I've been trying to figure out the galactic distances between planets in Warhammer 40,000. We know the Eye of Terror corresponds to Cygnus X-1 (approximately 7,300 ± 200 light years away from Earth). The book The Horus Heresy Book III - Extermination gives a map of the galaxy with a hex grid background. We can reproduce the fragment involving the Eye of Terror and Holy Terra below:
I have computed the center of mass for the Eye of Terror (assuming uniform density) and drawn a straight line connecting Terra to the center of the Eye of Terror. Along this line, I have marked distances which are multiples of the edge for a hexagon. Since the Eye of Terror is approximately 8.4 hex-sides away from Holy Terra, it follows that a hex grid side length is 869 ± 23.8 light years.
Result: The side of a hexagon is 869 ± 23.8 light years.
Now, for the interesting systems to the "East" of Terra:
Again, we use circles to mark "milestones" whose distance is 1 hex-side length.
We find that Prospero is sqrt(57) ≈ 7.55 hex-sides away from Terra which would be 6560.80 ± 179.686 light years away from Holy Terra.
The Davin system is about sqrt(156) ≈ 12.489996 hex-sides away from Terra, which would be 10853.80652 ± 297.262 light years away from Holy Terra.
If this is accurate, then the distance between Prospero and Davin is approximately sqrt(27) ≈ 5.196 hex-sides, which would be 4515.45645 ± 123.668 light years.
Anyways, I'm interested in Davin's distance from Terra because Guilliman (and friends) fought the Second Battle of Davin before rushing to Terra to relieve Dorn, Sanguinius, and the Khan. I've been curious about the distances and time intervals, mostly as an excuse to generate exercises in special relativity. Assuming the distances measured are radar distances, the proper speed for Guilliman's relief force would be about 3618c (i.e., 3618 times the speed of light) and relative to an inertial observer on Terra Guilliman's speed would be about 3618/sqrt(36182 + 1) ≈ 0.9999999618026812 times the speed of light.
Objection: These calculations all work in flat geometry, which we know is false due to General Relativity.
Response: While this is true, the curvature due to general relativity is so slight, the corrections would be negligible. One more compelling reason to use general relativity would be to observe the galaxy is rotating, which means we'd need to use a non-inertial reference frame. But this is so small as to be negligible.
A better objection would be the location of these planets are not in the direct center of the hexagons. Assuming a uniform distribution of offset within the hexagon, we obtain an uncertainty of 16*sqrt(2)*(1/3 - sqrt(3)/8)/(4 + ln(27)) hex-sides ≈ 314.864 ± 8.623 light years in the distances between any two planets.
(The math behind this is simple, just take M = \int^{1/2}_{0}\sqrt{y^{2} + \frac{3}{4}}\,\mathrm{d}y = (4 + \ln(27))/16 as the normalization constant for the probability distribution Pr(y) = \sqrt{y^{2} + \frac{3}{4}}/M, then we find E[y] = \int^{1/2}_{0}y Pr(y)\,\mathrm{d}y = \frac{16}{4 + \ln(27)}\left(\frac{1}{3} - \frac{\sqrt{3}}{8}\right) as the expected offset. Another approach would be to consider a uniform distribution over the hexagon, then find the variance. I believe this would give an uncertainty of 1.01958 side lengths ≈ 886.015 ± 24.2660, but I should work through this more carefully. For the uniform distribution on the unit circle, we would have the variance of the radius be 5/18, corresponding to 458.003 ± 12.544 light years.)
Objection 2: In Angel Exterminatus, Perturabo notes that Cygnus-X was a name "re-used" on a different celestial body than it was originally named, so all these calculations are wrong.
Response 2: True, Perturabo made that "observation", but it makes no sense. Cygnus-X is literally the first confirmed black hole that humans have discovered, and that would not be easily changed. It would be as if "Mars" nows designates Alpha Centauri.
Again, a better objection would be that I have confused Cygnus X with Cygnus X-1, which would just rescale the distance of a hex edge by about 46/73 ≈ 63%, or 547.59 ± 15 light years.
Another way to estimate distances is to first identify the center of the galaxy, then find Earth (holy Terra). This will be approximately 27,000 light years. (Our current best estimate is that the Solar system is a distance of 8.33 ± 0.12 kpc ≈ 27168.83 ± 391.3877 light years, see Eq (2) of arXiv:1611.09144.) As a consistency check, it appears that Terra is about hex edges away from Sag A*, which would make Terra about 22 hex edges away from (what I think is) Sag A*, which would be an estimated 19118 ± 523.6 light years away...which means we'd need to increase each edge length by about 42% to 1235 ± 17.8 light years. [Using this estimate: The Eye of Terra is 10,374 ± 149.52 light years from Terra, Prospero is 9324 ± 134.4 light years from Terra, Davin is 15425.15 ± 222.32 light years from Terra, and Guilliman would have traveled at a proper speed of 5141 ± 74 times the speed of light. This also fits the claim "Armageddon is 10,000 light years from Terra" better, I believe.]
It would be interesting to compare the various maps from the different books, to check if they are all mildly consistent with distances, and specifically to examine the locations of various "known" bodies (e.g., Armageddon is supposedly 10,000 light years from Terra — is this just poetic hyperbole, or an accurate order-of-magnitude estimate?).
