Adventures in hyperspace

The hypercube.  A geekly rite of passage, at least for geeks of a certain age.  The tesseract.   The four-dimensional cube.  Because what could be cooler than three dimensions?  Four dimensions!  Cue impassioned discussion over whether time is "the" fourth dimension, or such.

Cool concept, but can you visualize one, really "see" a hypercube in your mind's eye?  We can get hints, at least.  It's possible to draw or build a hypercube unfolded into ordinary three-dimensional space, just as you can unfold a regular cube flat into two-dimensional space.  Dali famously depicted such an unfolded 4-cube.  You can also depict the three-dimensional "shadow" of a 4-cube, and even -- using time as an extra dimension -- animate how that shadow would change as the 4-cube rotated in 4-space (images courtesy of Wikipedia, of course).

That's all well and good, but visualizing shadows is not the same as visualizing the real thing.  For example, imagine an L-shape of three equal-sized plain old 3D cubes.  Now another L.  Lay one of them flat and rotate the other so that it makes an upside-down L with one cube on the bottom and the other two arranged horizontally on the layer above it.  Fit the lower cube of that piece into the empty space inside the L of the first piece, so that the first piece is also fitting into the empty space of that piece.

What shape have you made?  Depending on how natural such mental manipulation is for you and how clear my description was, you may be able to answer "A double-wide L" or something similar.  Even if such things make your head hurt, you probably had little trouble at least imagining the two individual pieces.

Now do the analogous thing with 4-cubes.  What would the analogue of an L-shape even be in 4-space?  How many pieces would we need? Two?  Three?  Four?  Very few people, I expect, could answer a four-dimensional version of the question above, or even coherently describe the process of fitting the pieces together.

Our brains are not abstract computing devices.  They are adapted to navigating a three-dimensional world which we perceive mainly (but not exclusively) by processing a two-dimensional visual projection of it.  Dealing with a four-dimensional structure is not a simple matter of allocating more mental space to accommodate the extra information.  It's a painstaking process of working through equations and trying to make sense of their results.

That's not to say we're totally incapable of comprehending 4-space.  We can reason about it to a certain extent.  People have even developed four-dimensional, and even up to seven-dimensional (!) Rubik's Cubes using computer graphics.  It's not clear if anyone has ever solved a 7-cube, but a 3x3x3x3x3 cube definitely has been solved.

Even so, it's pretty clear that the solvers are not mentally rotating cube faces in four or five dimensions, but dealing with a (two-dimensional representation of) a three-dimensional collection of objects that move in prescribed, if complicated, ways.

From a mathematical point of view, on the other hand, dealing in four or five or more dimensions is just a matter of adding another variable.  Instead of (x,y) coordinates or (x,y,z), you have (w,x,y,z) or (v,w,x,y,z) coordinates and so forth.  Familiar formulas generally apply, with appropriate modifications.  For example, the distance between two points in 5-space is given by

d² = v² + w² + x² + y² + z²

if v, w, etc. are the distances in each of the dimensions.  This is just the result of applying the pythagorean theorem repeatedly.

Abstractly, we can go much, much further.  There are 10-dimnsional spaces, million-dimensional spaces, and so on for any finite number.  There are infinite-dimensional spaces.  There are uncountably infinite-dimensional spaces (I took a stab at explaining countability in this post).

Whatever intuition we may have in dealing with 3- or 4-space can break down completely when there are many dimensions.  For example, if you imagine a 3-dimensional landscape of hills and valleys, and a hiker who tries to get to the highest point by always going uphill when there is a chance to and never going downhill, it's easy to imagine that hiker stuck on the top of a small hill, unwilling to go back down, never reaching the high point.  If the number of dimensions is large, though, there will almost certainly be a path the hiker could take from any given point to the high point (glossing over what "high" would mean).  Finding it, of course, is another matter.

You can't even depend on things to follow a consistent trend as dimensions increase, as we can in the case of a path being more and more likely to exist as the number of dimensions increases.  A famous example is the problem of finding a differentiable structure on a sphere.

Since we can meaningfully define distance in any finite number of dimensions, it's easy to define a sphere as all points a given distance from a given center point (it's also possible to do likewise in infinite dimensions).  If you really want to know what a differentiable structure is, have fun finding out.  Suffice it to say that the concepts involved are not too hard to visualize in two or three dimensions.  Indeed, the whole field they belong to has a lot to do with making intuitive concepts like "smooth" and "connected" mathematically rigorous.   Even without knowing any of the theory (I've forgotten what little I knew years ago), it's not hard to see something odd is going on if I tell you there is:

• exactly one way to define a differentiable structure on a 1-sphere (what most of us would call a circle)
• likewise on a 2-sphere (what most of us would just call a sphere)
• and the 3-sphere (what some would call the 3-dimensional surface of a hypersphere)
• and the 5-sphere (never mind)
• and the 6-sphere

Oh ... did I leave out the 4-sphere?  Surely there can only be one way for that one too, right?

Actually no one knows.  There is at least one.  There may be more.  There may even be an infinite number (countable or uncountable).

Fine.  Never mind that.  What happens after 6 dimensions?

• there are 28 ways on a 7-sphere
• 2 on an 8-sphere
• 8 on a 9-sphere
• 6 on a 10-sphere
• 992 on an 11-sphere
• exactly one on a 12-sphere
• then 3, 2, 16256, 2, 16, 16, 523264, and 24 as we go up to 20 dimensions

See the pattern?  Neither do I, nor does anyone else as far as I know. [The pattern of small, small, small, big-and-(generally)-getting-bigger continues at least up to 64 dimensions, but the calculations become exceedingly hairy and even the three-dimensional case required solving one of the great unsolved problems in mathematics (the Poincaré conjecture).  See here for more pointers, but be prepared to quickly be hip-deep in differential topology and such.]   In the similar question of differential structures on topological manifolds, there is essentially only one answer for any number of dimensions except four.  There are uncountably many differential structures on a four-dimensional manifold.  So much for geometric intuition.

It's worth pondering to what extent we can really understand results like these.  Certainly not in the same way that we understand how simple machines work, or that if you try to put five marbles in four jars, at least one jar will have more than one marble in it.

Statements like "there are 992 differentiable structures on an 11-sphere" are purely formal statements, saying essentially that if you start with a given set of definitions and assumptions, there are 992 ways to solve a particular problem.  The proofs of such statements may use various structures that we can visualize,  but that's not the same as being able to visualize an 11-dimensional differentiable structure.  Even if we happen to be able to apply this result to something in our physical world, we're really just mechanically applying what the theorems say should happen in the real world.   Doing so doesn't give us a concrete understanding of an eleven-dimensional differentiable structure.  

That, we're just not cut out to do.  In fact, we most likely don't even visualize three complete dimensions.  We're fairly finely tuned to judging how big things are, how far away they are and what's behind what (including things we can't see at the moment) and what's moving in what direction how fast, but we don't generally visualize things like the color of surfaces we can't see.  A truly three dimensional mental model would include that, but ours don't.  Small wonder a hypercube is a mind-boggling structure, to say nothing of some of the oddities listed above.