speak of three spheres related to a polyhedron:

• The Circumsphere on which all of the vertices lie.
• The Insphere, which all the faces touch.
• The Midsphere, which all of the edges touch.

In a regular polyhedron we can identify these three spheres. But the space meshes we are dealing with are not regular polyhedra. Each of the models we are studying relates to the sphere it approximates in one of these ways:

• The Fuller (Geodesic Structure) Model to the Circumsphere.
• The Wester (Panel Structure) Model to the Insphere.
• The Yacoe (Geotangent Structure) Model to the Midsphere.

Up to this moment we have described the Space Meshes generation process as if it dealt only with the inversion of points in the z=1 plane on the sphere of radius = ½ applying the MINV transformation. But for each Model type these points subject to inversion represent something different. Their geometric interpretation will depend on the type of mesh we are trying to create:

• In a Geodesic Structure (Fuller’s Model), the points (lattice nodes) will be considered as the vertices of the triangular faces of the approximating polyhedron.
• In a Panel Structure (Wester’s Model), they will represent those points where the polyhedron’s faces touch the Insphere.
• In a Geotangent Structure (Yacoe’s Model) they will correspond to the points where the polyhedron’s edges touch the Midsphere.

The Fuller Model.

In the case of Fuller’s Model, only the interconnection mode of the set of points S in plane z=1 remains to be defined. For a Polyhedron with triangular faces, the connectivity diagram for the set of points would be given by the Delaunay Triangulation. The faces of the approximating polyhedron are generated by joining the points resulting from the MINV transformation according to the connectivity scheme thus obtained (Figure 8, right). This correspondence is especially evident in Figure 8 (left), where the Delaunay Triangulation of a set of points in plane z=1 reproduces the Schlegel Diagram of an Icosahedron. This solid will be generated by the stereographic projection of these points, in this way interconnected.

The Wester Model.

Regarding the Wester Model, two differences should be made clear: the representation of the edges and vertices is now the product of calculating the Voronoi Diagram generated from the set of points S in z=1. Now the approximating polyhedron’s vertices do not lie on the sphere. Instead, the point inverted from the z=1 plane represents a face’s tangency point with the Insphere. The corresponding point in z=1 is a generator P of the Voronoi Diagram. Vertex V’ on the face will be the point where the line connecting the center of inversion O with vertex V of the Voronoi Diagram intercepts the tangent plane (Figure 9, left).

The Yacoe Model.

The generating schema for the Yacoe Model will be a radial circle packing in plane z=1. Figure 10, left, shows the packing from which we will obtain the polyhedron shown in Figure 5, right. If we consider that the radical axis for two tangent circles is its common tangent [13], we can generate a planar tessellation SD (Power Diagram) from any packing of tangent circles (Figure 10, center and right). Being SD contained in plane z=1 and applying an inversion with center in the coordinate system’s origin and a power of 1 (Figure 11, left):

The tangency points T between the radical axes and the circles are transformed into the points T’ on the sphere E centered in (0, 0, ½) and radius = ½.
Edges a’1, a’2 and a’3, tangent to the sphere in points T’1, T’2 and T’3 (points which define a plane α that intersects the sphere), meet in point V’ (Figure 11, right). These edges are the generatrices of a cone circumscribed to the sphere, and this cone’s vertex (at the same time a vertex of the approximating polyhedron) is the pole of plane α with respect to the sphere E.


This way we demonstrate that, by means of the procedure we have proposed, the Delaunay Triangulation, the Voronoi Diagram and the Power Diagram generated from a circle packing, will generate polyhedra approximating the sphere that will in turn reproduce the Fuller, Wester and Yacoe structural models.

Approximating Other Quadrics: The Paraboloid Ω.

It has been proposed in [10] a very simple variation of this procedure that can lead to the design of structural meshes approximating other quadric surfaces. Let us consider now the projective transformation MHOM, expressed in homogeneous coordinates as:
where we refer as (X Y Z T) to the point transformed from (x y z t). If we consider a set of points S={P₁, P₂, P₃, … P_n} in plane z=1, the Revolution Paraboloid type structure is obtained by applying to S the following sequence of transformations:
Generation of the Hyperboloid and Ellipsoid of Revolution.A procedure such as described that includes the inversion of a set of points in plane z=1 and the projective transformation MHOM may also be applied [11] to a sphere whose radius is not equal to ½. The south pole of this sphere is (0, 0, 2z_c – 1). If inversion is applied from here, the inverse points of plane z=1 fall on the sphere CG. If the projective transformation MHOM is then applied the sphere will be transformed into the Quadric of Revolution CG. The nature of the resulting surface (Figure 12) depends on the projection center’s position:

• When zc =½, CG will be a Paraboloid (Figure 12, left).
• When zc <½, CG will be a 2 Sheet Hyperboloid (Figure 12, center).
• When zc >½, CG will be an Ellipsoid (Figure 12, right).

Skewed Quadric Surfaces.

Thus concluding that the variation of the center’s position along the OZ axis (and consequently the sphere’s radius) will result in different quadrics of revolution we will now examine the effect of keeping the sphere tangent to plane z=1 but removing two restrictions [11] [14]:

• its radius restricted to being equal to ½.
• its center restricted to being on the OZ axis

This new situation may be reduced to the former one simply by applying to this new system (the sphere and the set of points P in the z=1 plane) a sequence of displacement and scaling transformations before applying the projective transformation MINV. Inverse scaling and displacement transformations will return the inverted set of points P’ to the original sphere’s size and location. Then the MHOM transformation will yield a set of points P’’ which will lie on the surface of a variety of skewed quadric surfaces of which the mathematical characterization is a theme for further study.

The Quest for Regularity.

As we stated in the introduction to this paper, geodesic structure design methods such as were published by Clinton and others assure obtaining what we could describe, quoting Fuller, as a “generally spherical form” composed by “substantially equilateral triangles”. A procedure aimed at obtaining similar results by defining the generator points in z=1 has been the subject of extended deliberation among our fellow researchers.
The substitution of Euclidean metrics by Elliptic metrics has been experimented with certain success [4] in defining the point layout for the base triangle, distribution which can then be extended to the rest of the plane through Möbius transformations (Figure 14).

But the non-spherical surface approximating polyhedra generated from the same set of points in z=1 that will produce an acceptable tessellation of the sphere will display a great variety in face sizes. This lack of regularity is usually deemed as a disadvantage from both structural and aesthetic reasons. A line of research we are pursuing at this moment deals with the possibility of defining geometric distributions of the generator points that will result in an increased regularity of the non-spherical quadrics approximating polyhedra compensating in length and area its faces. Through the use of Hyperbolic Geometry reproducing the well-known layout of Poincaré’s Disk (Figure 15, left) a possible approach has been demonstrated. Although the goal of reducing the size and area variation in faces is approximated, further studies are needed to reduce the number of different polygons generated (Figure 15, right).

Other Spaces.

Research in relation to the problems posed in the previous section leads naturally to question the effect of considering spaces different from the Euclidian space. Recent papers [5] have described the use of pseudo-Euclidean or Minkowskian spaces.
The real radius spheres in pseudo-Euclidean space have the appearance of one sheet hyperboloids, those with purely imaginary radius are represented as hyperboloids of two sheets and those with zero radii take the form of cones of revolution. In addition to Euclidean and Minkowskian spaces [19], semi-Euclidean spaces can also be considered, in which circular and hyperbolic cylinders stand for spheres centered on the origin of coordinates. Applying the stereographic projection of the sphere associated to them will produce the planar models (Figure 16) proposed by Poincaré for the study of the Cayley-Klein geometries.

Introducing Complexity.

The fact of dealing with a set of points lying in a plane as the source for our 3D structures suggests that it would be a simple task to cut out flat patches [4] [14] of any desired form to obtain in turn the corresponding delimited section of the resulting 3D polyhedron. A variety of these 3D forms could then be assembled in such ways as suits the designer’s imagination (Figure 18). Globular Domes such as proposed by Otero [9] in 1992 using the more orthodox Parallel Trihedra procedure are an example of this kind of structures. But the formal possibilities derived from the freedom in selecting the 2D shapes for these patches are a plus.
This possibility coupled with the ability to generate a variety of skewed quadric surfaces points at a field of research in which mathematicians, architects and artists can join forces in the creation of up to now unsuspected structures, both utilitarian and aimed at the pure joy of the spirit. The Voronoi Diagram – Evolutionary Space [1] Art Exhibition held in Seoul in June 2006, in which along others, the outcome of this line of research was featured, served as a showcase for the wide spectrum of possibilities Computer Science offers for the generation of Architectural forms through Computer Science.
Figure 17. Techniques for the generation of quadric surface approximating patches.