9.7-a) Critique of Fuller's Dymaxion Map compared to B.J.S. Cahill's Octahedral

Why Cahill? What about Buckminster Fuller?

Evolution of the Dymaxion Map:
An Illustrated Tour and Critique
Part 9.7-a

by Gene Keyes
2009-06-15

Summary: I love Bucky, but Cahill's map is a lot better. Here's how.

CONTENTS
Click inside boxes to open other sections in separate windows.

Part 9.7-a
Learnability: synoptic view, globe and map

Thirty-six years ago in 1973-74, when I was still enamored of Fuller's icosahedral, and using it as the base map in my MA thesis, I tried my utmost to make the disparate pieces learnable and identifiable as global components. I assigned numbers to each triangle, starting at Antarctica, and spiraling to the North Pole. I assigned matching-pair numbers to each vertex, and matching-pair numbers to each triangle side. I drew a five-degree graticule on a 12-inch globe — what an exercise that was — , pushed in map pins at the vertex points, and strung red yarn into a spherical icosahedron, including the dubious split pieces. (All shown in Appendix 1 below.)

That whole endeavor, like Fuller's icosahedral map, was the very opposite of "keep it simple"; the opposite of Occam's Razor, that 'entities are not multiplied unnecessarily'; the opposite of Einstein's dictum to 'Make everything as simple as possible, but not simpler.'

If only I had known beforehand of the Cahill map's far greater elegance. The vivid superiority of the Cahill design is manifest when it comes to comparing an octahedral map to a globe. The similarity is obvious and easy to grasp. With Cahill's three cuts, the globe is divided into eight easy pieces:

• Each piece has an equatorial base and a polar vertex.

• Each piece has an identical graticule.

• Each piece has a readily-remembered portion of world geography.

• Each piece on a globe compares precisely to its flat-map counterpart.

A Cahill map is therefore not only a thing of beauty, but a teaching tool of unmatched simplicity and finitely-visible entirety.

A Fuller map is a thing of excessive complexity and perplexity; asymmetry and unlearnability:

• Most pieces lack either a pole or portion of Equator.

• No piece has an identical graticule.

• No piece has a readily-remembered portion of world geography (and four have nearly indistinguishable chunks of ocean).

• No piece on an icosahedral globe (which doesn't exist anyway, except for my home-made one below) compares handily to its flat-map counterpart.*

* Of course, there is no off-the-shelf Cahill globe either, but you (or any grade-school pupil) can easily bolden the two Great Circles traversing 20° W and 110° W, and voilà, a Cahill-type globe [technically, Cahill-Keyes, or Watermann, since we revised Cahill's original 22 1/2°.] And even if one did have an icosa-marked globe, like mine, you have to orient it 20 different ways to match a flat Dymaxion: i.e., it does not "handily" compare to a map, as stated above. See Appendix 1 below.

And as I stress over and over again, a Cahill world map had all the continents undivided and relatively undistorted, long before Fuller reinvented that wheel, in a much klutzier design.

(Fuller's derivative geodesic dome is a thing of beauty, but the principle is not suitable for a map. Fuller's whole-earth narratives, and World Game, are also a joy to behold, but can be done on a Cahill map.)

Because my criticisms overlap and interlink, especially in regard to learnability, let me restate an argument I made in Part 9.5 about the anti-metric aspect:

Fuller's map fails both to provide and to combine three fundamentals of whole-earth elementary education:

1) a synoptic globe-plus-map pairing;

2) a metric understanding of the global basis of a kilometer, a meter, and a millimeter;

3) a correspondingly easy-to-grasp set of metric scales such as 1/200,000,000 and 1/100,000,000 and 1/50,000,000 and 1/20,000,000, etc.

Of course, it also fails to provide a fourth fundamental of map learning: national boundaries (though this is not a function of Fuller's projection, only his predilection). I heartily agree that humanity needs to outgrow chauvinism and nationalism, but boundaries cannot be wished away, any more than a medical book can decline to discuss cancer. Temperature zones and remote image composites are all to the good, but are only single layers of a multiplex global mosaic. Fuller, however, tended to be so possessive of his patents and copyrights that he sometimes stifled development of his ideas, including the Dymaxion map.

His failure to provide one-and-five-degree resolution is not Fuller's alone; most single world maps and globes suffer from that shortcoming. But Fuller in his later versions had watered down his 1943 map's 5° graticule, to the faint 15° which would not so obviously reveal their mis-shapes. Cahill, on the other hand, had started with sub-standard 7 1/2° or 15° graticules, but by 1936 had graduated to 5° geocells: too late in his life, sadly, for these to be published. (Except above in Part 9.2, for the first time anywhere, and repeated one octant at a time in Part 9.7-b.)

Fuller emphasizes the closed-system nature of his map, an important criterion, unlike most other commonplace open-ended world maps — except Cahill, who also links all facets of the globe to one another in various possibilities. But in my own learning experience, from grade school to grad school, I also found any globe itself to be confusingly open-ended — until I absorbed the Cahill map + globe into my mindset: those eight equal pieces, flat or spherical.

It occurs to me that a globe without a Cahill-octahedral division is like a fat book without chapters; and that a globe / world map without a five-degree grid is like a long chapter without paragraphs.

The learnability of the Cahill world map is its near-identity with a globe so-marked into eight comprehensible symmetrical octants. The unlearnability of the Fuller map is its 22 facets, mismatched to a globe's meridians and parallels; not only asymmetrical as a whole, but asymmetrical and irregular within every single piece.

Appendix 1
The clumsy relation of Fuller's icosa-triangles to a globe

As emphasized above — and shown on the following page in Appendix 2 —, a Cahill octahedral has eight easy-to-grasp octants, each with a distinct content of land-mass and water. Each octant is easy to remember in its own right. The whole set in effect goes around the globe twice: once to show the northern hemisphere in four pieces, once for the southern, in four pieces. (Or, centered on the dividing meridian, the whole set of on-globe octants can be seen, two at a time, in four frames, rather than eight.) Orientation is easy to maintain, whether one takes a polar, equatorial, or temperate zone viewpoint. Each octant goes from pole to equator; each is unmistakable, by itself, or seen with a globe.

But with a Fuller icosahedral, one quickly becomes lost among the disparate, disjointed, disorienting set of 20 (or 22) triangles-plus-polygon. Four of them are all-ocean, and nearly indistinuishable from one another. Only eight have segments of the equator. Only four have parts of the poles.

Notice also, for instance, how Fuller's South America is divided over four different facets, whereas Cahill's South America is in two. Fuller's Australia straddles three facets; Cahill gets it all in one.

It is a puzzle, in the derogatory sense of the word. (One can assemble a Dymaxion map puzzle from its 22 pieces on the Internet, but that is only an amusement, not a learning experience as is, for example, a jigsaw map of separate U.S. states — or the Cahill octants themselves.)

Ironically, the fragmented components of a Dymaxion map are just the opposite of Fuller's whole-earth purview.

In the photos below, I have re-scaled a 12-inch globe to appear as if five inches, and a scale of 1/100,000,000. (Which happens to be ten times smaller than President Roosevelt's fifty-inch, 5º globe of 1/10,000,000.) The measurements are approximate, given the variables of perspective, and digital reprocessing, and how they appear on my monitor. However, it is done like this:

As shown in Part 8, "Notes on Scaling Dymaxion Maps", the [unstated] edge length of a Fuller triangle is 7,048.89 km. Thus, on a 1/100,000,000 globe, the edge length is 70.5 mm. I adjusted the pictures to depict (on average) a distance of 70.5 mm between the center of pin-heads (selected, depending on perspective). Likewise reduced are the edge lengths of the counterpart flat triangles, from 91 mm down to 70.5 mm. These are reprinted from my 1973 MA thesis, at 1/100,000,000, not their original scale of 1/77,500,000. As mentioned, I had numbered each facet as well as its edges and vertices, spiraling from South Pole to North Pole.

Source for all images in this series:

Rand McNally 12 inch globe;
5° grid and icosa triangles superimposed
by Gene Keyes, 1974; photos by GK, 2009

Facets from [1967] Honeywell foldup globe (see Part 6, Fig. 6.1)
R. Buckminster Fuller and Shoji Sadao, Cartographers
© 1954, 1967 by Buckminster Fuller
Cut and numbered by Gene Keyes, 1973; scanned by GK, 2009.

Fourth of four almost indistinguishable ocean pieces

Because there are so many illustrations in Part 9.7, I am putting Appendix 2 on the next page. Like the Fullerized globe above, the following is an image set of a 5° Cahillized globe and its flat octant counterparts.

Go to Part 9.7-b, Appendix 2
Learnability: synoptic view, globe and map
(continued)

Appendix 2
The close relation of Cahill's octants to a globe