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.
Warmup: visualizing a 3-sphere
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?
Basic example: visualizing a 4-sphere
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.
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.)
Visualizing in 5 dimensions
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.
Warmup: visualizing a 3-sphere
As a warmup to the warmup, let’s visualize a 2-sphere S², a.k.a. a regular old
plumbussphere (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?
Basic example: visualizing a 4-sphere
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.
Harder example: 1,3 saddle point
[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?