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Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere

Edward S. Popko and Christopher J. Kitrick
A K Peters/CRC Press
Publication Date: 
Number of Pages: 
[Reviewed by
Jeff Ibbotson
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 George Bernard Shaw once observed that “the British and Americans are two great peoples divided by a common tongue”. Having read several of Buckminster Fuller’s books on Synergetics as well as being trained to write professional mathematics, this reviewer previously concluded that the same was true for design thinkers and mathematicians. And yet they often study the same subject matter when it comes to geometry. This lovely book is written to supply a bridge between these groups and does a wonderful job appealing to all quarters. 
The treatment begins with a nice introduction to spherical polygons and the connection between choosing points on a sphere and constructing a subdivision grid. This is the main target of the book – to examine the coordinatizing process for spherical surfaces. This theme is returned to throughout the book and six basic means of dividing the surface are examined in detail. The next few chapters exhibit some interesting examples of the breadth of applications of this process. These include “kissing problems” (mutually tangent spheres), structure of pollen grains, virus geometry, climate modeling, celestial catalogues and so on. Two applications which stand out for their unusual quality are: (a) placing of dimples on golf balls, (b) Uber’s “hexagonal hierarchical geospatial indexing system”! 
The story aspect which ties the book together concerns the work of R. Buckminster Fuller and his invention of “Synergetic Geometry”. Fuller was an unorthodox thinker who developed a vision of the universe in terms of new concepts which united design thinking, geometry and the process nature of the universe. Along the way he created a number of concepts such as “vector equilibrium”, synergy and omnidirectional halo. The meaning of these terms is often somewhat elusive (probably since Fuller’s understanding developed in time through his interactions with his students). The root of his quest is tied to the creation of geodesic domes – structures which exhibit stability, rigidity and tension and provide efficient means of enclosing space. Along the way to discovery, Fuller created the “Dymaxion projection” which allows for better maps of the earth’s surface through use of polyhedra and the projection of their vertices onto the globe. In the case of the dymaxion map it is the vertices of the cuboctahedron which serve as starting point. The projection of edges supplies a geodesic network which is then supplemented by altitudes of the faces and then a system of lesser (non-Great) circles which are mutually perpendicular. The 25 edges created allow for a single right triangle (which Fuller calls the “LCD”) that tessellates the sphere. This system did not, in fact, constitute a sufficient framework for a geodesic dome and so Fuller quickly changed the model to start with an icosahedron. The development of these ideas is explained along with Fuller’s teaching and work at Black Mountain College  and the North Carolina College of Agriculture and Engineering (Raleigh, NC). M. Bromberg described Fuller’s pied-piper presence this way:
Bucky was highly intuitive and seductive with an audience. With thick glasses and fleeting stares, he seemed to quickly 
find the soul of the people he was with. He seemed to be a thinker and con artist at the same time, creating
an atmosphere of ‘we’re off to see the wizard’.
An unfortunate side of Fuller’s character led him to ignore the important “disciples” who really made the process of dome-building work: J. Fitzgibbon, Duncan Stuart, Shoji Sadao, Kenneth Snelson and others played key roles in the transition from Icosa-31 (the system of coordinate lines described above) to the eventual hexagonal-pentagonal panel design and the use of the triacontahedron (which has 30 diamond faces, 32 vertices and 60 edges). They are treated here much more fairly. The breakthrough design of “Weatherbreak” (1950) outside Montreal produced a dome which took 48 man-hours of work to set up and weighed 1140 pounds. This stiff and lightweight structure attracted the attention of the military and agriculture and made clear the connection between portability, strength and lightness in Fuller’s vision of building living spaces on “spaceship earth”. The story of his continued success with various other large installations (the 1953 Ford Rotunda Dome, the famous installation at the 1967 Expo in Montreal and Spaceship Earth (1981) at the Disney World Epcot Center) are all included here with details aplenty.
The development of a subject that mathematicians would recognize as spherical geometry begins in Chapter 4. The altitude-azimuth coordinate system is described and terminology developed: spherical polygons, caps, lunes, dihedral angles, spherical excess and defect all appear. There is no formal presentation of spherical trigonometry (although the law of cosines appears in a few places) and although the imagery of surface normal vector fields and the rudiments of the Gauss mapping are explained in picture form there is precious little analytic geometry or calculus. The order of presentation is also nonstandard – kissing problems are discussed before Platonic and Archimedean solids, dual polyhedra are only briefly mentioned and symmetry operations on polyhedral and spherical polyhedral occur with no mention of groups. Schlaffli symbols are not mentioned until Chapter 8 and the curious “triangulation number” appears with a hand-wave toward the Euler characteristic but not much more. Any potential instructors planning on using this as a text for an upper-level geometry course should take a close look at these differences. On the one hand, the text is that much more accessible to the casual reader although the reasoning sometimes used is quite different than one might expect in a mathematics text. 
The final chapters take a deep dive into the six different subdivision schema and focus on honeycombs and icosahedral models, rotegrities, nexorades and reciprocal frames. The subject of machine learning appears in a discussion of “self-organizing grids”. Ideas are presented alongside some algorithms utilizing matrices and optimization. These algorithms are presented in pseudo-code and it is assumed that the reader knows how to implement them in an appropriate programming language. Several appendices address stereographic projection and coordinate rotations in 3-D (again with pseudo-code). This reader found the writing fairly dense in spots and the increase in synergetic terminology made it slower. 
The strength of this book lies in its illustrations. They are beautiful and draw one into a deeper examination of the actual surface geometry of a sphere as well as that of the immersion of the sphere in three-dimensional space. This central dichotomy is present all the way through the book and constitutes a type of “picture explanation” of spherical trigonometry. Is this equivalent to the customary development in most books? I don’t think so (the motivation of navigating on the surface of the earth is not addressed at all) but this does represent, I think, another way of visualizing and working with geometrical ideas on a spherical surface. I would certainly recommend the book to anyone interested in Fuller’s work as it supplies a more stream-lined and less jargon-filled account. If the aim of the book is to excite students and present Synergetics in a more traditional context then by any measure this has been a successful endeavor. 


Jeff Ibbotson is the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.