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Cahill-Keyes 1975
Why Cahill? What about Buckminster Fuller?

Evolution of the Dymaxion Map:
An Illustrated Tour and Critique

Part 9.2

by Gene Keyes

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

Click inside boxes to open other sections in separate windows.
1) Introduction and
Background Notes

2) 1943:
Split Continents

3) 1944:
Whole Continents

4) 1946:
The Dymaxion Map Patent

5) 1954
Whole Continents

6) 1967 ff:
Later Editions and
World Game Versions

7) 1995 ff
Dymaxion Maps
on the Internet

8) Notes on Scaling Dymaxion Maps
9) Critique:
Dymaxion Map Compared to Cahill

9) Critique: Seven Design Flaws of Fuller's Map as Compared to Cahill's
9.1) Layout assymetrical
9.2) Graticule irregular
9.3) Korea distorted
9.4) Scalability poor
9.5) Anti-metric edges
9.6) Globe fidelity poor
9.7-a) Learnability poor
9.7-b) Learnability poor
9.8) Conclusion

Part 9.2
Total Irregularity of Graticule

In order to appreciate how badly disjointed the Dymaxion map's graticule is, we first consider the total graticular regularity of the Cahill octahedral — notwithstanding my major criticism of Cahill's map, that its prime meridianal division at 22 1/2º W, results in too many half degrees throughout. (This was the main reason why I developed the Cahill-Keyes version, divided at 20º W. As well, Steve Watermann independently made the same decision.) In Fig. 9.1.1 on the previous page, the half-degrees are not apparent, because the geocells are 7 1/2º. Nevertheless, all of Cahill's maps have smooth and identical graticules in each octant. Although the half-degree drawback is visible in the illustrations below, their graticular regularity, symmetry, and identicality is manifest.

Fig. 9.2.1 below: In 1936, near the end of his life, Cahill drafted a "penultimate" version of Equal Area Variant, "C". But it went unpublished. I am debuting it below, reduced to 1/100,000,000, one-quarter the size of an original that was 1/25,000,000. Unlike his earliest Butterfly versions of 1909 et seq, this was executed with a 5º graticule (except 2 1/2º x 5º at the edges). Notice not only the graticular identicality, but the smooth and regular curves throughout:
For a smaller picture size, click in it once; to restore full size, click in it twice.
B.J.S. Cahill octahedral Butterfly World Map, 1936 version C with 5 degree graticule

Source:  Cahill archives, Bancroft Library, item #57, University of California, Berkeley.
Hand-spliced from separate Xeroxes and digitally retouched by Gene Keyes.

Fig. 9.2.2 below: Next, consider a Dymaxion icosahedral map at 1/100 million. Because most Dymaxion maps degrade their graticule to 15º, we cannot see very well how they stack up against the neat array of 5º cells in the Cahill above. (But keep reading...) Meanwhile, the helter-skelter displacement of meridians and parallels is nonetheless evident, and differently scattered in each facet:
For a smaller picture size, click in it once; to restore full size, click in it twice.
Dymaxion map, Erica Gaba, from Wikipedia

Source: Dymaxion Map as drawn by Eric Gaba, Wikimedia Commons
(svg file converted to jpeg by Gene Keyes, and reduced on my monitor
to 1/100,000,000 and triangle edges of 70.5 mm, from its original scale
of 1/80,000,000). 

Fig. 9.2.3 below: Next we examine closer-up the same area as shown by Cahill, and then Fuller: namely, East Asia, Korea, and Japan, at 1/10,000,000, a scale 10 times larger than Figs. 9.2.1 and 9.2.2 above (or click to see them half-size at 1/20,000,000). (I have rotated both maps "right side up" for comparison, even though I concur with Fuller et al that there is no such thing as right side up, only habit. And convenience of lettering direction.)
For a smaller picture size, click in it once; to restore full size, click in it twice.
Cahill 1936 Butterfly World Map, East Asia close-up

Source: Cahill archives, Bancroft Library, item #57, University of California, Berkeley.
Hand-spliced from separate Xeroxes and digitally retouched by Gene Keyes
and enlarged to 250% of original's scale of 1/25,000,000.

Fig. 9.2.4 and 9.2.5 below: Now we reconsider the Grip-Kitrick edition of the Dymaxion Map and its half-degree graticule. With its anonymous latitude and longitude, the overly-fine mesh seems quite innocuous. But in Fig. 9.2.5, I have superimposed and identified a 5º graticule, as well as some 1º geocells. As will be seen, this indicates how badly a Dymaxion map disrupts the graticule. In these examples, I have enlarged the scale (on my monitor) from 1/21,000,000 to 1/10,000,000 (or click to see it smaller at 1/20,000,000). Source note for both items: see under Fig 9.2.5.

First, Fig. 9.2.4 below:
For a smaller picture size, click in it once; to restore full size, click in it twice.
Dymaxion map, Grip-Kitrick, enlarged excerpt, East Asia close-up

Fig. 9.2.5 below: And here is how any map should truthfully portray its graticule, if we are to make a judgment regarding its fidelity to a globe at the same scale (line boldness is exaggerated for emphasis here; not that a map would actually be done in this manner; but a 5º and 1º grid would be plainly discernable):
For a smaller picture size, click in it once; to restore full size, click in it twice.
Dymaxion map, Grip-Kitrick, enlarged, with 5 and 1 degree graticule superimposed
Now we can appreciate the dreadful gash in the 5 x 5º geocell set comprising 120 - 125º E and 35 - 40º N. It is this same rip which tears Korea upward by 60º from the Asian mainland, giving it an east-west orientation, rather than north-south

Also apparent is the fact that along the triangle edges, the meridians and parallels do not have good continuity. Fuller states in his book Synergetics [Vol. 1], p. 701, that his "transformational projection model demonstrates how the mosaic tiles ... [go from curved to flat surface] without interborder crossing deformation of the mapping data." Maybe in principle; but at least in the very precise Grip-Kitrick rendition, this is not so. It is one more disadvantage of having a different graticule in every single facet, instead of Cahill's identical graticule in all octants.

Source of Figs. 9.2.4 and 9.2.5: Dymaxion Sky-Ocean World,
Grip-Kitrick Edition of Fuller Projection
© 1980 by Buckminster Fuller
excerpt scanned by Gene Keyes from personal copy,
and adjusted (at least on my monitor*)
from its actual triangle-edge length of 335 mm.and original scale, 1/20,700,000
to a triangle edge length of 352 mm, for a scale of 1/20,000,000.
In Fig. 9.2.5:
5º and partial 1º graticule and numbering
superimposed by Gene Keyes

*Distance between vertex and west coast of Japan should be 95 mm; otherwise, adjust your settings or do the math (as explained here).

Fig. 9.2.6 below: We are not finished. Now consider the Cahill map with numbering added to the meridians and parallels; and compare their smooth continuity, to the Fuller map's breaks, bends, and disruptions: Notice above, for instance, the ordeal of 125º E, or the "unparalleled" northward swing of the 40th parallel, which, of course, should be heading due east; then recheck those same lines below in Cahill:
Cahill Butterfly Map, East Asia close-up, with graticule numbering added

Source: Cahill archives, Bancroft Library, item #57, University of California, Berkeley.
Hand-spliced from separate Xeroxes and digitally retouched by Gene Keyes
Enlarged to 250% of original's scale of 1/25,000,000, with numbering added
by GK.

Let me now reprise in smaller compass how East Asia should look on a world map of good fidelity to a globe: taking Cahill as the gold standard. These segments are reduced to 1/50,000,000. The important thing is that both have a 5º graticule, and that is what enables tell-tale comparison to a globe of 5º: Let us say: a ten-inch 1/50,000,000 globe.. Due to the lack of such globes (except FDR's), I am showing a computer-drawn 10-inch orthographic (globe-like) map (Fig. 9.2.7), and then a photo of my own 10-inch globe, onto which I improvisationally pencilled 5º geocells, and marked with Cahill-Keyes octants (Fig. 9.2.8):
Fig. 9.2.5-b: Grip-Kitrick 1980 Dymaxion Map, with 5º geocells superimposed by Gene Keyes:

Fig. 9.2.6-b: Cahill's 1936 Equal Area Variant "C", numbering added by
Gene Keyes.
    Dymaxion map, Grip-Kitrick, East Asia, with 1 and 5 degree graticule added (smaller version)
Cahill octahedral world map, East Asia close-up, with graticule numbering added (smaller version)
Fig. 9.2.7 and 9.2.8 below: Cahill (Fig. 9.2.6-b above) comports well with a globe.
Fuller (Fig. 9.2.5-b above) does not:
(More images of this Cahillized map and globe combo in Part 9.7-b.)

Orthographic map projection, East Asia, marked with Cahill-Keyes octants

Source: Orthographic projection from
R.L. Parker's "Supermap" program, output in 1975,
and octants added, by Gene Keyes

Fig. 9.2.8 below: Picture of a 10-inch globe with the same 1/50,000,000 scale as above. (More about this globe in Parts 9.6, and 9.7-b.)
Cahill-Keyes 10-inch globe marked with 5 degree graticule  
Source: 1975 Replogle globe, with 5º geocells added by Gene Keyes 2009;
photo by GK

Having shown how badly ruptured the graticule is in the East Asia area, I next extend my remarks about how the Dymaxion map contorts not only Korea, but also Norway.

Go to Part 9.3
Contortion of Korea and Norway

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Text cc. 2009 by Gene Keyes; Cahill-Keyes Map c. 1975, 2009  by Gene Keyes