The flatness of Middle Earth is a textbook[1] exercise in differential geometry. We are given the distances between any pair of four cities in Tolkien's fantasy world: Umbar, Hobbiton, Erebor and Dagorlad. The object is to check whether those distances are consistent with a flat space.

The table below gives the distances in miles.

Umbar Hobbiton Erebor Dagorlad
Umbar 0 1112 1498 780
Hobbiton 1112 0 813 960
Erebor 1498 813 0 735
Dagorlad 780 960 735 0

There are 6 distinct pairs of cities. The cosine theorem and 5 of the distances can be used to estimate the 6th. On a perfectly flat surface the estimated distance should be equal to the actual distance. The calculation gives a relative error of about 1%, which means that that middle earth is most likely flat.

This exercise can be taken a step further, by asking what the radius of curvature should be for the distances to agree. This can be solved using the spherical cosine theorem. If the deviation from flatness is small, then perturbation theory can be used. In this case, the relative discrepancy between the estimated distance and the actual distance should be of the same order of magnitude as the ratio between the actual distance and the radius of curvature. According to this, in order to correct that 1% error, the curvature radius has to be 100 times greater than the typical distance (about 1000 miles). This translates to about 25 times the radius of the earth.

Reference Edit

  1. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley (1972) page 7 figure 1.1