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Tuesday, 27 February 2007

"But if you can't picture it..."

The other day, I was talking to a couple of friends of mine - a literary theorist and a historian/political scientist - about the (apparently) recently proved Poincaré Conjecture. The Poincaré Conjecture involves a statement about the 3-sphere - that is to say, something like the usual sphere, but one dimension bigger. In order to visualise a 3-sphere, you would need to picture it in four dimensions, just as the usual sphere (2-sphere) is usually considered to be a two-dimensional surface, embedded in three-dimensional space. The idea of four dimensions can provoke puzzlement among humanities students of a sort to startle a blasé mathematician for whom such things are second nature. My friend the student of history and politics asked, with a slightly concerned look on his face,

"Can mathematicians actually picture four dimensional space?"

"Roger Penrose says he did it - briefly - once," I said, grinning [1].

"No, but are there people out there who can actually..."

"Not that I know of."

My historian friend was relieved. My literary theorist friend was confused. "If you can't picture it," he asked, "how could you have any intuition about it? I mean, you could just say whatever you wanted about it and no-one would be able to refute you."

The first way that students of mathematics start dealing with four (or more) dimensions is fairly simple. If you're in two dimensions, you can describe any point with, say, an x co-ordinate and a y co-ordinate.

To go to three dimensions, you need a third co-ordinate in, say, the z-direction (use the previous picture but imagine the third axis coming out of the screen towards you.). To go to four dimensions, add a fourth number, and there you have a way of describing four dimensional (flat/Euclidean) space.

It gets more complicated when you start dealing with curved surfaces, but the idea is similar. If, earlier, you wondered why we consider the surface of a sphere to be two-dimensional, think of it this way. The Earth is approximately a sphere. We can describe positions on the Earth's surface with two co-ordinates: latitude and longitude. Two co-ordinates, two dimensions. Exact definitions might worry about the poles, where there is no real longitude to speak of - it could be anything - but there are fairly simple ways of getting around that which need not concern us here.

Besides the question of four dimensions, there is a broader principle raised by my friend's comment. The extent to which intuition plays any role at all in the basic system of mathematical proof is arguable. This is a fascinating topic in its own right that deserves a post of its own, so for now I shall simply promise to address it later.

[1] He says it in The Emperor's New Mind.


Alon Levy said...

Actually, now that I think of it, the explanation of why a sphere is two dimensional is even better than I thought it was.

Just to be sure, you're sending this to the Carnival of Mathematics, right?

Lynet said...

Well, either this one or one of the other couple of maths posts I've got in mind, yeah :)

Anonymous said...

With all due respect, I strongly disgaree. There are plenty of people who can visualize 4-dimensional geometry. I've blogged about this before, and don't feel like doing it again. But google these names to start:

Alicia Boole Stott

Ludwig Schläfli.

"H. S. M. Coxeter, Regular Polytopes"

This will be a better thread once you've done this basic search, and you will find some cool things about interesting people.

-- Prof. Jonathan Vos Post

Lynet said...

Although I left space for the possibility that there might be people out there who can visualise four dimensions that I haven't heard of, I admit it never occured to me to look for them.

Thanks for the information. With all due respect, however, I feel unduly patronized by your tone.

Anonymous said...

I apologize for my tone, Lynet.

I blogged on this at such great length once that I annoyed the blogmistress, and I was inadvertantly rude because of my unhappy memories of that.

I used to be able to visualize 4-dimensional geometry as a precocious child Mathematician, but the ability faded to my now merely being able to sometime vaguely visualize what once was clear "in my mind's eye."

My son, about to graduate university at age eighteen with a double B.S. in Math and Computer Science considers it one of my few failures as a parent not to have been able to teach him how to visualize 4-D geometry.

It is fascinating to read postings between two people who CAN so visualize.

See my:

Table of Polytope Numbers, Sorted, Through 1,000,000

-- Prof. Jonathan Vos Post

Anonymous said...

Actually, I've been struggling the past few days with a 4-dimensional geometry problem where my visualization is good enough to pose the problem, yet is almost but not quite good enough to clearlyt see a solution, which I'm therefore approaching by brute force.

The problem is find the hypervolume of the convex hull of a set of points of known coordinates in Euclidean 4-space.


The messy way is to pentatopalize (4-D version of 2-D triangularize and 3-D tetrahedralize) the convex hull into 4-simplexes, and then calculate the hypervolume of each pentatope by the Cayley-Menger determinant (generalization of 2-D Heron's formula). This invokes the 4th coefficient in -1, 2, -16, 288, -9216, 460800, ... (Sloane's A055546).

If V_4 is the 4-D hypervolume of the general 4-simplex, then:

(V_4)^2 = ((-1)^4)/((2^4)(4!)^2) det B where B is the 6x6 matrix determined by d(ij) = the L2-norms of edges between vertex i and
vertex j.

[the spacing might not work well in your font]

B =

0..........1..........1..........1 ..........1..........1
1...0...(d_12)^2 (d_13)^2 (d_14)^2 (d_15)^2
1...(d_21)^2...0...(d_23)^2 (d_24)^2 (d_25)^2
1...(d_31)^2 (d_32)^2...0... (d_34)^2 (d_35)^2
1...(d_41)^2 (d_42)^2 (d_43)^2...0 ... (d_45)^2
1 ...(d_51)^2 (d_52)^2 (d_53)^2 (d_54)^2 ...0

Anonymous said...

Very exciting and (as you'll read) BIG news on visualizing a higher dimensional geometry, see:

News about E8
Posted by John Baez

The exceptional Lie group E8 is a marvelous 248-dimensional monster, with mysterious connections to the octonions and string theory. Here’s a nice webpage about a new calculation involving E 8:

* American Institute of Mathematics, Mathematicians map E8.

Wonderful stuff -- even if nobody alive can actually visualize one of the most complicated structures ever studied: the object known as the exceptional Lie group E8.

-- Prof. Jonathan Vos Post

Anonymous said...

Oh, and see John Baez's diary for the full story, with diagrams, which starts:

"Richard Block... is an expert on Lie algebras, and there's a fun story about him and Murray Gell-Mann, the physicists who one the Nobel prize for inventing 'quarks'. Actually quarks were part of a mathematical scheme which Gell-Mann called the Eightfold Way, because it was based on the 8-dimensional Lie algebra su(3)..."

Worth the reminder that multidimensional geometry is beautiful for its own sake, but mysteriously tells things about the fundamental Physics of the universe we live in, and maybe all possible universes.

Well, "now" as of mid March 2007. The hotlink will change later.

-- Jonathan Vos Post