©1975, 1980, 2010 by Gene Keyes

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Abstract

We now get into what Cahill called the "intensive mathematical details", which must fulfill my four essential design criteria for the Cahill-Keyes master map:

1) Visual fidelity to a globe in each and all of its eight equal octants, as well as showing each and all of its 64,800 one-degree geocells (except for the sliver-like 3,600 at the poles between 90 and 85°, drawn with 1° latitudes, but 5° longitudes );

2) Proportional geocells in all adjacent ranks and files, none being grossly out of sync with its neighbors, despite calculated shrinkage toward the octants' centers, and other minor adjustments along the octants' outer meridians.

3) 10,000 km lengths for each of its octants' three main jointed edges, the (rounded) metric distance from pole to equator, being one-quarter of the earth's circumference, ca. 40,000 km, and:

4) An M-shape Master-Map profile whose eight octants fit exactly in a km-gridded rectangle whose span is 40,000 km, representing that very circumference.

Caveats:

1) "Fidelity" is not the same as "accuracy" in the geodetic, equal-area, or equal distance sense, but rather, an overall likeness to a globe — to its continents, and to its graticule — better than any other entire world map.

2) "10,000 km" in principle is the metric distance from pole to equator, but the earth is not a perfect metric sphere, nor are these map's edges a perfect metric match, being 10,042 km, due to design constraints explained below.

3) A computer monitor cannot show you maps at their proper scale and dimensions, in both width and height. My hard copy map originals and printouts are the right size each way; but not on a monitor. (See demonstration here.) I wish monitors were otherwise, but if you want to get the scales and appearance just right, you'll have to print one or more of the pdf files linked on p. 6 . Note well that while computer-drawings are extremely accurate, monitors are merely loosey-goosey.

4) Until the digital graticule presented below, the Cahill-Keyes map has existed only as hand-drawn prototypes at scales of 1/20,000,000 and smaller, plus a one-degree octant at 1/10,000,000; plus some 17 square-meter test panels (out of 496) for the 1/1,000,000 Megamap graticule (see photos here), plus two 1/1,000,000 test panels, of Denmark, and Canada's Atlantic provinces. (Plus earlier offset-print specimens, shown on this website as jpegs.)

5) My academic field is world politics; I am not a mathematician. While this map does not embody geodetic precision nor cartographic formulas, I had to do a lot of trigonometry in my quest for proportional geocells across the entire face of the earth on the length and breadth of the Cahill-Keyes world map. (And Mary Jo Graça's Perl programming for it, next page, adds a further degree of mathematic nicety.)

Those design criteria in turn imposed some constraints, and indicate why the map is done in a certain way, and not some other way. (Cahill himself had evolved at least five different projection variations within his basic framework, but for reasons explained elsewhere, I started from scratch on a different path.)

Besides the four main criteria stated above, three other considerations had to be met; again, differing in detail from Cahill:

5) That the octants be entirely symmetrical: all six end segments be equal to each other, and all three main mid-segments likewise equal one another.

6) That the end segment polar incision be no longer than 17° of latitude;

7) That the polar zone parallels from latitudes 89 to 75 be drawn as circular arcs.

My design had to simultaneously satisfy all seven desiderata, none of which were present in Cahill's versions. It is their combination which establishes the character of the Cahill-Keyes — and entailed such difficulty in doing it. As Cahill himself wrote, "simplicity in most things is achieved by a circuitous route through all manner of complications." (1912, p. 164)

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Example of a design constraint

Note that a 10,000 mm scaffold-triangle altitude is required to produce an M-profile Cahill-Keyes map with a span of 40,000 mm [km]. (See Notes on Scaling Cahill and Cahill-Keyes Maps.) But the inscribed octant's altitude (scroll down to Fig. 2) is ipso facto shorter than 10,000 km, while the triangle sides are longer than 10,000 km here in Fig. 1.

Fig. 1: Scaffold triangles and M-profile of Cahill-Keyes map. Original drawing by GK; reproduced in ClarisWorks 5 by Mary Jo Graça.


For example: what happened when I tried to attain a 3-segment side that was exactly 10,000 mm within a scaffold triangle whose altitude is 10,000 mm? Well, I was able to do that after a lot of trial-and-error calculation, and found that the octant's altitude must be 9,033.3625 mm, instead of my chosen 9,060 mm. But it resulted in the end segments being somewhat too long, requiring an unacceptable stretch at the octants' outer meridians in two of the polar latitude degrees between 75° and 73°: namely, 167 mm per degree, or 150% of any latitude's nominal 111.111 mm, instead of my already excessive 130 mm there. Likewise, the one-degree latitude-lengths on the outer meridians from 0° to 15° would have been stretched to 124.5 mm, i.e., 112%, instead of my already excessive 121 mm. In addition, the shorter altitude of the "exactly 10,000 km" octant imposes still greater compression on the central portions of the graticule.

Therefore, I reluctantly decided to let the "proportional geocell" principle prevail over the "exactly 10,000 km" principle. Nevertheless, the quarter-circumference octant edge is only 0.4% larger than ideal, while I thus avoid grossly stretching some geocells up to 150%. (More details are in the "supple zone" notes below.)
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From 1980 to 1984, I computed all the x-y coordinates of all 8,100 geocells of an entire Cahill-Keyes Octant Graticule (minus the 450 polar spiky ones, drawn @ 1 x 5°), copying the 5-decimal-place numbers by hand into tables, from a TI-30 scientific calculator, and doing others with a BASIC program into a fanfold printout.

This was well before the Internet. Now that I have revived this project for my website, after a 25-year dormancy, I boggled at keyboarding so much data.

However, my companion Mary Jo Graça has patiently written a Perl program (p. 2 below) to re-calculate all those x-y coordinates and output them in HTML tables here (p. 3). Next, she exerted her coding skills to prepare macros (p. 4 and 5) using the Perl data, which could plot the map graticule via a vector drawing program. In doing these digital versions, I have been the architect, as it were, and Mary Jo the engineer.

At first we were using an outdated 1993 Mac version of TurboCAD, but its exported .dxf files proved unusable, and it could not do very large scales. So we switched to OpenOffice.org Draw 2.0 (OOo for short; it is cross-platform, and free.). Though more difficult for Mary Jo to program its macros, it could output image files in either .pdf or .odg form.*

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* Impetus to try a computer version of the Cahill-Keyes map came from Clay Cole in February 2010, who started one in AutoCAD. But that is a pricey program for PCs, and our machines are an old Mac G3 Biege OS 9, and a pair of cheap Asus eee 701 netbooks with Linux (Xandros).

If you are doing this map in a vector-draw or CAD program, then one size fits all, depending on how big your monitor is, and how much you can zoom in or out. But in 1980, when I began hand-drafting the 1/1,000,000 megamap on separate square-meters of millimeter-grid cross-section paper, I compiled tables that were adaptable two ways: 1) for a single octant template (upright and tilted 30° to the left.) at 1/10,000,000, in a grid of 1,000 x 1,200 millimeters; i.e., just over one square-meter; 2) and for the scaled up Megamap itself, whose single-octant template would occupy 62 square meter sheets within a grid of 10,000 x 12,000 millimeters: (See diagrams below, and Photos of the Cahill-Keyes Megamap Prototypes.) That in turn is the prelude to a hoped-for 20 x 40 meter exhibition piece (65 x 130').

Therefore, numbers in tables 1 and 2 below on this page (for the hand-drawn version) are stated as coordinates of a millimeter grid that is 10,000 x 12,000 mm for the 62-sheet 1/1,000,000 template. (Note that x-line 0 starts 3,000 mm above the bottom of the grid, so as to intersect point G, which is the octant's central meridian, at the Equator's midpoint.) When I was simultaneously drafting the ten-times-smaller octant at 1/10,000,000, I would move the decimal one place to the left; i.e., I would read only the first column, [before the (*) digit]. (That would still yield a pretty big wall map, 2 meters high by 4 meters long, if the octant template is reproduced eight times.)

4) Orientation, computer-drawn

Deferring to Mary Jo's automated re-orientation of the graticule's x-y coordinates (p. 2 and 3 below), I have omitted most of my original whole-octant tables — except the ones given here, which specify the original design framework and segment lengths (as shown above). They remain the same in the subsequent computer-drafted version.

For a 1/10,000,000 map, read first column of digits only. For a 1/1,000,000 map, append (*) column's digit. Other decimals are for optional precision: e.g., if a 1/100,000 map, append first digit from four-place decimal column; for a 1/10,000 map, append the second; etc. Decimal places are for more exact computation, not visible to naked eye, until they start to cumulate.

Meridians are constructed in three zones:

Polar (ALB in diagram above),

Middle-Sector (LBDJ), and

Equatorial (JDEI).

All meridian angles change from one meridian to the next at a constant rate:

Polar zone meridians start at vertex A; all 90 (or 91, including both outer sides) are drawn as one-quarter of a standard 360° polar grid, i.e., 90°, spaced 1° apart. (From the Pole to parallel 85, meridians are spaced 5° apart, but all 1° parallels are drawn.)

Middle-Sector meridians start at vertex M (not shown before they meet their polar counterparts); all 90 are likewise drawn, except that their angles are 2/3° apart, i.e., 90°, 89 1/3°, 88 2/3°, 88 1/3°, etc., reaching the outer angle of 30° relative to the x-axis.

Equatorial zone meridians do not have a common vertex, but start at equisected points along the Equator, changing their angles @ 1/3° from 75° at edge I J to 45° at edge DE, and meet their Middle-Sector counterparts at joints calculated by trigonometry.

(Clay Cole suggested that the Equatorial zone meridians could meet at point "X" of an overlapping (unshown) 30° isosceles triangle, whose vertex is far beyond the main scaffold triangle. with x-y coordinates (at 1/1,000,000) of -4489.4388 mm and 16,436.1902 mm, outside the given drafting grid of 10,000 x 12,000 mm. However, Mary Jo Graça recalculated and confirmed with CAD my original conclusion that there is no common meridian vertex if the Equator's longitudes are all identical. Using a point "X" origin for the Equatorial meridians would result in somewhat differing longitude widths, contrary to the design principles of this map.)

Parallels, too, are constructed in the same three zones:

Equatorial supple zone:

From P 0 to 15 on M 45, there is a maximum stretch to 121.0627 mm per degree (vs. the ideal 111.111 mm), diminishing proportionately to the same latitude lengths on P 30 to 0 as in the Middle Sector.

Note: Parallel 15 meets joints J and D, of the middle and equatorial segments along M 45, but P 73 is 6.0360 mm below joints L and B of the middle and polar segments along M 45.

Parallel 15 was a special (and difficult) case in the design of the map, a kind of a fiat or "skyhook" construct to knit the graticule between its upper and lower portions. When hand-drawing it, I had equisected the meridians, as stated, but had connected the points between M 35 through M 20 with a spline, to smooth out P 15 and lower parallels, so that they were unlike the angular joints on the Equator (P-0). However, when it came to doing the same parallel with a computer, Mary Jo devised a circular arc (replicated at either end of the parallel) centered at 2786.8887, 1609.0110 in MJ's half-octant coordinates, with a radius of 5760,8557 mm. See further explanation, p. 2 below.

Summary of supple zones:

Overall effect on the three Great-Circle outer edges of the
Cahill-Keyes Octant Graticule

Besides geocell proportionality, the Cahill-Keyes design emphasizes the 10,000 km distance on each main side of the octant, embodying the metric principle of 1/4 of a great circle from pole to equator, on a perfect metric sphere where each degree of latitude (and longitude only at the Equator), is ca. 111.111 km.

However, the earth is not a perfect metric sphere, nor is this map. When I say 10,000 km, i.e., mm, as the octant edge, I am rounding from 10,042.7 mm, a design constraint as explained above.

The equator-longitude widths on Cahill-Keyes are equal across the board, and around the world: 111.5847 mm (vs. 111.321 in reality, and 111.111 in principle).

But the two Great Circle meridians bounding all the Cahill-Keyes Octants, (the same 10,000 km as the Equator), are finessed somewhat by the two "supple zones" along M 45. While most of the lattitude lengths (i.e., the Middle sector, P 73 to 15), each 110.4278 mm, are just a bit under 111.1111, they are a little too long in the supple zones at P 75 to 73 (130.9883 mm per degree), andP 15 to 0(121.0627 mm), and somewhat too small in the Polar zone itself, P 90 to 75 (104 mm).

Nonetheless, the overarching aim is always to minimize adjoining geocell disparity across the map as a whole, while keeping all eight parts visually attuned to their counterparts on globes, and to the metric principle of 10,000 kilometers along any edge of an octant.

8) Latitude lengths per 1 and 5 degrees
in Cahill-Keyes Octant Graticule
(central and outer meridians)