Spikey Spheres

I find some value from picturing 4D objects as 3D objects moving and changing size and or shape through space. In the ball example. At T:1/4 and T:3/4 I see the same 3D ball shape with different balls. At time 0, 0.5, and 1 they are gone.

As such it's natural for a 4D where to fit in the middle as it can be centered at T:1/2 and you just need to find the intersection point in 4D space where the balls touch.

Now, this does not work for rotations, but it's better than nothing.

However there is a 4-dimensional sphere (3-sphere) that you can easily access: the possible rotations of any object in 3-dimensional space is topologically and metrically isomorphic to a (half of a) 3-sphere, in the same way as the rotations of an object in 2-dimensional space are isomorphic to a 1-sphere (circle).

Think of an object’s current orientation as the “north pole” of our abstract 4-dimensional sphere. Rotating the object around any axis pushes you toward the equator, which you reach once you have rotated the object halfway around (the equator is made from all 180° turns, and is topologically a 2-sphere). Continuing to rotate the object takes you towards the “south pole”, which is where you get if you rotate your object 360°. To get back to the north pole, rotate by 360° again.

Cf. https://en.wikipedia.org/wiki/Versor https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati... and https://en.wikipedia.org/wiki/Rotation_group_SO(3)

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One other way you can get at the 3-sphere is by taking a stereographic projection into all of 3-dimensional space (plus a point at infinity). If you have some 3-dimensional shapes drawn on the surface of your 3-sphere, these will get distorted under the stereographic projection, but locally angles will be preserved, just like a stereographic projection of the 2-sphere onto a piece of paper.

>A teacher of mine told us, "Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere, you're wrong. It's just not possible."

Your brain is a computer. If a computer can manipulate 4D hypersphere, so can you.

You can't visualize it, but that's a different thing. You can't visualize a 3D sphere either, only 2D projection.

I think he was wrong. It's pretty hard but not impossible. You don't visualize them directly as 3D objects though, you visualize 3D objects with metadata attached everywhere. While you navigate the space in your mind's eye you have an extra layer of information that flows with you.

Consider Alicia Boole Stott. She created cardboard models of the various cross-sections of the 6 regular polytopes. Including the 24-cell which has no equivalent in 3D, the 120-cell and the 600-cell which are super complex figures. I think if you read her story you'll also be convinced that she indeed visualized the figures. [1]

I spent quite a lot of time earlier this year on this visualization exercise on the 4 simplest regular polytopes while working on an toy application to manipulate 4D objects and tesseractic honeycomb in VR.

You can have a good grasp of where every edge go and the angle between every faces. I could count the number of vertices and edges of a 16-cell or 24-cell just by navigating around the figure in my head. The tesseract and 16-cell are good starting points.

1: http://www.sciencedirect.com/science/article/pii/S0315086007...

I've been thinking about the 4D visualization problem and I am becoming increasingly convinced we can develop some cool tools to help wrap our mind around it using VR goggles.

There are two ways to draw a 3D object in 2D. One is to draw slices, or projections. The other is to use tricks of "perspective" (points of infinity and shading) It is very hard to visualize a 3D object simply from the slices model.

When we try to draw 4D objects in 3D, we often simply construct models that are slices of the 4D objects and then try to extrapolate back in our minds what the 4D object "looks like". This is even more difficult to wrap our brains around.

I postulate that we can develop similar tricks of perspective and shading that will help use see a 4D object if we draw on a 3D canvas. I don't think we can do so on a 2D canvas and make any sense of it. However, I'm hopeful that using VR goggles and drawing in 3D might be the trick.

I've only really just started thinking about this idea. I have a ton of books on constructing perspective for 3D objects on 2D surfaces. Much has been written about it since the Renaissance. It is only in the last couple of years that we've had an effective and accessible 3D canvas to work with. (constructing hundreds of sculptures is not really practical)

I don't think a lot of work has been done with VR goggles yet for 4D visualizations.

> you get a Gaussian distribution as the dimension goes to infinity.

Wow, that's really cool. I wonder, I'm sure you could prove this in more direct ways, but is it also a consequence of the central limit theorem?

It should be clarified that this fact refers to a ball of volume 1 in N dimensions. Since the volume is held constant, as the dimension goes to infinity, so does the radius.

Sorry for the nitpick, but I assume you meant that the chance goes to 100%, rather than infinity.

That makes sense now that you mention it. The equation for the surface area of an n-sphere goes up as r^n, and the volume growth at any given radius is equal to the surface area at that radius.

I've heard this phrased as "oranges are not a fun fruit in high dimensions".

Exactly! Now consider that deep networks that classify images are tasked with getting reliable statistics in very high dimensional spaces. Conv nets are being forced to map from high dimensional spaces down to a very low dimensional categorical decision. This is why weird classification errors you see in "adversarial examples" keep popping up. The learned classification boundaries are very spiky, and it's easy for the world to fall in between the spikes. More data can't solve it (at least not practically) because there are way too many gaps between the spikes.

A more tractable approach is to learn to use dynamics for perception rather than ("just") statistics. The dynamical physics of a ball rolling is much simpler (lower dimensional, more tractable) than a statistical view of millions of differently illuminated pixels hitting a camera.

(A colleague of mine has a blog post on this issue of "statistics and dynamics" at http://blog.piekniewski.info/2016/11/01/statistics-and-dynam...)

I've heard this phrased as "objects in high dimensions are mostly surface", which makes clear that it's not that there's a less dense "core" but that every interior point is near the surface.

That seems very intuitive if you think about the difference between a line segment and a circle. In 1d space a segment that extended S1: from 0 to 1 covers half the space of one that extends from S2: 0 to 2.

In x,y space you have that same 50:50 ratio at y=0. But, for every y > 0 the ratio decreases as a higher percentage of S1 is used up just reaching that y height than S2, until all of it is used up and the top segments are 100% s2.

In x,y,z space you get that same picture at z=0. But at any z > 0 more of that S1 is used to than the S2 segment so the ratio must be lower than in x,y or x space.

Moving to 4D and up the same simple idea just continues.

> I am not sure I would interpret this observation as spheres that are getting more spikey. Maybe what gets more spikey is the actual box, so the sphere has it easier to touch or even extend outside of it.

I think that this is not correct. In n dimensions, the cross section of a unit n-cube near the edge is a unit (n - 1)-cube, just as your intuition tells you. However, the cross section of a unit n-sphere near its intersection with the unit n-cube is a tiny, tiny (n - 1)-sphere (the radius of a cross section at height z = 1 − 𝜀 goes to 0 as n goes to ∞); this is the 'spikiness' that the author is discussing.

This comes down to the definition of (hyper)volume. We use a (hyper)cube as our unit, but that’s somewhat arbitrary.

We could alternately define (hyper)volume using a standard n–simplex as the unit, then we would say the volume of an n–ball → ∞ as n → ∞.

here is a way to intuit that:

- because (1-ε)^n/1^n goes to zero if n goes up, most of the volume of a cube gets concentrated near its surface.

- to go from a n-dimensional cube to a n-dimensional sphere, you have to cut away most of its surface (everything except for 2n 'caps' near the points with coordinates (0,0,0,...,0,0,±½,0,0,0,...,0,0,0)), and hence, most of its volume.