Cahill 1909

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Gene Keyes home page CahillKeyes 1975 
Notes on Scaling Cahill and CahillKeyes Maps
by Gene Keyes Reformatted and updated 20110609 Expanded and revised 20090307; 20090228 Earlier drafts 20070405, 19831116, 19831109, 19830529, 19820627, 19820623 Both diagrams below in this article were originally drawn by Gene Keyes I stumbled into a quagmire attempting to scale
the copies of various Cahill maps I have on hand, and will record these
notes before I forget how I resolved the problem. [Here referring mainly
to Truncated Octahedron 9sided octants, or nonagons: not the complete triangles
of the Gnomonic or Conformal variants.]
It turns out that CahillKeyes maps bear a single nominal scale, whereas Cahill maps can have up to five scales (including two nominal), depending on the criteria used. [See appendix 1 below.] By design, the CahillKeyes octant perimeter is approximately in scale to its nominal globe: the equator and the two outer meridians of the octant are each 1/4 of a great circle on a 100.425% perfect metric sphere. (More details here.) Hence on CahillKeyes, the equatorial scale and the nominal scale are the same. However, the Cahill octant only has oneway symmetry: not threeway symmetry like mine. Cahill’s nonagon has shorter meridianal midsegments than equatorial, whereas mine are all equal. Likewise, the endsegments on Cahill octants are unequal, north / south / equatorial. For a given Cahill octant and a CahillKeyes octant, each with a 90 mm altitude, Cahill’s has an equator of 93 mm, v. mine of 100. (e.g., p. 202, J.Assoc.Eng.Soc.) Note, however, that the optimal [or optimal / nominal] basis of comparison is the “Scaffold” altitude of the circumscribing equilateral triangle, and hence these are not otherwise comparable octants, Cahill’s having a scaffold altitude of ca. 104 mm v. mine of 100 mm (i.e., 10,000 km). Therefore, my chosen basis of comparison is the span between the outer vertical sides of 4 octants in an arch, or 8 octants in an “M”, representing my [Coherent World Map System] spans of 20,000 or 40,000 km respectively, or 2 or 4 lengths of the scaffold triangle altitude. I will call this a Comparison Scale (which may vary a little from other ways of interpreting a given map’s scale). Thus, the spreadout Mprofile represents the circumference
of the earth, 40,000 kilometers: the world unfurled. (Maximum shrinkage
/ distortion in a CahillKeyes map ocurs along the central meridian of each
octant, but in assemblies of four or eight octants, the nominal relation
of width to earthcircumference is preserved.) See appendix
2 below.
The Comparison Scale occurs in two usages: (1) An enlarged or reduced version to accord with a specified scale, often 1/200 M, so that any two or more maps may be compared regarding their distortion, graticule, etc. (2) A recalculated original scale of the map, based on its 4octant span as divided by 20,000 km. (Since the Comparison Scale happens to be an equatorial scale as well, other nonCahill, nonButterfly world maps can be likewise adjusted by setting their equator within a 200 mm frame.) Given that the M shape is not the prevailing mode of Cahillian maps, it is sufficient to use the 4octant subset, for calculating the map’s existing scale, and/or the percent increase or decrease needed to put the map into a size where all examples can be compared: say, 1/200 M, where the 4octant length is 100 mm and the full 8octant length is 200 mm). Thus, with 100 mm as a numerator, we can use a calculator with a 1/x button to get the full 200 mm, 1/200 M conversion percentage, by using the length of the 4octant span, representing half the entire map. For instance, Dahlberg’s Cahill illustration is 48 mm across a 4octant span. Entering 48 mm in a calculator with a 1/x button produces a 208.3% enlargement factor, resulting in the Comparison Scale of 1/200 M. Whereas the map’s unchanged original scale — as derived from the Comparison Scale — is 1/416 M: 2 x 48 = 96 mm / 40,000 km [earth circumference] = then 1/x = 1/416.6 M. [Leaving aside zeroes in calculator entry and readout.] Comparison calculation: take a CahillKeyes 8octant span of 200 mm, divided by 40,000 km = 0.005, then 1/x = 200 [1/200 M] Note that the same results obtain when using the 4octant span, representing half the earth’s circumference: thus, Dahlberg’s Cahill, 48 mm / 20,000 = 0.0024, then 1/x = 1/416.6 M. CahillKeyes, 100 mm / 20,000 = 0.005, then 1/x = 1/200 M. For a given 1/100 M map, a Cahill version by the other
four scaling criteria (below) might be 15% larger than mine at the same
scale: in principle, mine shrink in visàvis a scaffold
triangle, whereas his stretch out, especially his Conformal and Gnomonic
versions.
The KeyesComparison Scale posits that the width of a rectangle enclosing an entire world map [Mshape or comparable] at 1/100 M be 400 mm, representing a metric Earth circumference of 40,000 km. (Or 200 mm for a 1/200 M map; etc.) Likewise, a square enclosing a fouroctant arch, representing half the earth and half a metric earth circumference of 20,000 km, is deemed to be 200 mm if 1/100 M, or 100 mm if 1/200 M, etc. (Because the Butterfly shape is not as wide as the Mshape, we must use the halfamap, fouroctant arch, to align with a 20,000 km measurement.) With sizeadjusting software, a map image can easily be fitted into such a rectangle, and imputed with the given Comparison Scale, even if its scale might vary somewhat by other ways of measuring.
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^ APPENDIX
1:
Five different scales of a Cahill map with truncated octants (EqualArea), depending on criteria:
1) Main Scale would be according to preponderant lengths per
10° of latitude, divided into 11.1. (Say, at 4050° N or S, at
22.5° E or W from outer meridian.)
2) Natural Scale would be per Map Scale Indicator (a special ruler from Memorial University of Newfoundland Dept. of Geography), based on 1 statute mile, 1 kilometer, 1,000 feet, or 1° of latitude. 3) Nominal Scale 1 would be length of 1/3 of octant perimeter divided into 100. (Equatorial.) 4) Nominal Scale 2 would be length of the (truncated) altitude divided into 100. 5) Comparison Scale (or KeyesComparison Scale) as detailed herein: the span between the outer vertical sides of 4 octants in an arch, whereby the altitudes of the circumscribing triangle are deemed to be 100 mm for a standard scale of 1/100 M, and the width of the arch is adjusted to 200 mm. (Or a 50 mm altitude for 1/200 M; etc.) While not the right place to discuss minutiae
of the CahillKeyes design, this diagram indicates the relation of its nonagon
to a circumscribing "scaffold" triangle. Each main side has three segments
constructed to jointly equal 1,000 km. (Further elaborated here.)
