To show how the existence of World Wallpaper violates the continuity assumption, start with a particular path representing one of Pete's world tours, on the globe. Then, look at the corresponding path on World Wallpaper. Here is a key idea. If World Wallpaper exists, then any actual path has a unique corresponding path on World Wallpaper. The continuity assumption already appears.
Here is the path we will use. Assume Pete's world tour takes him from Irvine, California, to New York, New York, to Paris, France, to Perth, Australia, to Tokyo, Japan, and back to Irvine, California. This is a perfectly valid trip in which the rock star travels from Irvine to Irvine. In other words, on completion of his trip, he returns to his original starting point.
Appearance of the Continuity Assumption: If the collection of tangent vectors describing Pete's trip on the spherical Earth changes slightly, then the collection of tangent vectors on World Wallpaper change only slightly as well.
We do something with Pete's trip that brings out a property of our being on a globe. The phrase we use: We will shrink Pete's trip from Irvine to Irvine. That is, picture Pete taking a series of trips, each starting and ending at Irvine. Each, however, is a little shorter than the previous trip.
For example, instead of going all the way around the world, as on his first trip, the loop of Pete's second trip goes up to the north. Instead of going to New York, he goes over Newfoundland, then to Helsinki, to Novosibirsk and down to Irvine. The 3rd trip would go still higher. Yet, by the 6th trip it would look like Pete was really taking a trip near the North Pole. By the 8th trip it appears he is making a little loop around the United States. Finally, on his last trip, he wouldn't move at all. He'd just stay home. Total length of his first trip: 24,800 miles. Total length of his last trip: 0 miles.
[Figure 9: Include these trips on a map.]
We aren't suggesting Pete makes these trips. He just thinks about making them. Then, he compares each of these trips with the corresponding trip on World Wallpaper. Each of the ten trips on the globe has a corresponding trip on World Wallpaper.
[Figure 10: Attempt to draw the first three trips on a representation of World Wallpaper.]
For any trip that Pete could take, along this shrinking collection of trips, there would be a corresponding trip on World Wallpaper.
What will happen when we arrive at the final shrunken trip? Look at the trip on World Wallpaper. It begins and ends at two different points, both representing Irvine.
What happens to the beginning and endpoint of the shrinking trips? Do they remain the same?
Answer: The two endpoints on the World Wallpaper trip never move. On World Wallpaper, Pete will be starting and stopping at the same distinct points, both representing Irvine. His trip on World Wallpaper always seems to go through a distance stretching at least 24,800 miles. Yet, the trips on the globe---representing the earth---seem to shrink to nothing. The final trip covered 0 miles; it went nowhere.
Pete believed his representation of his tour on the flat World Wallpaper was the same as the actual trip taken on the spherical Earth. After the continuous shrinking of each of the trips, the trips are obviously not equal. Assuming the existence of World Wallpaper gave a final trip of 24,800 miles long while the other trip (on the globe) has no length. The representation of the trip on a sphere is mirroring the reality of traveling on the Earth.
The two trips do not agree. Therefore, one must be wrong. World Wallpaper does not exist. World Wallpaper violates the second map constraint, the continuity assumption.