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Geodesic Math: Errata and Criticism
by Bob Burkhardt
Last revision: January 26, 2016
Reference
Hugh Kenner, Geodesic Math and How to Use It,
Berkeley: University of California Press, 1976 (first edition) and
2003 (just a new printing apparently, no corrections).
Preface
Over 25 years after its first publication in 1976, Hugh Kenner's
pathbreaking and still very useful book, Geodesic Math,
was rereleased. Here are problems I've found or others have
found in the first edition or the reprinting. 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 wellwritten 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 (19232003) was Professor Emeritus of English
at the University of Georgia. He is the author of dozens of wellknown
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.
Also useful might be a third section entitled ”Notes”
which contains additional explanations of material in the book
which seems obscure or can be simplified. This hasn't been done.
Errata

In the preface (“What This Book Is”), Kenner states the mathematical
prerequisites for understanding the book as “algebra and highschool 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.

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.

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 tprisms beyond 3, 4 and 5
is incorrect. See
http://bobwb.tripod.com/synergetics/photos/x6prism.html.

In Chapter 2, the first sentence should read “The threestrut 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 threeprism's sides
are folded along their long diagonals.
Thanks to John Braley for pointing this misstatement out.

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 6strut Tensegrity”.
Fuller, who developed this structure in 1949, called it a
tensegrity icosahedron.

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.

In TABLE 3.1, I get some slightly different values than he does:
For the TCuboctahedron,
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 t_{1},
I get a value of 0.5951 rather than his 0.5983.
For the short tendon t_{2},
I get a value of 0.5204 rather than his 0.5168.
For the TIcosadodecahedron,
for the long tendon t_{1},
I get a value of 0.5433 rather than his 0.5442.
For the short tendon t_{2},
I get a value of 0.5025 rather than his 0.5016.

In the appendix to Chapter 3, his derivation uses an approximation
to get a formula for ED^{2} 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.

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).

In Chapter 5, in Kenner's inventory of the triangles in Diagram 5.8,
he should say there are five different shapes rather than
four. He is missing the three isosceles BCC triangles. He does
correctly state the sum is 16, but the shapes he lists only add up to 13.
Thanks to Gerry Toomey for pointing this out.

Also in Chapter 5, in the paragraph where Kenner mentions Diagram 5.10,
the second sentence refers to “eight Valence4 vertices”
(my emphasis) for the octahedron. He should refer to six Valence4
vertices. Thanks to Gerry Toomey for pointing this out.

In Chapter 7, Kenner encourages you (on p. 48) to make
an icosahedron with twenty struts, but I think you'll need
thirty.

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.

In Chapter 8, Diagram 8.2 on p. 55 is incorrectly drawn. In the
last paragraph on this page, the text correctly states that φ increases
“as we turn counterclockwise from north.” However Diagram 8.2
shows φ increasing as we turn clockwise from north. One way of
fixing the diagram would be to reflect the 3D figure in a mirror and change
the label “EAST” to “WEST.” Thanks to Margaret Glynn for pointing
out this problem. More is said on this coordinate system in
Chapter 9, and a comment below points out another
peculiarity of Kenner's presentation.

Also 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 10, 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.

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.

In Chapter 11, in the fifth paragraph, the last word should be
“icosahedral” rather than “icosahedra.” Thanks to Gerry Toomey for
pointing this out.

In Chapter 12, in the second paragraph, the pattern in Diagram 12.1
is rotated 120° each time rather than 60°.
Thanks to Gerry Toomey for pointing this out.

In Chapter 12, Eq. 12.15 should read
Class II θ = arctan(2(x_{2}^{2} + y_{2}^{2})^{0.5}/z_{2}).
The correct formula is used to compute the tables.

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
x^{n} + y^{n}/b^{n} = 1
is invalid. This equation doesn't appear explicitly in the chapter, but
it appears to be the generalization of the third equation (not numbered)
of the chapter which Kenner has in mind. I think it would be better to
rewrite the first equation (again not numbered) of this chapter using absolute
values so:
x^{k}/a^{k} + y^{m}/b^{m} + z^{n}/c^{n} = 1. This would
make the generalization of the third equation:
x^{n} + y^{n}/b^{n} = 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.
This alteration, the use of absolute values, has implications for succeeding
equations as well. The substitution x = sin θ and
y = cos θ now results in absolute values for the
trigonometric functions. sin^{n} θ now
becomes sin θ^{n};
cos^{n} θ now
becomes cos θ^{n};
tan θ now becomes tan θ; etc.
In the positive quadrant (for the 2D examples), where x and y are both greater
than zero, or the positive octant (for the 3D examples), where x, y and z are
all greater than or equal to zero, this change has no effect. This is the
region where Kenner takes all his computational examples, so he has no trouble.
It does affect the other quadrants or octants, and the change needs
to be made in situations where globally valid equations for all quadrants
or octants are desired.

In Chapter 13, Eq. 13.3 should have parentheses like
are present in Eq. 13.1.

In Chapter 13, Eq. 13.5 should be
r_{1} = {E_{1}^{n1}/(cos^{n1}φ + E_{1}^{n1}sin^{n1}φ)}^{1/n1}.
I've added a subscript “1” on the last n exponent.

In Chapter 13, Eq. 13.6 should be
r_{2} = {r_{1}^{n2}E_{2}^{n2}/(E_{2}^{n2}sin^{n2}θ + r_{1}^{n2}cos^{n2}θ)}^{1/n2}.
I've changed a φ to θ and added a subscript “2” on the last n
exponent.

In Chapter 13, Eq. 13.7 needs an additional pair of brackets to make
it clear that
(E_{1} cos θ)^{n}
is part of the denominator of the fraction. The equation should read
r_{2} = {(E_{1}E_{2})^{n}/[(E_{2} sin θ)^{n}(cos^{n}φ + [E_{1} sin φ]^{n}) + (E_{1} cos θ)^{n}]}^{1/n}.
Thanks to Dave Warner for pointing out that this equation had a problem.

In Chapter 13, in the paragraph following Eq. 13.7, Kenner says
“quadrant,” but “octant” is what is meant.

In Chapter 15, in the paragraph entitled “Rotating the Coordinate System”
(p. 108), Kenner references Eqs. 18.1 and 18.2, but, seeing as such
equations don't exist, Eqs. 14.1 and 14.2 must have been meant.
Thanks to Gerry Toomey for this observation.

In Chapter 17, in the second paragraph, Kenner writes:
“The builder of a wooden frame needs axial angles to know how to cut his struts
where their ends meet. Hub design requires axial and face angles.”
I thank Gerry Toomey for the following comment on Kenner's statement:
“True but obviously incomplete. Face angles are also needed by the builder of
a wooden frame who wishes to sheathe the triangular frames. Also, hub builders
will generally need to know radial angles in addition to axial angles. Radial
angles define the distribution of struts around a hub. Unlike the face angles
of triangles converging at a hub, radial angles always sum to 360 degrees.”

In Chapter 17, the angle AXO of Diagram 17.4 does not in general
represent what Joe Clinton calls the partial dihedral angle. The only time it
gives the correct result is when OX happens to be perpendicular to BC, and in
general this will not be the case. Thanks to Gerry Toomey for pointing this
out. Gerry gives the following formula for the partial dihedral angle value:
arcos{[c  (a((a^{2} + c^{2}  b^{2})/(2ac)))]/[a sin(arcos((a^{2} + c^{2}  b^{2})/(2ac))) tan(arcos a/2)]}

In Chapter 17, the formula for computing the floor dihedral angle is
incorrect. The adjustment is marred by a conceptual error in
Diagram 17.6 which draws the line from the reference sphere center to
the reference sphere surface. The line should actually go to the midpoint
of the floor chord (both ends of which touch the reference sphere)
about which the angle is being computed. Thanks to Gerry Toomey for
pointing this out. Gerry suggests the formula
Angle AXO  arcsin{[cos θ]/[1  (a/2)^{2}]^{1/2}}
where AXO is calculated using the corrected formula cited above,
θ is the floor latitude and a is the floor chord length.

In Chapter 17, the next to the last paragraph (not counting the final
list and single sentences as paragraphs) ends with the phrase “a 3ν octa
structure is just commencing to display outward curvature” which seems like it
should be replaced with something like “the floor radius of a 3ν octa
structure is just beginning to decrease.” Thanks to Gerry Toomey for pointing
to the problem here.

In Chapter 17, the last procedure for computing nonspherical
dihedral angles is flawed. Thanks to Gerry Toomey for pointing
this out. He has an alternative procedure that works, but it is
too involved for me to go into here.

In Chapter 23, the θ entry for 4ν = 4,2
(8ν = 8,4; 16ν = 16,8) should be 63.4349488
rather than 63.4343488. Thanks to Gerry Toomey for pointing this out.

In Appendix 1, item #1 gives the correct numerical value for
the icosahedron median, but the incorrect formula. There should be
no division by 2. So the formula should read
arc tan(τ  1).
Thanks to Gerry Toomey for pointing this out.

Also in Appendix 1, the first sentence if item #2 states:
“Greatcircle arcs from these division points meet
the triangle's left and right sides at 90°.” In fact, neither of the
two arcs Kenner is referring to intersects the left and right sides of
the triangle at 90°. Gerry Toomey thinks Kenner means they intersect
the “vertical median” at 90°, and gets thanks for pointing
this erratum out.

In Appendix 2, the HP program for computing dihedral angles is
flawed since it uses the incorrect formula embodied in Equation 17.7.
Thanks to Gerry Toomey for pointing this out.
Criticism

In the preface (“What This Book Is,” p. viii), 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.” First, I do not believe all
geodesic domes are a type of tensegrity, though some people would disagree with
me on this. For more details, see
Definition and Classification of
Tensegrities.
Second, “Fuller constructs” would be better written as just “constructs.”
While Fuller did coin the word tensegrity, and contributed substantially
to the development of the technology, Kenneth Snelson also contributed
substantially to the development of the technology, as Kenner concedes
at the end of his preface.

Chapter 6 (“Rigid Tensegrities”) uses the socalled (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 headtohead 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 shoehorn in an illfitting
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 benttofit
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.

Appendix 2 and Appendix 3 look pretty obsolete at this point,
but maybe some current HP calculators are backward compatible
with the HP65, or perhaps simulators can be found on the Internet 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 1005,
Shirley, MA 01464.
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