Equestrian Geometry · 10:51pm Apr 2nd, 2014
The geometry of Equestria, or rather the entire planet (Equus) itself, is troubling. First lets examine what we know.
0) Light travels in a straight line unless acted on by an outside force. We can't take Equestrian physics as the same as ours down to the last bit since there's magic (magic could be sufficiently advanced technology, but that is not in the spirit of the show) but if this is not true then we can hand wave any observations we make, saying, 'that's just how light travels.' Of course, these straight lines are by no means necessarily Euclidean.
1) Equestria is round. A quick Google search will reveal that there are globes in Equestria. While this could be an elaborate hoax, that seems unlikely.
2) Eternal night is possible. Surely Nightmare Moon would have at least stopped to think otherwise.
3) The sun and moon are manually raised and set. If Twilight's visions, Celestia, and Luna are to be believed, then a thousand years ago Luna prevented the moon from setting (if necessary) or from being set by Celestia. Further, we have seen both in the sky at the same time - which is very odd when the accompanying skies are considered - but further suggests that the alicorns are responsible for their movements in some way.
4) The sun and moon are raised and set ridiculously fast. A view of filly Twilight watching Celestia raise the sun reveals that the Equestrian sun rises in a few seconds during the time it would take ours hours (we do not know how long Equestrian days/years are, but we have wild mass guessing that suggests days are approximately the same and years are slightly longer [based on Twilight's planner]). I can not remember when it occurred, but I remember seeing a scene where Celestia sets the sun just as quickly while Luna raises the moon. It seems valid to assume that the moon is also set at a similar speed.
5) Equus is everywhere (presumably) approximately Euclidean (at least on the surface and low altitudes). While it is possible that in universe the geometry is distorted and merely sanitized for us, the Euclidean axioms are all satisfied in the same sense that they are in our universe (see classical vs modern physics).
6) Presumably the moon is a physical place one can go to. Nightmare Moon was reportedly banished there with a 'mare in the moon' effect supporting that. Also, the manner in which ponies speak of the moon, including the alicorns, suggests this as well. While it is possible that it is an elaborate trick, I am going to throw Occam's Razor at this one. Presumably, the sun and stars are similarly situated.
So, a few quick deductions before we move on to further discussion.
7) The current celestial body(s) is visible from everywhere on Equus. This is implied by (2) and (4) with a dash of (6). If eternal night is to be taken seriously then there must be nowhere where it is eternal day because that kind of defeats the point. Personally, I'd be hanging out in eternal twilight in between, but whatever.
Further, there is the troubling presence of (4). With the celestial bodies exchanging that quickly, a huge portion of a round world would never see day or night. Equestria's climate suggests it is a fairly large country considering it has the 'Frozen North' directly above it and the 'Badlands' and the hot desert 'San Palomino Desert' to the south. While we can not take Equus as an Earth analog due to the planned weather cycles and the irregularities with the sun and moon, These climates should not occur anywhere near each other.
Also, the globe seen in the Golden Oaks Library combined with the official map of Equestria suggests that the country must cover a sizable portion of the globe. While it is possible that it solely occupies space bathed in full day and full night, it is unlikely. Then somewhere in the country must be experiencing the effects (at least moderately) of reduced day and night intensity which we do not see in the show despite having footage of the north, south, east, west, and central Equestria.
Additionally, with (6) we assume that sunlight and moonlight are real things obeying the laws of whatever physics exist in universe and not an elaborate illusion of some kind that could break the deductions we have previously made.
8) There is space not visible from Equus. This follows from (7) and (3). From (3) we get that the sun and moon are actually raised and set and thus out of sight. Since the missing celestial orb must go somewhere, it follows from (7) that it is not visible from Equus and thus is somewhere not visible from Equus (perhaps inside it). I reluctantly admit it is possible that the sun and moon are recreated each day/night, however a pony capable of that would not get their flank handed to them by Chrysalis (Discord is fine since he cycled the day/night too). Concern over Sombra is also evidence against that as well.
9) The sun and moon are never inside Equus (at least in the process of their normal cycle). By (5), Equus is locally Euclidean and obeys the plane (or space in this case) separation axiom. This means that for the sun or moon to enter the inside of the planet it would have to either pass through it or teleport into it. First consider the former. Surely, this would be important enough to bear mentioning in the canon (perhaps not in the show but somewhere) it the sun itself sank into the planet everyday. Since this has not happened, we can assume it is not the case.
Next consider teleporting into Equus. We have seen the sun/moon raised and set over the horizon (whatever that means in Equus for we have not discussed it yet) which would require careful continuous teleportation of the sun/moon to the horizon location. This is theoretically possible, although needlessly complicated compared to utilizing the space in (8). Further weak evidence points away from this in that we have never seen a partial teleportation in canon. There are plenty of mundane uses for that (a bad example is slicing bread without a knife). Also, as in (8), there should be no reason a pony capable of such a feat would lose to Chrysalis. This weak evidence against leads us to us Occam's Razor to discount the possibility (As a note, the geometry does not become Euclidean if this is the case; we argue this only to get to (10)).
As a note, we are assuming that the sun is not allowed to move extra-dimensionally. That is to say, the in-universe is perfectly capable of being fully observed from 3-space or with additional equipment to view n-space. This is again an application of Occam's Razor since we have no evidence to suggest otherwise (teleportation is magic and magic is allowed to cheat). At least not yet.
(10) There is space outside Equus not visible from Equus. Further, it is reachable. This is a direct result of (8) and (9).
Now that we have something to work with, we can start thinking about what axiom system and model can be used to describe Equestrian geometry, if any. Once determined, we could also answer the question 'What does Equus look like from the moon?' which is bound to be interesting. An intuitive guess would suggest that all of Equus is visible at any given time during the night from the moon.
Continued here: Equestrian Geometry - A Model
The conclusion you came to is very similar to what is described in Sharing the Night. Where the space encompassing the planet is non-Euclidean in nature such that the night sky looks identical from every point on the planet.
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The sky need not look identical from every point. Sharing the Night was the original inspiration for me to take a closer look at this, although it only gives us a few conclusions without observational evidence. I'm hoping to develop a full model and axiom set but things are looking less than consistent for poor Equestria without simplifying the problem with extra dimensions.
2000327
It would be near impossible to make such a model without extra dimensions.
Ah, I've read at least a couple stories that dealt with this cosmological issue by assuming exactly that-- that the sun and moon essentially entered another dimension when not in use. Several things on the show (such as that weird hallway Twilight ascended in) imply the existence of such spaces in canon.
Deciding not to assume against it does make things more complicated. But not impossible. I'm going to question some of your other assumptions, such as that the Moon and Sun aren't visible from any part of the planet when 'put away'. It's possible that they're just not visible from anywhere Nightmare Moon cares about. Maybe the other side of the planet (opposite wherever Celestia designates noon) is ocean or wasteland or otherwise void of life that could appreciate her night sky. Then, when Nightmare Moon talks about eternal night, she means for everywhere (and everyone) that matters.
Eh, don't think of it as losing to Chrysalis. Think of it as losing to weaponized love. If there's one thing the show is consistent about it's that emotion-based magic channeled properly trumps everything else-- whether it be friendship through the right artifacts, love through Chrysalis and Cadenza, or hatred through Windigo.
As an aside, I really don't think Sombra runs off hatred. The two emotions I find most likely, given the episode are either greed (which is why his obsession with crystals outlasted the rest of his mind and explains his need to dominate) or fear. I think fear is more likely. His nightmare door certainly works with it. More interestingly, the crystal ponies respond to the mere mention of his name with terror-- because he kept them terrified. No one said it had to be the caster's fear. Imagine Sombra controlling the Crystal Heart, with the power to broadcast emotions into the populace, filling the entire Empire with blind, senseless fear, then tapping into the fear of all of his subjects to feed his magic. In a world where emotional magic is the king, it's a powerful strategy.
EDIT: This would also explain canon Twilight's aptitude with it. We've never really seen her given to hatred, but she has displayed incredible fits of neurotic worry, enough to easily power all but the greatest fear-magic without needing to terrify people.