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Equestrian Geometry - A Model · 5:57am Apr 12th, 2014

Second part in a series on Equestrian Geometry
First part located here: Equestrian Geometry

Two things.

First, I am honestly surprised I have managed to find a model that satisfies the requirements we set forth. I have created a simple program that you can use to visualize straight lines inside of the space visible from Equus. You can download it here Visualization. If you don't have Visual Studio the debug executable should work. You might also be able to build with Mono. As a note on the program, it is set up from the perspective of moving the planet around, but the planet can be thought of as the just an arbitrary point on a line. While I have not given it much thought, it is possible (but unnecessary) that we could extend the model to work with the planet outside the parabola we will discuss rather than inside it.

Second, canon does not provide us with enough information to choose a specific model, or even a specific set of properties in this model we will discuss. However, the fact that we can find a model at all is good news and will make it much easier to create some axioms. As a forewarning, for similar reasons we will likely be unable to choose a complete set of axioms (for those of you screaming Godel's Theorem, geometries in general can be complete, eg Tarski's Axioms) without picking one (or more) axioms (or perhaps we will need to strengthen one axiom) to add that do not contradict canon but neither are given evidence for by canon.

Alright, now the model. To simplify, we will consider the 2D version of our geometry (it is straightforward to extend it to the 3D case as paraboloids or revolution also have a focus). We will start very zoomed out to observe the grand scheme and then take a close up look at Equus to see what's going on there.

Choose a (Euclidean) parabola. The exact shape and direction are irrelevant for the general model. For understanding, draw the focus and draw the (Euclidean) line through it parallel to the directrix (again, the Euclidean one) and call it the reflectix. Next, choose a location inside the parabola to fix Equus (which is approximately just a point at the distances we are considering at the moment). While it is possible to place it on or below the reflectix, it makes more sense to place it above it.

Figure 1 - A very simple instance of this model with a line (really a ray)

The plane of our model is the ordinary Euclidean one. Our points are also ordinary Euclidean points.

A line is defined (inside the parabola, we will discuss lines outside later) as follows. Given a point and a (Euclidean) direction draw the (Euclidean, as it will be until the end of this definition) line containing this point heading in the direction given until it intersects with the parabola. Note that we will describe half the line (a ray) here, but the other half is constructed in the same manner. Next, draw the line segment from the point of intersection with the parabola to the focus.

Next draw the line segment originating from the focus with the same angle as the previous line segment coming in on the opposite side of the reflectix. Figure 2 shows the case for when the line segment is above the reflectix.

Figure 2 - The angles that must be matched

Finally, draw the ray from the new point of intersection with the parabola upwards parallel to the axis of symmetry.

Now consider the odd case. The (not Euclidean) line lying on the axis of symmetry. The half of the line constructed by the downward direction is as normal, but the upward ray is different. It never intersects with the parabola and simply continues on to infinity. This is undesirable, but unavoidable so we will simply accept this as a valid line as well. As a note on our physical interpretation, this is an infinitesimal small section of the universe that can not be observed so we can ignore it (it remains so in the 3D case as it is still a line).

For clarity, I will reiterate that despite our figures showing lines (rays) originating from Equus, any point inside the parabola is a valid point to pick to originate a ray in the same manner. Further, there are points and lines outside the parabola. We will discuss them in further detail at a later time as this section is longer than I wish.

Next, let's check that this model has the properties we wanted. We need only check (1) to (6) as the ones after are derived from them. (0) is automatically satisfied.

(1) is clearly satisfied. The 2D case uses a circle and the 3D case uses a sphere.

(2) requires us to define the path of the sun and moon (briefly note that (3) and (4) are satisfied if (2) is satisfied as they are observations on variations in speed on their paths). They do not travel in a straight line or a circle nor an ellipse. Consider if they travel in a (Euclidean) ellipse centered on the axis of symmetry such that part of it lies inside the parabola (all below the focus, but near it) and part of it lies outside. Given a large enough sun/moon (there is no reason they can not be the same size), it will intersect a large enough swath of lines that every side of Equestria will see a sun/moon of some size (variation is potentially large, although that is a discussion for later when we develop a distance function). It is important to note that no part of the sun/moon should intersect the focus because then the entire sky would be the sun/moon while it is in overlapping it. As a comment, we can place the stars at some arbitrary distance (preferably a large one) above Equus and as all lines (inside the parabola) go off to infinity in that direction, we are guaranteed that all of the sky will be filled with stars. This is extraordinarily difficult to show in an image, so at this juncture it will either have to be taken on trust or but examination using the program or pencil and paper.

Take a moment to consider that in our model, as far zoomed out as we are, when we consider Equus a point then the lines emanating from one side of a (Euclidean) line to the other are the lines of the visible sky (see Figure 3).

Figure 3 - Shows several lines in the sky from the point of Equus
where the (Euclidean) perpendicular is roughly 30 degrees left of the
axis of symmetry including the left and right extremes

Admittedly, this discussion has been very hand wavy. I am more interested in providing a model not for the sake of performing rigorous proof based on it, but to help both myself and everyone else visualize the shape the MLP universe could have. I do not know if there are other models that are significantly different, but this serves as one. If I were to point out the major flaw of this particular model, it would be that the angles are not in any way conformal away from Equus (and perhaps not even there, although the size of Equus is so small that they are approximately so).

Penultimately, we comment that (6) is also satisfied as the moon/sun has a path that allows it to be a real thing and satisfy all of the other properties we desire.

And lastly, a curious observation about the model. Equus is visible from Equus. However, it is far enough away and small enough that at worst it will appear to be a star and at best not be visible at all (for it only emits reflected light). Astrophysics are unfortunately far outside my specialty.

Next time we will start on axioms!

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Comments ( 9 )

This is awesome stuff! Looking forward to more.

I knew I would see you around here! When using started reading this story, I thought of how you wrote your Twilight. The similarities are shocking, actually. Inspiration maybe? XD

Similarities to the cosmology? No, I'm drawing from Medieval European and West Asian ideas of astrology and geomancy.

Do you mean his story Seeking Power? I've got it favorited and am going to read it as soon as it's finished, so I don't know about the similarities there. If I had to guess, I would say it's a case of two analytical minds coming to the same conclusions on how to project Twilight's show-cannon personality into a more mature narrative. Now I want to read it more than ever!

Maybe I'll break my rule about starting unfinished stories for this one.

I have one problem with this... is the surface of the universe smoothly reflective or projective, or changes on the number of light bounces?

I mean, I have a serious problem with the first part of constructing the ray as you described - where it bounces off the paraboloid at a very random angle that happens to let it cross the focus.

I mean - say, a star is shining "far above". Its light can reach Equus directly from above, or bypass it, reflect from the paraboloid, pass through focus, hit the paraboloid again and... on the first hit it should have acted as a mirror, retaining falling angle=reflected angle rule. But on the second one the surface should appear matte, with the image not reflected in one direction but projected like on a cinema screen - only that way it will be visible from any arbitrary angle and not from one corresponding to the second reflection angle.

on a separate note, I choose a different approach in a fic that will probably never see the daylight...
Equus is a small segment of a Dyson Sphere - a small material part of sphere radius 1AU which in its bulk is a magical construct (I call it pellicle). Sun and Moon are launched on a ballistic curve each morning and evening, and then they travel under Equus, through ancient mechanisms, recharging from the real Sun. They can be controlled mid-flight although it takes significantly more energy than just the "morning launch". Obviously, inhabitable Equestria is mostly "flat earth" but other planets still circle the sphere and the "real Sun" in the middle - Venus and Mercury are not known.

That sounds like another working model. Anything that satisfies the assumptions in the first part will work equally well (although how Equestria ended up on part of a Dyson sphere, who knows). The one I provided is rather clunky, but it works (that's often how research goes; the first example is awful but sufficient to show existence, and then it's refined from there). I never really got around to it, but I am reasonably certain there are less contrived models that illustrate the assumptions at work.

Honestly, I think you can just say the universe is a parabola and call it good, but I haven't checked that. The moon/sun should still be visible from everywhere on Equus except the very top at the north pole, which we haven't actually seen a view of during the show.


(although how Equestria ended up on part of a Dyson sphere, who knows).

Equestria is multi-thousand-years-old Earth, with humans making one big mistake - creating Discord, who choose to remix Earth in shape of its children tales.

Okay, I'm having a lot of trouble grasping this. It's a cool model, but I feel like I'm missing half of the information required to understand this (I really hope it's not because I'm blind). Below is where this concept is currently in my head.

Equus is the spherical planet noted above.

The lines are rays of light? (With the horizontal line merely being a reference plane.)

Is the parabola is the edge of the universe? How does this universe edge behave when the light hits it? If it was like a mirror, the angle going from Equus to the focus doesn't make sense.

The focus: is it just a big spherical mirror or something? All I'm imagining right now is a big radar dish.

The stars are presumably really high up in the y axis or something?

Where is the sun and the moon? You said it travelled in an ellipse, with it's centre on the axis of symmetry, slightly below the focus, partially leaving the universe. Are they not in the model? If they were, what plane would the ellipse be on? I'm guessing the XY plane, but would it work equally as well on the XZ plane?

The lines are just lines. Light does travel along them, but they are just straight lines.

The parabola is the edge of the universe in a sense. It's more appropriate to say that the universe turns back around into itself. It doesn't have an edge, it's just that at some point if you keep going straight in a Euclidean sense, you'll find yourself going some other way (again, in a Euclidean sense). This particular model (it's not the only one, I'm sure) doesn't behave like a mirror at the edges. This is a very contrived model that satisfies the assumptions simply to show that such a model does, in fact, exist.

I don't know what you mean when you ask if the focus is a big spherical mirror. It's just a point in space. The only thing special about it is that all straight lines pass through it. It would be a good approximation to think of this model as a big radar dish.

The stars are indeed very high up in the y-axis.

Here is a link to another discussion of a simplified (meaning less contrived) model which has diagrams showing how the sun and moon move. You can also follow the chain of replies up and down for further discussion of the model. Note that it doesn't matter what plane the sun and moon are on in three-space (indeed, they don't actually have to move completely within a single plane, but neither is there a reason for Celestia/Luna to move them otherwise).

Thanks for the reply. Apparently, I read the first and last third of that conversation, somehow skipping over the middle third. This definitely explains it a lot better. Thanks

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