HIDDEN CONSTRAINTS IN DRAWING TRIPS ON A MAP

For Pete's map to be valid, it must satisfy two fundamental properties. Pete stated these succinctly using language he learned researching maps.

  1. Distance Hypothesis:
    To represent actual distances on the Earth a map would scale distances.
  2. Continuity Assumption:
    If the traveler moves a short distance on his or her trip, the traveler's corresponding location on the map must change a similar small amount.
The ideal would be that no trips (from a starting point to a destination) drawn on the map can violate either of these qualities.

One large Mercator map, however, violates both constraints. From any globe, Pete saw that Antarctica appears more than twice its actual size. Thus, it violates the distance hypothesis. The map also fails the Continuity Assumption, as the trip from India to Hawaii attests (Figure 6).


Figure 6: One Map's Violation

Pete saw he couldn't help but violate the Distance Hypothesis; he was picturing a round object---The Earth---on a flat space. Thus, he decided to relax the first quality, if the distance distortion was minor. Trying, however, to impose the Continuity Assumption, forced him to put a right-side-up Mercator map next to an upside-down Mercator map.

[Question: How well does Pete's second attempt at World Wallpaper satisfy Q_1 and Q_2?]

The Distance Hypothesis asks for an accurate distance representation. Since World Wallpaper is a collection of Mercator maps one should ask, "Does a single, 450-year-old Mercator map represent distances accurately?" Most common Mercator maps have Antarctica covering the entire bottom of the map. Also, Greenland appears as large as Africa. Greenland and Antarctica surely aren't that large: Greenland is just a fraction of Africa's size (Figure 7), and Antarctica is a small circular continent.

Africa Greenland Figure 7: Africa and Greenland's Relative Sizes


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