One trick I thought of for thinking about high-dimensional spaces is to put multiple dimensions on the same axis: Consider the vectors in R² from the origin onto the unit circle. Lengthen each into a axis, each going infinitely forward and backward, each sharing all its points of R² with one other, all of them intersecting at 0. Embed this in R³, then continuously rotate the tip of each axis into the new dimension, forming a double cone centered at 0. Rotate them further until all tips touch, forming a single axis that contains the information of two dimensions.
You can now have an axis contain the information of any R-vector space, and visualize up to 3 at a time. Of course, not all mental operations that worked in R³ still work.
I don't get what you're saying. Do you mean, you define a map f: R² -> R by f(v) = abs(||v||) (and then you map this to the z-axis)?
"each sharing all its points of R² with one other"
I don't know what this means; I think you're saying each axis overlaps with the axis from its antipodal point.
Could you give an example of something that's difficult to visualize, but it's easier with this method?
Trying to picture the warmup is already hard enough for me, so I'll start with asking questions about that and revisit the rest later:
(Exercise: what are the lines of latitude? What about longitude? Can you picture the north and south pole in different placetimes, and the corresponding equators?)
I expect that the longitude are the lines taken by choosing one in the middle sphere, and tracing the line following it on both side? As for latitude, if I use the analogy of the 2-sphere, each circle in the film is one line of latitude; so maybe each 2-sphere in this film is a latitude?
Also, I don't understand what you mean by your last question. In the 2-sphere version, the poles are only visible at time 0 and 2 and the equator is only visible at time 1.
Yeah, in my interpretation of "latitude line" it's not really a line, but rather "those set of points that are exactly distance d from the south pole". Which as you say are the 2-spheres that are visible at each instant of time.
>and tracing the line following it on both side
Sorry, I don't know what you mean by this. I think you're saying that we pick a longitude line in the equatorial S², and then draw the S² that passes through the south pole and that longitude line. That's also what I had in mind for longitude lines. We could also interpret longitude lines as great circles (rather than "great S^(n-1)" where we're considering S^n). Then it's actually a line (I mean, a geodesic, i.e. a great circle), and it looks like two antipodal points on the S² at each instant, never seeming to move (except that they ride along with the expansion and contraction of S²). These describe a great circle passing through the north and south pole. There's an S²'s worth of these lines, since the equator is an S² (analogous to there being a circle's worth of longitude lines on S²).
>Also, I don't understand what you mean by your last question
Edited the post. I mean to say, what if you put the south pole somewhere else. One point of the exercise is to get you to picture other great circles and "great S²s".
(I misused "longitude line"; I guess that normally refers to half a great circle. So, we'd sweep out half of a great S². Which... bleh. I'd rather think about great circles.)
Interesting. I visualize a 3-sphere as a "filmstrip" of 3d film, i.e. a line of 2-spheres of increasing and then decreasing size. This gives up continuity, but something has to give to use the human visual system for higher-D spaces. I find mapping a space dimension to time gives me little idea about the simultaneous spatial structure (even the "visualize a 3d object by viewing its 2d cross-sections one after another" version seems quite hard to me).
Lexicographic ordering is one useful way to collapse 2 dimensions into one, I just prefer to collapse 2 spatial dimensions into 1 spatial dimension that way. (As mentioned in the comments, another way is to treat the 2d as a complex number and then keep the magnitude, perhaps representing the argument as a rotation about the new axis, or using color (hue works well). Anyone know any others?)
Yeah, that's a good method. And then a 4-sphere is a dynamic movie of these 3d filmstrips. I seem to default to picturing a 3-sphere as spheres that overlap (and I just know that they don't actually intersect because they're separated in the 4th dimension). I'm idly curious whether your 2-spheres intersect, or if they're lined up side-by-side but separate.
Actually now that I think about it more it seems like the fimstrip visualization is better for the things I was using "groundhog day" for. E.g. the 1,3 saddle point level curves make good sense as a movie of 3d filmstrips, and likewise the 2,2 saddle point. That I came up with groundhog day instead of filmstrip is a bit of evidence that there's something easier about it in some cases, but next time I'll try starting with filmstrip.
They're separate, and equally spaced (like actual film). That means that the difference in radius between the first and second 2-spheres has to be much larger than the difference between the middle and next-to-middle ones. I don't visualize more "frames" than I need for whatever I'm doing, though fewer than 5 doesn't really work, so I think most often I use 5. You can still get an "all on top of each other" (2d) "view" by making the 2d spheres semi-transparent and looking at the filmstrip from one end.
It actually extends okay into a planar grid of 3d frames for 5d; less well to 6d (things start "occluding" others too much) but maybe still sometimes useful. You can even add meta-film and sort of get it up to 9d. Anything beyond that I don't find it possible to actually see any variations in all the dimensions at once (I'd REALLY like to have an intuitively meaningful visualization of the Leech lattice, but 24d just doesn't seem possible with any technique I can think of...)
In my experience / opinion, the biggest problem with these techniques is that rotations that are partly in one "level" of the visualization and partly in another really aren't natural... of course, for the special case of a sphere, rotational invariance means that doesn't matter :-)
Look at the Connection Machine CM-1 and CM-2 (http://tamikothiel.com/cm/cm-design.html) for a really cool physical realization of this, btw.
In the "Warmup" section, I would make the first paragraph (dealing with visualizing S2 in R2 (space) x R (time)) not a paranthetical, and remove the "If the following paragraph is too hard" part. I expect most people who may read this (at least those who will benefit the most from it) will find the exercise of visualizing an object they are already familiar with a good intuition pump for using the technique to explore a less familiar conceptual space, so that section should be treated as a main part of the text.
I'm glad you wrote about this topic, since it is something that people often incorrectly think is not possible to do, and I may share more thoughts once I have finished reading this.
You're right, fixed, thanks. (Post was written hastily, since it was a priori unlikely I'd finish it.)
Here's a tool for visualizing some 5 dimensional things. Poetically, the idea is to fold multiple "time dimensions" into the one actual time dimension using a sort of lexicographic ordering. More literally, we play a short 3d movie (each instant is a 3d scene) over and over, varying something about it across each playthrough; each short playthrough traverses the fourth dimension once, and the variation across playthroughs is the traversal in the fifth dimension.
As a warmup to the warmup, let's visualize a 2-sphere S², a.k.a. a regular old plumbus sphere (hollow ball). A sphere S² can be seen as bunch of cross-section circles stacked on top of each other. Or, instead of stacking them in space, we can "stack them in time" by playing a 2d movie: blank screen -> point -> rapidly expanding circle -> circle gradually reaching its widest, then reversing (at the "equator" of the sphere) -> circle rapidly shrinking to a point and disappearing.
Now the warmup: how to visualize a 3-sphere S³, which "lives in four dimensions"? A 3d cross-section of S³ is a 2-sphere. Each cross-section S² fits in our 3d world ℝ³, but how to deal with the fourth dimension? We can use time. We play a movie forward. At each moment, we have a 3d scene. The scene starts off as empty space. Then for an instant, there's a single point p. The point immediately becomes a rapidly expanding sphere S² (centered at that p). The sphere expands, slower and slower, until it reaches its widest; then it starts shrinking. It shrinks faster and faster until in an instant, it shrinks to a point and disappears.
As landmarks, say the whole movie starts at time 0 and lasts until time 2, and at its widest the S² has radius 1. Then we have a picture of S³, with radius 1; its center is at point p at time 1; and it has a "south pole" at point p at time 0, with corresponding north pole at p at time 2, and with equator being the sphere at time 1. Exercise: what are the "lines of latitude"? What about "line of longitude"? Instead of putting the south pole at point p at time 0, can you picture the south pole in some other placetime, and visualize the corresponding north pole, equator, and latitude and longitude lines?
Can you picture a 4-sphere?
The method I'm describing works like this: each cross section of S⁴ is an S³, with a radius depending on where we take the cross-section. Each S³ can be pictured as a movie of a point appearing expanding to a sphere, then contracting back to a point and disappearing. The S³ movie for S³ with smaller radius has the point appearing later, the sphere S² expanding to a smaller maximum radius, and the sphere contracting and disappearing sooner. So we're going to play a movie-2 across time-2. At each point in time-2, we quickly play a movie-1 across time-1, which pictures S³ with a radius that depends on the time-2. As time-2 progresses, the radius of the S³ starts at 0, quickly increases, then slowly peaks at 1 at time-2 = 1, and then decreases until it hits 0 at time-2 = 2.
[This section assumes more math. See https://en.wikipedia.org/wiki/Morse_theory#Basic_concepts and the linked video.]
So, I was watching this video, and wanted to mentally picture critical points. (Link is to the point where I started thinking about this; you could watch more to get more context.)
For surfaces, it's straightforward. For 3-manifolds, it's harder, but we can still get by in 4 dimensions, so we don't need multiple time dimensions. For example, a 2,1 saddle point (two dimensions have positive 2nd derivative, the other one is negative) can be pictured as a movie of two surfaces deforming towards each other until they touch at a point and form a "wormhole".
What about 4-manifolds? A 4,0 critical point is just an S³ that appears out of nowhere (across time-2, which I'm identifying with the level-set parameter, and generally time-2 I think should be "the thing really being varied"), so it's just the first half of the S⁴ visualization.
Here's what a 1,3 saddle point looks like: at first (for a fixed time-2), we have a movie-1 in time-1 that's a 2-sphere hanging around for a while, then suddenly contracting to a point and disappearing; then some moments (in time-1) later, an S² reappears and then hangs around. As time-2 progresses, it's the same movie, but there's less and less time between when the first S² disappears and when the second S² reappears. At some point in time-2, the time-1 movie has S² shrink to a point, and then immediately without disappearing, reexpand as S². A moment in time-2 later, the sphere never (in time-1) shrinks to a point, and instead just shrinks in radius and then reexpands.
Can you picture a 3,1 saddle point? (Bad hint: ought emit esrever.)
Can you picture a 2,2 saddle point?