Intrinsic View of $\\mathbb{S}^3$

Intrinsic View of $\mathbb{S}^3$

We are often accustomed to viewing the three sphere through stereographic projection - but this is not what it would look like to an inhabitant. Here, we will try to understand what it means to "look around" inside a Riemannian manifold using 𝕊3S3 as a guiding example.

What do we mean?

What does $\mathbb{S}^3$ look like from the inside? Due to the fact that $\mathbb{S}^3$ can be realized topologically via a one-point compactification of $\mathbb{R}^3$ its easy to get a non-metric picture: the three sphere looks like $\mathbb{R}^3$, but any two paths which head away from the origin forever “meet at infinity”. Via stereographic projection we can even make such a thing precise and draw amazing pictures of objects inside of the three sphere. For instance, the hypercube to the right.

If you think enough about the three sphere you become quite accustomed to dealing with these pictures, and can use them quite profitably as aids in reasoning. Recall the pictures we drew of the Hop fibration, and how thinking about them allowed us to see how to construct other Seifert vibrations of the three sphere.

However, these pictures are not an accurate portrayal of what $\mathbb{S}^3$ would look like from the inside. Our view of the hypercube’s 1-skeleton and of the Hopf fibration would be all but unrecognizable to an actual inhabitant of the three sphere. The way things “look” is fundamentally a geometric property as we will see, and the violence of stereographic projection almost never preserves that picture.

To see what I mean here, let’s think quick about what goes into looking at something. Light bounces off of an object and travels to directly to your eyes along straight lines…but of course this is only an approximation. We know that even in the real world light doesn’t travel “straight” (whatever that means) but instead follows geodesics in spacetime. Thus, geodesics themselves look perfectly straight because light runs parallel to them! Thus, we would expect that if we were to get dropped off inside of a curved world, the objects that look straight to our eyes would be those that well approximate geodesics. And so we can see just how flawed our pictures are: each circle in the Hopf fibration is a great circle in $\mathbb{S}^3$ and thus a geodesic. But Hopf circles almost never look straight in our drawings!

The Visual Sphere

In order to construct more accurate renderings of $\mathbb{S}^3$ let’s back up and think a little more precisely about how we would like to model vision. While technically incorrect it is useful to think of vision the way the ancient Greeks did - that our eyes are beaming out light which then hits objects so that we can see them (this is the way modern computer retracers draw imagery however).

If we are standing at a point $p$ in some Riemannian manifold $M$, we will model our visual field by the unit sphere of directions at that point in the tangent space. Given some direction $v$ in $UT_p(M)$, the points of $M$ that we see when looking in that direction are exactly the points on the geodesic $\gamma$ through $p$ with direction vector $v$, a curve we can easily describe using the exponential map:
$$\{\exp_p (t v): t\in\mathbb{R}^+\}$$
(Recall $\exp_p(w)$ is by definition the point in $M$ distance $\|w\|$ along the geodesic through $p$ with initial direction $\hat{w}$). If there is some object that we want to look at in our manifold, say a subset $X\subset M$, the size of this in our visual field is just the solid angle in $UT_p(M)$ of directions for which the associated geodesics nontrivially intersect $X$.

To describe this region explicitly we take the preimage of our object under the exponential map to get its image in the tangent space, and then we can project it onto the visual sphere by the standard “divide by the magnitude” projection.
$$\mathbb{P}^+\left(\exp_p^{-1}(X)\right)$$

This picture produces exactly what we would expect when we apply it to Euclidean space: projective geometry originated out of the desire to get perspective correct in artwork, and under the canonical identification of $T_p\mathbb{E}^3$ with $\mathbb{E}^3$ the exponential map is actually the identity - thus the map above becomes nothing but positive projectivization!

What does $\mathbb{S}^2$ look like?

We’ll start off with a case that we can visualize completely: that of the 2-sphere. Say there is an ant on the sphere looking at some other small object:

The angular size of this object in the ant’s field of vision can be found by looking for the geodesics which just glance the edges of it:

Already we can see something interesting is going on: as the object moves away from the ant at first it appears to shrink in size, as we would expect from our experience in Euclidean space:

But as the object continues to increase in distance from the ant past a certain point, quite counterintuitively its size appears to grow! This is because the positive curvature of the two sphere causes initially diverging geodesics to converge again, and so tracing backwards the geodesics which just glance the object we see that they were diverging even more originally than before!

In fact, objects on a unit sphere of distance $\pi/2+x$ from an observer appear the same size as objects of size $\pi/2-x$! And so, as the object nears the antipodal point of the ant on the sphere, it continues to grow in size until it fills the ant’s entire field of view: no matter which direction it looks, that geodesic ends up hitting the object.

But this isn’t the only surprising visual effect - if we analyze the above story closer we can see that the ant also sees a surprising amount of the sphere at different distances. From our experience in Euclidean space we are well aware that you can never see more than half of a sphere at any given time (you can only see a smaller portion than this if the sphere is close to your face, and that proportion limits to half as the sphere recedes to infinity). This qualitative behavior seems to hold true for objects near the ant: the further away they get the greater proportion the ant can see. However once the object has passed a distance of $\pi/2$ and appears to grow in size, the proportion of it that we can see keeps growing, way past half!

Thus while objects at distances of $\pi/2-x$ and $\pi/2+x$ take up the same proportion of the visual field they don’t look identical: the ant can tell that an object is nearby if it can see less than half of it, and faraway when it can see more than half of it!

Playing ball in $\mathbb{S}^3$

Let’s try and tell this story one dimension up, inside of $\mathbb{S}^3$. Working out the exponential map (if you’re brave!) or using spherical geometry, its easy to see the same general phenomena hold. Throwing a ball in the three sphere would be a wild experience - to our brains trained for Euclidean space the ball would appear to recede for awhile then slow down and reverse, coming back straight at us with increasing speed! All the while we would see the surface of the ball distorting, so that when the ball appears to be right in front of our face if we can keep ourselves from flinching, we would see 90% of its surface. Right when our brains would be telling us that the ball was going to hit us it would momentarily appear to engulf our entire world: the sky in all directions would be replaced with a view of the baseball, as though its skin were turned inside out. The ball would then continue on its journey past our antipodal point, shrinking in size once again. We would now see less than half of the sphere and everything would appear flipped around and upside down, like when you see your upside down reflection in a concave mirror.

The ball, now approaching the backside of your head would shrink in size until it was a distance or $\pi/2$ away, and the proportion of the ball that we could see would continue to grow, even past this point when the balls apparent size was growing once more. This time as the ball fills our vision we should be worried: it is about to hit us, just from behind!

This story gives us some sense of what its like inside of the three sphere, but no real global picture. We'd really like a way to see what it looks like when all the great circles of the Hopf fibration are nice and straight. And for that, we'll have to come up with an explicit projection.

Perspectively correct projection

4D Polytopes

Now that we can create accurate intrinsic views, let’s make some comparisons with the views we are used to. Let’s draw a hypercube on the three sphere (a tiling of $\mathbb{S}^3$ by eight equal cubes), stereographically project it while rotating $\mathbb{S}^3$, and then view this from the point $(10,10,10)\in\mathbb{R}^3$ to get a familiar picture:

Now, drawing this perspectively correct gives a quite different view:

The first thing of note is that you can only see a portion of the cube at any given time - and this isn’t an artifact of the path we have taken in $\mathbb{S}^3$ - there is no “zooming out” if you live inside of the three sphere as moving around will just move you from one cube into an isometric one (compare this to the image under stereographic projection above: one of the cube faces is unbounded and moving through that one in $\mathbb{R}^3$ allows you to view the entire 1-skeleton of the hypercube at once). Also, the edges look straight! This is of course what we expected, as the edges are arcs of great circles and thus geodesics in $\mathbb{S}^3$, but its good to see that it really happens.

Just for fun here’s a side-by side comparison of these two perspectives for the four-dimensional generalization of the tetrahedron and its dual.

The Hopf Fibration

Here's what we've been waiting for: a walk inside the Hopf fibration. We'll proceed slowly though to make sure that all aspects of the pictures make sense. First, lets see what a single great circle looks like inside of $\mathbb{S}^3$

It does indeed look straight! Let's move on to a couple of geodesics inside of the three sphere:

What do the tori of circles inside the Hopf fibration look like? Each fiber twists about the torus once in each direction, and being a geodesic, should appear to be a straight line. However the torus surface itself is intrinsically Euclidean, and so should appear negatively curved to an inhabitant of the three sphere. A potentially good analogy is that of a hyperboloid of revolution in Euclidean space - this surface is ruled by Euclidean straight lines which appear to twist about it, and the surface itself appears to be negatively curved.

Let’s check our intuition: here I have drawn four tori worth of circles in the Hopf fibration, and placed the viewpoint as far from them as possible. Here's a quick video of what it looks like if you were to spin around and inspect your surroundings:

Ignore the weird part here where the fibers are getting cut off when they get near you - this is just an artifact of how this image was plotted. But cool! They locally do look a lot like ruled hyperbolic paraboloids!

For a final visual, were going to look at the view inside of $\mathbb{S}^3$ if you drew a couple of tori from the Hopf fibration, rotated all of the three sphere by a fixed-point free family of rotations (rotations of $\mathbb{R}^4$ which rotate simultaneously in two orthogonal planes). Here’s the standard stereographic view - note that the two tori appear to switch places, the inside becomes the outside and vice versa:

And, here’s the intrinsic view:

Here's an interesting question: while the tubes here representing geodesics do seem to follow straight lines, their surfaces don't look like that of a cylinder, but instead look slightly negatively curved. Here's a hint: these tubes are actually equidistant surfaces from the geodesics that lie at their cores.