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Reference

Hugh Kenner, Geodesic Math and How to Use It, Berkeley: University of California Press, 1976 (first edition) and 2003 (just a new printing I think -- I haven't paged through a copy yet).

Preface

Over 25 years after its first publication in 1976, Hugh Kenner's pathbreaking and still very useful book, Geodesic Math, has been re-released. Here are problems I've found in the first edition. I've marked minor changes using bold face text. If any of these have been fixed in the reprinting, or you think I'm incorrect, please let me know. If you write me about other problems, I will consider putting them here along with an acknowledgment to you for bringing them to my attention.

I have separated the remarks here into two sections, Errata and Criticism. The first section contains items which I think would be universally agreed to be errors. The second section contains items which are a matter of opinion. All the quibbles aside, Geodesic Math is a very well-written book, and it taught me a lot about tensegrity and geodesics. The constant encouragement to build models should be taken seriously.

Amazon.com reports "Hugh Kenner (1923-2003) was Professor Emeritus of English at the University of Georgia. He is the author of dozens of well-known and highly regarded books of literary criticism, and is also the author of Bucky: A Guided Tour of Buckminster Fuller (1973)." He was known for his studies of modern authors, James Joyce and Ezra Pound in particular. Inquiries from Bob Sanderson and Bill Beecham prompted me to start this list. Eventually I plan to include a third section entitled "Notes" which will mainly be additional explanations of material in the book which seems obscure or can be simplified.

Errata

1. In the preface, Kenner states the mathematical prerequisites for understanding the book as "algebra and high-school trig". However, certainly the tensegrity part of the book (Chapters 1, 2, 3, 4 and 6) requires an understanding of some differential calculus concepts. People who don't know differential calculus, but are comfortable with algebra and trigonometry, can understand the geodesic chapters without understanding the tensegrity chapters. That being said, for people who do have a background in calculus and are interested in tensegrity, the tensegrity part of the book is well worth reading. Even if you don't know calculus, but are interested in tensegrity, there are things of interest which you can glean from those chapters. Thanks to Bob Sanderson for pointing out Kenner's omission of the differential calculus requirement.
   
   
2. In the last sentence of the preface, it is stated Kenneth Snelson was "unknown to Tony" (Tony being Anthony Pugh), but a consultation of Pugh's Introduction to tensegrity shows this is not the case as there are several places there (pages ix and 3 among others) where Snelson is specifically cited. Thanks to Val Gómez Jáuregui for pointing this out. Most likely Kenner's statement just needs to be qualified a little better and there probably was a substantial period of time when Pugh was unaware of Snelson's contributions.
   
   
3. In Chapter 1, at the very end of the appendix (p. 10 in my first edition of the book), his claim that there are no regular t-prisms beyond 3, 4 and 5 is incorrect. See http://bobwb.tripod.com/synergetics/photos/x6prism.html.
   
   
4. In Chapter 2, the first sentence should read "The three-strut Tensegrity described in Chapter 1 is asymmetrical, having triangular ends and folded rhomboids for sides." My American Heritage Dictionary of the English Language defines "rhomboid" as "a parallelogram with unequal adjacent sides" and says "rhomboidal" is the corresponding adjective. This is a planar concept, and what's being referred to here is not planar, hence my substitution for the word "rhomboidal" in the sentence above. The folded rhomboids corresponding to a three-prism's sides are folded along their long diagonals. Thanks to John Braley for pointing this mis-statement out.
   
   
5. In Chapter 2, on p. 14, Kenner states "The rightward and downward forces, represented by the heavy arrows, may be regarded as equal, like all the other sets of forces in the system. They are also at 90° to one another." I don't find the reasoning here clear at all. I prefer the calculus derivation for computing tendon lengths. You can find this in Chapter 4. Kenner never gives this structure a name besides "symmetrical 6-strut Tensegrity". Fuller, who developed this structure in 1949, called it a tensegrity icosahedron.
   
   
6. In Chapter 3, at the bottom of p. 20, Kenner's description of Diagram 3.1 is incorrect. It should say the equator, the Greenwich meridian, and the 90° meridian. The 180° meridian, which Kennner incorrectly references, is on the same great circle as the Greenwich meridian and so is not suitable here for what he is trying to describe. Thanks to Dave Welsh for pointing this out.
   
   
7. In TABLE 3.1, I get some slightly different values than he does:
   
   For the T-Cuboctahedron, I get 56.25° for the "Dip angle δ". In this case, γ is the dihedral angle of the tetrahedron, cos^-1(1/3). For the long tendon t₁, I get a value of 0.5951 rather than his 0.5983. For the short tendon t₂, I get a value of 0.5204 rather than his 0.5168.
   
   For the T-Icosadodecahedron, for the long tendon t₁, I get a value of 0.5433 rather than his 0.5442. For the short tendon t₂, I get a value of 0.5025 rather than his 0.5016.
   
   
8. In the appendix to Chapter 3, his derivation uses an approximation to get a formula for ED² which he doesn't seem to recognize as such. The angle between EC and AD is slightly more than ι. This is because the plane determined by DEC is not orthogonal to the radius going through A.
   
   
9. Also in the appendix to Chapter 3, my copy shows Eq. 3.4 as dip = sin/2(δ/2). It should be dip = (1/2)sin(δ/2).
   
   
10. In Chapter 7, Kenner encourages you (on p. 48) to make an icosahedron with twenty struts, but I think you'll need thirty.
    
    
11. Also in Chapter 7, on p. 53, second paragraph from the bottom, second to the last sentence, Kenner says there are four great circles determined by the six edges of the tetrahedron, but there are only three. In addition he neglects the reflective symmetry of the tetrahedron (each edge and the midpoint of the opposite edge determine a plane of reflective symmetry) which determines another six great circles. So the tetrahedron actually has as many symmetry circles as the octahedron (13), and the two great circle configurations are equivalent.
    
    
12. In Chapter 8, at the top of p. 56, Kenner says “the derivation of this last figure will be explained later,” “last figure” meaning the value of 54.7356⁺ at the bottom of p. 55. However, he never appears to get around to the referenced explanation. In Chapter 9, on p. 64, there is an equally cryptic claim that “we have learned” about the derivation of this value. It appears there is a circular reference here, and the value never gets explained. The value can be expressed as arccos(sqrt(1/3)). It is possible to derive it from Diagram 8.2 on p. 55 by replacing the spherical triangle with a planar equilateral triangle, remembering that a ray from the origin (where the person is standing) through the center of the spherical triangle will also go through the center of the planar triangle. Techniques similar to those used on p. 64 can be used to derive the desired angular value from this modified version of the diagram. Thanks to Dave Welsh for asking where the vague reference on p. 64 referred to.
    
    
13. In Chapter 9, his discussion of spherical coordinates reverses the use of φ and θ that I have seen in calculus book discussions of this coordinate system. He is consistent anyway.
    
    
14. In Chapter 12, Eq. 12.15 should read Class II θ = arctan(2(x₂² + y₂²)^0.5/z₂). The correct formula is used to compute the tables.
    
    
15. In Chapter 13, the section marked "Superspheroids" (p. 90 in the first edition), the fifth sentence I think requires a calculus argument since if the exponent (I've given it the label "n") is really exactly zero, the equation xⁿ + yⁿ/bⁿ = 1 is invalid. I think it would be better to rewrite the equation using absolute values so: |x|ⁿ + |y|ⁿ/bⁿ = 1. b is a positive constant. As n goes to infinity, a rectangle is approached (height of 2b, width of 2, centered about the origin). At n =1 , it is a diamond with the same dimensions. As n goes to zero, crossed line segments are approached. For any positive exponent value, x stays between 1 and -1 and y stays between b and -b. Values outside these ranges will never satisfy the equation.
    
    
16. In Chapter 13, Eq. 13.5 should be r₁ = {E₁^n₁/(cos^n₁φ + E₁^n₁sin^n₁φ)}^1/n₁. I've added a subscript "1" on the last n exponent.
    
    
17. In Chapter 13, Eq. 13.6 should be r₂ = {r₁^n₂E₂^n₂/(E₂^n₂sin^n₂θ + r₁^n₂cos^n₂θ)}^1/n₂. I've changed a φ to θ and added a subscript "2" on the last n exponent.
    
    

Criticism

1. In the preface ("What This Book Is"), Kenner states "In fact, Fuller's geodesic domes constitute a special case of a larger class of Fuller constructs called Tensegrities, and the way to an intuitive understanding of domes is to understand Tensegrity first." I do not believe geodesic domes are a type of tensegrity, though some people would disagree with me on this. To some extent, Kenneth Snelson certainly deserves credit for the development of tensegrity, and David Georges Emmerich was a pioneer as well.
   
   
2. Chapter 6 ("Rigid Tensegrities") uses the so-called (by others, not by Kenner) deresonated tensegrity spheres to make a connection between tensegrity structures and geodesics. While I think deresonated structures and their rotegrity relatives are interesting, I do not believe that these structures qualify as tensegrities. With the deresonated structures, once the gap between two head-to-head struts gets closed and they get nailed to a strut slightly inside of them, the structural stability stems from the continuity of the struts between triangles. It is true that in most realizations where the strut needs to be bent slightly to make connections, it is pulling up on the strut underneath. However I believe this pulling is not essential to the structure and if the strut were permanently bent to fit (by say heat treatment) before being put in place, the structure would still be as stable without the strut exerting an upward force on the inside strut it is attached to. I think it is important not to confuse the tensions and force needed to shoe-horn in an ill-fitting component with the essential tensions that are innate to tensegrity. I should also state that this is somewhat armchair speculation on my part. Though I have put together geodesic models and many tensegrities, I have not put together a deresonated structure with bent-to-fit components.
   
   That being said, I think this chapter makes an interesting transition between tensegrities and geodesics. Just because there is a connection between two concepts, doesn't make one a subset of the other. For example, even though you saw a caterpillar become a butterfly, you wouldn't be wise to go around calling a caterpillar a type of butterfly or vice versa.
   
   
3. Appendix 3 looks pretty obsolete at this point, but maybe some current HP calculators are backward compatible with the HP-65, or perhaps simulators can be found on the net somewhere. Certainly there are geodesic calculators around.
   
   

Contact Information

I am interested in your comments and questions. If you aren't hooked up to the Geodesic listserv, please direct your comments and questions via email to bobwb@juno.com or via Postal Service mail to me at Box 426164, Cambridge, MA 02142-0021.

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