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

Evolution of the Dymaxion Map:
An Illustrated Tour and Critique

Part 5

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

Part 5:
1954: The Third Dymaxion Map:
Icosahedron with Whole Continents
(The Raleigh Edition)

In 1954, Fuller finally supplanted the cubo-octahedron with the now-familiar icosahedral version  I will save my comments on the map itself for the critique in Part 9. Suffice to say here that it presents even worse symmetry than either version of the cubo-octahedron. Whereas those earlier two at least preserved the facet shapes, in this one, a pair of deformities have been added to the otherwise identical 20 triangles of a Platonic solid: (1) an even split down the middle of one triangle to fill out Antarctica; and (2) a deplorable V-shaped split to save Japan and Korea from a different slice.

When the Raleigh Edition was first published in 1954, it was accompanied by a very big broadsheet in very small print, which I am enlarging via OCR-HTML below. The map was drafted by his architectural partner Shoji Sadao, who later told me that at the time they were very optimistic about its prospects for success, but that it simply did not catch on. As Fuller himself had mentioned in that 1954 broadsheet, "Because it is a mathematical revolution and mathematics are fundamental, its broad acceptance will be fundamentally slow."

In April 2009, the map publisher ODT, and the Buckminster Fuller Institute collaborated to re-create and reproduce the 1954 Raleigh, because the original artwork had been lost. ( http://www.pr.com/press-release/142209 )

Fig. 5.1 below:: Reprint of 1954 Raleigh Edition of the Dymaxion Map, available from the Buckminster Fuller Institute, and ODT.
1954 Raleigh edition Dymaxion map reprint, smaller
New printing also available in several formats from ODT:

Fig. 5.2 below: As co-produced by the Buckminster Fuller Institute (bfi.org), and ODT (odtmaps.com): portion of folded wall size facsimile reprint of 1954 Dymaxion Map, Raleigh Edition. [Principal] scale as stated on map: 1/47,500,000. Caption:

The Raleigh Edition of Fuller Projection
R. Buckminster Fuller & Shoji Sadao, Cartographers
Published by the Student Publications of the School of Design,
North Carolina State College, Raleigh, N..C., U.S.A.
Copyrighted 1954       U.S. Pat. 2,393,676
1954 Dymaxion map Raleigh edition, excerpt, full size

Source: excerpt from personal copy, scanned at its full size by Gene Keyes.
© 1943, 1944, 1953, 1954, 1967, 1971, 1980, 1980 by Buckminster Fuller

Dymaxion Airocean World
Fuller Projective Transformation

by R. Buckminster Fuller

[Raleigh, North Carolina, 1954; l large folded sheet, 13x19", 33x49 cm, to accompany the Raleigh Edition of the Dymaxion Map ]

Reformatted  in HTML and with introductory note by Gene Keyes, 2009-03-28
Fuller's published works are a mish-mash, to say the least: this piece does not exist on the Internet or in his anthologies, but as a large broadside sheet in very small print. It accompanied, and expounds upon, the 1954 Raleigh Edition of the Dymaxion map, when he first switched from a cubo-octahedron version, to the much more familiar current version of an icosahedron. He mentions the latter's characteristics, but does not elaborate on the reason for that change, except for its better depiction of a trans-Arctic perspective.

With this 1954 model, Fuller has yielded, and gone icosahedral: all triangles (plus a polygon cut-out), instead of the previous square-plus-triangle mix. As Fuller himself said elsewhere, "Don't try to make me consistent. I am learning all the time." (Quoted by Bucky expert C.J. Fearnley, cjfearnley.com.)

An alternate printing of this big folded sheet in small print also includes an abridged version of his 1944 article "Fluid Geography", omitting its anti-icosahedral passages. See further explanation at 
Part 3 of my review, "Fluid Geography" excerpts.

I was disturbed to see how — in that Cold War, pre-ICBM era —, Fuller touted his map’s supposed accuracy for aiming [future] intercontinental missiles at Russia, though he also looked forward to a time when that would not be necessary. In later years, Fuller was much more outspoken in favor of “livingry” instead of “killingry”.

Since 1917 Mr. Fuller has been evolving his Airocean World Map and new projection method — searching for the best way of seeing the traffic pattern of the Air World's "Town Plan." In 1927 he made his first map "sketch" of the "World Town Plan" with full forecast and illustrated documentation of the present day — 1954 — world airline routes in his book 4.D, privately published in Chicago. In 1938 he published an improved map "sketch" of the Airocean World as the end-papers of his book Nine Chains to the Moon, Lippincott, publishers. In 1940, Fortune Magazine published his world map "sketch" in a presentation of Fuller's "World Energy Map." In March 1943. Life Magazine published "The Dymaxion World," Fuller's first true disclosure of his mathematical projection invention. The Life edition portrayed accurately but awkwardly the Airocean World — for the sinuses intruded several of the continental masses. In the spring of 1944. Neptune Magazine published an improved orientation by Fuller of his true mathematical transformation. This avoided sinuses intruding the continents and provided a world map greatly improved over the Life edition. Unfortunately, however, it frustrated inspection of over-the-Arctic flight, for a sinus intruded the Arctic Ocean.

In the present 1954 Raleigh Edition of Fuller's true mathematical projection presented herewith and published by the Student Publications of the School of Design, North Carolina State College, he has reoriented the sinuses in such a manner that all the previous objections have been overcome.

Due to its inherent advantages in respect to astronomical observation. aerial mosaicing and comprehensive world triangulation by great circle grid — while containing within its continentally unsinused surface all the major, shortest air routes between the most people — the Raleigh Edition of the Fuller Projection comprises a world-around Airocean strip map of approximately invisible dimensional distortion and as such will probably in due course be commonly recognized as the best single world map of the Air Age. Because it is a mathematical revolution and mathematics are fundamental, its broad acceptance will be fundamentally slow.

The Dymaxion Airocean World Map and its projective transformation strategy was glimpse.conceived in 1917, but required what has turned out to be over a third of a century of mathematical exploration in what Mr. Fuller named "Energetic and Synergetic Geometry," to bring it at last to its present condition of: "satisfactory." It is satisfactory because it is the least distorted means of studying at one glance the total synergetic significance of Airocean economics and the alternate strategies for integrating all phases and states of energy behavior resources toward the highest operative advantage of all world people.

To confound those who "know all about geography" — from a Mercator projection — and who say that we go from the U.S.A. west to the Orient and east to Europe, Mr. Fuller has discovered the "Dymaxion Equator." The Dymaxion Equator is a great circle running approximately due east and west through a point on the Pacific Coast of the US.A. about two hundred miles north of San Francisco. The Dymaxion Great Circle through this point has "50 - 50" as its North Pole, i.e., 50 degrees East longitude by 50 degrees North latitude, and 130 degrees West longitude by 50 degrees South latitude as its South Pole. This Dymaxion Equator runs from Cape Canaveral, Florida, through the U.S.A. to Cape Mendocina, California, then due west through a point 130 degrees West longitude by 40 degrees North latitude. north of Midway Island (far north of Hawaii), north of Wake Island, passing over the seventy-five mile northern neck of the island of New Guinea, thence through the Malay Straights north of Australia, thence across the Indian Ocean running due west through a point 50 degrees East longitude by 40 degrees South latitude, and thence just south of Cape Hope, South Africa, and thence through the South Atlantic just north of Brazil and thence returning to the U.S.A. at Cape Canaveral, Florida, having gone completely around the world on one great circle course without touching any other continent than North America and having passed over twenty-one thousand statute miles of open ocean waters. In the Southern Hemisphere of the Dymaxion Equator lie only "Greater Texas," Central and South America, Australia and Antarctica. In the Northern Dymaxion Hemisphere dwell 93% of the human family.

The Fuller Comprehensive World Projection — or Transformation — is contained entirely within a plurality of great-circle.bounded triangles — or quadrangles — of constant, uniform modular subdivision whose identical length edges — shown as steel bands in the illustrations on this page — permit their hinging into flat mosaic-tiled continuities at the planar phase ot the transformation and thereby permit a variety of hinged-open complete flat world mosaics.

As in skinning an animal, a fruit. or a vegetable to provide a flat skin stretch-out, the development of a flat map of the complete world involves arbitrary piercing of the world ball's surface map-skin, thereby making one or more holes or gashes from which to start the stretching-out and peeling-off process of the skin until it is liftable from off its ball center. After the data has been further stretched it may be laid out as one or more flat map sections. If the skinning is accomplished in separate peelings and those sections have curved peripheries they may be associated only tangentially, e.g., as "gears" or "fans," which destroys the chance of forming a continuous one-surface comprehensive world map.

To provide a continuous one-surface world map — while peeling off the sections of the globe — the transformation must be such that the pieces have straight and matching edges when peeled off and flattened out.

Unlike any of those of its cartographic predecessors which present the whole spherical world surface data within a unit flat surface map — a few cases of which are shown on this page — the Fuller Projection maintains a uniformly modulated and constant length great circle boundary scale in closed — 360 degree — equilateral periphery controls.

The Fuller Projection operates in a series of transformation stages. It first subdivides the total world surface into a plurality of great circle bound polygonal zones. Next it transfers the data froni the sphere's surface in separate mosaic "tiles" corresponding to each of the great circle bound polygonal zones of step one. While migrating as a zonal mosaic tile, each tile (independently) transforms internally from compound curvature to flat surface by methods shown on this page and later described. Each tile transforms entirely within the respective polygonally closed uniform symmetrical containers, allowing none of the data to spill outwardly in perverse distortion.

When all of the tiles have transformed independently from spherical to planar, their straight polygonal edges — unaltered in length during the transformed-in-transit migration — permit re-association in a variety of continuous data mosaics.

The Fuller Projection has the relationship to all other known projection methods that a mechanically fillable and sealable set of flasks bear to hair combs, hair pins, fish bones, and star spurs, as instruments of liquid transfer.

The Fuller Projection not only contains its surface data increments within uniform and linearly unaltered closed great circle arc polygonal peripheries — all the way from its spherical to its planar positions — but also uniquely concentrates all of its spherical angle excess. It concentrates the excess into 360 degree symmetry within equilateral polygons. This means that the compound curvature subsides by symmetrical internal concentric contraction into a flattened condition entirely within its neither elongating nor shortening peripheral integrity. The internal subsidence of
the Fuller Projection is in contrast to all predecessor projective methods for showing the whole world within one unitary surface. All the predecessors disperse spherical excess by outward "fanning"-i.e., by stretching out to flattened condition.

Spherical excess is the amount of angle by which the three internal angles of spherical triangles exceed the constant internal angular sum of 180 degrees — which characterizes all planar triangles. The internal angles of spherical triangles always add to more than 180 degrees and the larger the spherical .triangles the greater the "excess." Convince yourself of this infraction of your planar thinking by drawing a meridian of longitude which is a great circle from the North Pole to the earth's equator, which is also a great circle. Meridians and the equator always intersect each other at 90 degrees. Draw a line along the equator a quarter way around the world, which is 90 degrees. Then draw a line returning by meridian 90 degrees to the North Pole. You will have completed a spherical triangle whose three angles are each 90 degrees and add to 270 degrees, or 90 degrees "spherical excess."

Because the area of a circle of radius "2" is approximately four times the area of a circle of radius "1," when the dimensional variables of a projected surface are outwardly distributed, the proportion of total map area that is in relatively greatest distortion exceeds by the ratio of three to one the relatively least distorted areas. When — as in the Fuller Transformation — the spherical surface subsides inwardly by symmetrical contraction, then the proportion of the map which is in relatively greatest dimensional distortion is less by a ratio of one to three than its relatively undistorted area.

How the Fuller Projection transfers the spherical data to the planar — employing only great circle coordinates while all other projections employ a progressively complex admixture of great circle and lesser circle coordinates — is shown in the two columns to the right. [Below. —GK]

At outset of the Fuller Transformation — from spherical to planar condition — radii of the sphere of reference which penetrate perpendicularly each spherical surface coordinate point of the comprehensive great circle grid, separate from one another at their respective internal ends — at the center of the sphere — and each and all remain constantly perpendicular to the transforming, internally shrinking surfaces
throughout the transformation, and their original uniform lengths also continue as "constant" throughout the transformation.

Because of the constant perpendicularity of the Fuller Projection's radii to the transforming surface, the Fuller Projection greatly simplifies celestial calculations. All astronomical phenomena always occurs in outward perpendicularity — zenith — to the Fuller Projection's internal spherical coordinates. On transformation to planar grids the astronomical data always remains in identical perpendicular zenith to the corresponding coordinate positions in the planar phase of the Fuller Transformation.

Because of this property, the Fuller Transformation would be more suitable than any other projection for a comprehensive planar mosaic of a total covering-set of world-around aerial photographs. Aerial photographs are always taken from zenith positions and at constant altitude, or radius distance from the earth's spherical surface.

The Fuller Projection, consisting of great circle bounded triangles of any angular magnitude, can transform the comprehensive geographical data of the world from spherical to planar by employment of either the spherical tetrahedron, spherical cube, spherical octahedron, or the spherical cubo-octahedron and its alternate, the icosahedron, or any development of these four respectively. It is a discovery of Energetic Geometry that there are no other spherical triangular grid bases other than the above.

However, it is a matter of Dymaxion cartographic strategy that: — the greater the number of great circle polygonal zones employed in the transformation the less the spherical excess to be subsidingly concentrated within each zone surface respectively, and therefore the less the residual distortion distributed to each of the planar mosaic aspects of the whole world's reassembled surface when arrayed in one continuous fiat "skin."

The Fuller Projection is patented and represents the only method by which the whole world data can be transferred from the spherical to the planar — within an all great circle grid, triangularly, quadrangularly, multi polygonally, or all two or three together. Because of a hemisphere's polar symmetry to its opposite polar hemisphere the total inventory of great circle grid triangles in the comprehensive world grid is always even in number, therefore adjacent triangles may always be associated in total or partial quadrangular pattern-phases without increase in vertex count.

Because automatically guided long distance fiight of aircraft or missiles most profitably follows the great circle — or shortest spherical distance courses — the guiding of aircraft or missiles can only be operated accurately on a world-around scale when referenced to a world-around triangular great circle grid survey. The Fuller Transformation will in due course become the appropriate cartographic device for plotting and planning such world encircling aeronautics.

It is no secret to the Soviets and it is well known in cartographic circles that because both the North and South American continents have been triangularly surveyed and gridded and the triangulation was carried across to Africa and thence into Europe and the Near East, India, and Australia, to satisfy the needs of World War Two, and the triangulation published, that the Soviets now know exactly where the U.S.A. and all its strategic resources are located, and that because the triangulation has never been extended into the Soviet Union and much of Soviet dominated Asia, we do not know with mathematical accuracy where most of the Soviet's resources are. We do not know within a vital margin of several precise miles.

But, the surveyed triangulation around the Soviet Union and Soviet dominated Asia ever closes in. Some day when the basic causes of lethal threat have evoluted to non-critical magnitude — the direction in which they now fortunately accelerate — through swiftly multiplying industrialization — then the inventory of controlled missile programming will be converted into controlled cargo and passenger routings to any point half way around the world in one hour.

It is to be noted that because the geometric constants or controls of all conventional projections are predicated upon a three dimensional coordinate system — comprised of an admixture of great circles with a variety of non-uniform lesser circles — that the constants which provide the original reference controls inherently are broken open and their finite quality converted to infinite — in respect to some part of their transformation data.

It is also to be noted that the internal earth lines formed in the intersection of all three dimensional coordinate planes of all predecessor world projection methods represents a hodge-podge of lengths and angles-of-incidence to the earth's external surface. When the surfaces are stripped of the earth and arranged in projected planar condition; these lines and angles look like a runover porcupine.

The lack of coincidence of three. dimensional coordinate-radials with the spherical radii and non-uniform radial length of the three-dimensional coordinate and non-perpendicular incidence of three-dimensional radials upon the spherical surfaces, has caused a heterogeneity of angles and lengths in respect to all conventional projections, which in turn has added frustration and entirely unnecessary awkwardness and complexity to the trigonometric problems of navigational science.

This unnecessary awkwardness of three-dimensionality has also promoted the calculus in "blind" calculations, where visual transformations might otherwise have accrued to a simplified multidimensional spherical trigonometry (which latter is rarely taught in school or college). The much simplified spherical trigonometry, plus a permeative topology, plus quanta and wave mechanics, plus thermodynamics, plus chemical structures, integrate as Energetic and Synergetic Geometry, which sum totally is no more difficuit than is the visible reading of this map, which is visible Energetic and Synergetic Geometry — nor reading the Energetic-Synergetic Geometry illustrations of its transformation processes.

The Fuller Projection great circle grid employed in the Raleigh Edition of the Dymaxion Airocean World is that of the spherical icosahedron, chosen because the latter has the largest number of identical and symmetrical spherical triangles, and therefore the least "spherical excess" of all the possible symmetrical triangular great circle bound mathematical cases.

The spherical icosahedron was also chosen because its controlling arc boundary of 63 degrees 26 minutes 05.816 . . . . . seconds was just adequate to the triangular spanning of the maximum continental aspects encountered in the unpeeling of the earth's data — within twenty symmetrical great circle bound control triangles — spanning those continents in such a manner that all twelve vertices of the spherical icosahedron grid lay in the open waters of the continuous world ocean. As a result, this peeled strip map contains all the world's continental contours respectively. The Raleigh Edition of the Fuller Projection presents the whole data in such a way that there is no visible discrepancy in the relative area sizes and no displeasing distortions of the shapes.

The map may be cut out around its periphery and bent on its main triangular edges and the exterior edges brought together — closing all exterior sinuses — thus making a continuous or finite surface and constituting a planar faceted icosahedron transformed from the spherical icosahedron.

When the surface is thus closed — proving itself to be afinite continuity — and the resulting icosahedron is compared to a globe of the world, the relative shapes and area sizes will be found to be entirely faithful to the spherical globe's relative sizes and shapes.

Because the Raleigh Edition of the Dymaxion Airocean World gives the continental stretch-out over the North Pole without continental contour sinuses and also avoids sinuses in the Arctic area, it will probably be as appropriate to future air voyaging as was the Mercator map appropriate to the square rigged east-west sailings with the Trade Winds — closely paralleling the Equator, around which the Mercator projection was least distorted.

Typical of the comprehensive information that may be quickly witnessed by the Raleigh Edition, is that the jet stream of the northern hemisphere whirls its 300-400 miles-per-hour course approximately over the green-to-blue area of the central target-like pattern, giving the Soviets on Cape Deshneva in northeastern-most Siberia a continuous jet-assist catapulting in a three hour flight to Chicago, while our own Greenland or Labrador springboards, though jet stream serviced, are not assisted in the correct direction to reach any of the Soviet "Chicagos."

This statement is no "comfort" to the Soviets who already know this and have been too long exceedingly comfortable in this respect due to our own prevailing public ignorance of dynamic Airocean geography, we being as yet blinded by old-plate-stocked Mercator projection vendors, and by the historical east-west orientation inertia. It is hoped that a new north-south dynamic world orientation will be aided by the Dymaxion Airocean World.

The map used by the United Nations on their official symbol, though disadvantaged by an awkward, old-fashioned projection, was a vigorous step in the direction of corrective orientation.

Our western "ramparts" of Hawaii may be seen on the Fuller Projection, Raleigh Edition, to be only a little less remote to the "show" than is Robinson Crusoe's Island. All the old-world-conceived pre-Airocean strategies of controlling the waterfronts from "down under,"— which took us into Europe's "soft belly" via Africa, and into Japan via Australia in World War Two, instead of throwing our science into winning an over-the-North-Pole competence, — as yet shows up in our present focus on Formosa which, as can be seen on the Dymaxion Airocean World Map, is looking in the opposite direction from that in which lie both our potential destruction and our potentials of salvation, and most importantly of all, our potentials of greatest world service and competence which alone through demonstration of responsibility, adequate knowledge and technology can accelerate our preoccupation in directions that could eliminate the basic cause of utter disaster.

Dymaxion map diagram, pt. 1

Dymaxion map diagram, pt. 2 of 3
Dymaxion map diagram, pt. 3 of 3

GK note: Fuller gives another, and still more technical, elaboration of what he calls the “Triangular Geodesics Transformational Projection”, in Synergetics [Vol. 1], (NY: Macmillan, 1975) p. 699-724.

However, it has so much more to do with spherical trigonometry than with mapping and geography, that I am not including it here in my critical review. It contains the above illustration, and more discussion of the rods-through-a-triangle idea, but no Dymaxion maps.

From 1954 till now, over half a century, the Raleigh format has remained the "classic" layout of the Dymaxion map, with very few variations. There have been other editiions, but likewise very few. The next two parts show as many as I could find, both hard copy (since 1967), and online (since 1995).

Go to Part 6
Later Editions and Large World Game Versions

Go back to Contents
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Text cc. 2009 by Gene Keyes; Cahill-Keyes Map c. 1975, 2009  by Gene Keyes