# Fuller's World Map: Distortion

Below is a table I composed in an attempt to compare the area distortions between Fuller's world map and the Snyder, Robinson and Van Der Grinten world maps.

A quick word or two about area distortion is in order here. First, there are different ways in which to calculate the area distortion of a world map. The method I used to calculate the area distortion of Fuller's map for this table is not exactly the same as was used for the other two world maps. For Fuller's map I simply divided each named land mass into smaller spherical (and planner) polygons (no more than 20 polygons per area.) This only gives me a ruff indication of the area distortion. My best guess as to the error in these numbers is +-2%. That is, in the table below, Greenland is listed as having 18% of its total spherical area missing on Fuller's world map. (18% "missing" is listed as minus 18%.) (Not including the area missing due to a change of scale, of course.) My best guess, knowing how I calculated this number, is that this value is really somewhere between -20% and -16%. Not having calculated the area distortion percentages for the other two world maps, I have no idea what kind of error is associated with the numbers listed. The Robinson and Van Der Grinten map's area distortion percentages were taken from a National Geographic Society illustration reprinted in Science News, Oct. 22, 1988. Snyder's projection method is an equal area project and so has all zeros in the table.

Area Distortion Percentages and Distortion Range
NAME FULLER SNYDER ROBINSON VAN DER GRINTEN
Africa -19% 0% -15% +8%
Antarctica -19% 0% N/A N/A
Australia -18% 0% -12% +37%
China -17% 0% -5% +61%
Greenland -18% 0% +60% +554%
South America -18% 0% -15% +14%
U.S.A. -19% 0% -3% +68%
"old" U.S.S.R. -20% 0% +18% +223%
Distortion Range 4% 0% 75% 546%

The important thing to note is that Fuller's map has a much smaller range of area distortion percentages than either of the of the Robinson or the Van Der Grinten maps. This means that Greenland, say, will not appear to be unnaturally large on Fuller's world map.

It should be pointed out that Fuller's map projection method is not the only projection method which will give a small range of area distortion percentages. As you can see, Snyder's equal area projection method gives the smallest possible range: zero. See my comments under Visual comparison of Fuller's map to the gnomonic and Snyder's equal area projections.

Another feature of Fuller's map, and Snyder's map when Fuller's orientation of the icosahedron is used, is that Antarctica is seen in its entirety, as it would appear on a globe.

Note that this comparison does NOT mean that the Robinson and the Van Der Grinten world maps are "bad" maps. However, as tools which are to be used to display all the world land masses without grossly distorting their shapes, they fall far behind Fuller's world map. As tools to be used for other purposes, they may be just the right maps to use.

I have determined an exact equation which can be used as part of a computer program to create Fuller's world map. This equation also allows for the computation of the area distortion of Fuller's map in a different, some might say a more "exact" or "mathematical", way. There are two different ways to calculate "exact" area distortion at a point on the map in the cartography literature. One is called the Tissot's indicatrix method and the other is to calculate the first Gaussian fundamental quantities. The calculation of the area distortion by the Tissot's indicatrix method is being computed by another group.

I have calculated the area distortion of Fuller's map at any given point by calculating the first Gaussian fundamental quantities. Since I don't think displaying a rather involved equation would be of interest here, I'll only make a comment about the results. I find that the greatest area distortion for Fuller's world map occurs at the icosahedron's face center and the minimum area distortion occurs at the icosahedron's vertices.

If you are interested in learning about the first Gaussian fundamental quantities and calculating the area distortion of Fuller's map at any given point, check out the book Map Projections For Geodesists, Cartographers and Geographers, by Peter Richardus and Ron Adler, North-Holland, 1972. You will also need my exact transformation equations for Fuller's world map. See About my "exact" transformation equation.