Good Math/Bad Math

Thursday, April 06, 2006

Dimensions

This is another of my sort-of ranty posts. I was listening to the radio the other day, and heard an interview of Lisa Randall. Professor Randal is a mathematical physicist who's work focuses on the implications of extra dimensions. Things like string theory have suggested the possibility that our universe has more than the three spatial dimensions + time, but that we are unable to perceive them. The physics of this are fascinating, but I'll leave physics discussions to people better qualified to talk about them. What bugged me was the host's inability to grasp the concept of what a "dimension" is; and that's a problem that I've seen in a lot of discussions.

Bad Sci-Fi has, for a long time, babbled about "alternate dimensions", or "parallel dimensions", where by "dimension" they mean, roughly, "Universe". That is a wretched misuse of the term, which has almost no connection at all with the real meaning of the term. That's what the host of the radio show thought dimension meant, and no matter how many times she pointed out that that was wrong, he just couldn't get past the scifi misdefinition of dimension. (He kept asking questions like "Would there be other dimensions be just like ours, except that the Nazis won WW2?")

Mathematically, dimensions have a lot to do with a field called topology. Topology is, in my experience, the most mind-bendingly difficult field of mathematics around. Fortunately, you don't need to know much about topology to understand what a dimension is.

I'm running in circles a bit here, but there's a reason for it. Dimensions are hard to explain; they're such a fundamental part of our normal experience that we don't really consider what they mean in our perception of the world - and yet the ways to define them without going into the formal math of things like topology need to connect to that subconscious comprehension.

So, to stop goofing around, what is a dimension?

One way of saying it is that it's a direction. Think of a two-dimensional plane: You can move anywhere on the plane - between any two points - by moving either forwards or backwards in two, perpendicular directions. Think of our three dimensional experience of the world: you can move between any two points in three dimensions by moving left/right; up/down; or forward/back. There are three fundamental directions, and thus three dimensions.

The problem with saying it's a direction is that it implies a few things that aren't necessarily true:
  1. Our idea of direction is connected to our three dimensional world - and even within that, it's mostly connected to two dimensions. We mostly think of directions as north/south-east/west; sometimes we add up-down. Directions, intuitively, are things that exist within our three-dimensional range of perception. But adding another direction beyond those three spatial directions - what does that mean? It's adding a direction that we can't see, that we can't point in, that we can't move in. It just doesn't fit with our basic intuitive sense of what direction means.
  2. Direction implies that a dimension is straight - that is, that moving in a direction, even if it's a direction beyond our three dimensions, has the properties of motion in a euclidean space: things like the fact that you can't get closer to something by moving away from it. Or you can draw parallel lines separated by some distance in that direction. Dimensions aren't necessarily straight or continuous.
A better way to explain, but a slightly less intuitive one is to not separate dimensions quite so much. The set of dimensions in a space is the number of pieces of information that you need to identify a unique location in that space.

On a plane, you can put down a coordinate grid, and identify any point on the grid with only two numbers. In a region of three dimensional space, you can specify any location in that space using only three numbers. But getting beyond three, we start to have some trouble.

Try thinking about the colors on your computer screen. Basically, for each pixel on your screen, there are three lights that can be turned on in different brightnesses: one red, one green, one blue. The color of each pixel is determined by the brightnesses of three lights for a pixel: if all of the lights are completely off, the pixel is black; if the blue light is on very bright, the red light is on halfway, and the green light is off, you'll get a bright bluish-violet; if the red light and the green light are on full brightness, and the blue light is on halfway, you'll get a sort of pastel green. So the colors of your computer screen have three dimensions.

Try thinking about the flavors of food. It's commonly asserted that we humans have the ability to perceive sweet, salty, bitter, and sour flavors. Every flavor that we can experience is some combination of sweet, salty, bitter, and sour. Therefore, flavor has four dimensions.

Try thinking about how to describe a personality. There's a common personality test (the Meyers-Briggs) that measures a personality in terms of four traits: extraverted vs introverted; sensing vs intuitive; thinking vs feeling; and judging vs perceiving. That's a four dimensional description of personality.

Now, try looking at this post. I start off in the beginning by spiraling around the subject of dimensions before changing direction and actually talking about what it means. You can say that there are two different dimensions in this post; and one of them is not straight. It doesn't move in a straight line; it squiggles around, never getting very far from where it started.

Most versions of string theory say that the extra dimensions are all curled up in tiny little loops with a particular kind of topology - just like my discussion of dimensions starts off with little loops. Professor Randal thinks otherwise; that the other dimensions might be just as big as the spatial dimensions we perceive. But either way, the dimensions are the mathematical/topological sense of how many bits of information we need to locate a single point in the universe.

25 Comments:

  • Multiple "dimensions" are used all the time in economics. For example, David A Hsieh and William Fund have a seven factor (dimension) model for the returns of portfolios of hedge funds. Based on the model a particular hedge fund (or fund of funds) could be plotted on a 7-dimensional graph, if such thing existed:

    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=778124

    "Throughout our analysis, we model the risks of FoFs using the seven-factor model of Fung and Hsieh
    (2004a). These factors have been shown to have considerable explanatory power for FoF and hedge
    fund returns.5 The set of factors consists of the excess return on the S&P 500 index (SNPMRF); a
    small minus big factor (SCMLC) constructed as the difference of the Wilshire small and large
    capitalization stock indices; three portfolios of lookback straddle options on currencies (PTFSFX),
    commodities (PTFSCOM) and bonds (PTFSBD), which are constructed to replicate the maximum
    possible return to a trend-following strategy on the underlying asset, all in excess returns;6 the yield
    spread of the US ten year treasury bond over the three month T-bill, adjusted for the duration of the ten year bond (BD10RET) and the change in the credit spread of the Moody's BAA bond over the 10
    year treasury bond, also appropriately adjusted for duration (BAAMTSY). We use a linear factor model
    employing these factors to calculate the alpha of FoFs."

    By Anonymous steve, at 2:26 PM  

  • FWIW this paper, superficialy similar to Grier's, breaks the data set into three distinct periods which they state:

    "We divide up the past ten years into three distinct subperiods
    and demonstrate that the average FoF has only delivered alpha in the short second period from
    October 1998 to March 2000."

    Now these guys are respected researchers, not hacks like Grier and employ something called the modified Chow test, with which I am not familiar but is some form of a Chi-squared test, to asses the validity of the break points

    By Anonymous steve, at 2:31 PM  

  • Hey MarkCC,

    With the pixel example you used, I'd actually consider the pixels to be represented in 5 dimensions:

    Red brightness
    Blue brightness
    Green brightness
    *AND*
    x position on the screen
    y position on the screen

    Otherwise, great post. I'm greatly enjoying your blog. Thanks for all the interesting math insights.

    By Blogger Big C, at 3:34 PM  

  • Hey Mark,

    What would it mean for our perception of dimensions for them to be "curled up" or very large? Would curled up just imply that any motion we made would pass through the entirety of the dimension so quickly we wouldn't notice? (I'm thinking of king kong walking on a loop the size of a pinhead...he'd go around so many times in one step it would hardly be meaningful to notice what his loop coordinate is. This isn't literal, of course, but you get the general idea) What would a "big" dimension mean?

    -M

    By Anonymous Anonymous, at 4:05 PM  

  • I too have always been annoyed by the use of "alternate dimensions" for "alternate universes reachable by means of extra-dimensional travel" (not confined to just "bad" sci-fi) and it is sad to seen how it keeps keeps people from being able to properly understand even the pop-sci versions of string theory.

    I have also found it weird that versions of String theory presented to popular audiences felt the need to insist that their extra-dimensions where "wrinkled" (like the surface of an orange or something) with wrinkles of plank-length or so, and that is why we can't perceive them. It doesn't seem so wrong to think we are 3-D creatures living in the cross section of an 11-D (or whatever) universe like in Flatland. Glad to see that there is disagreement on this.

    By Blogger JP, at 5:24 PM  

  • This comment has been removed by a blog administrator.

    By Blogger MarkCC, at 5:28 PM  

  • anonymous:

    What it means for the dimensions to be "curled up" or "very large" is actually a pretty serious open question.

    One version of the "curled up" is that the total distance you can move in the extra dimensions is so incredible small - on the order to the planck length - that they just aren't directly visible or tangible to something as big as us. Another verison is roughly that we "fill" the extra dimension completely, so we can't move in it, and so we don't perceive it.

    The big extra dimensions is more interesting, but it's a newer idea, and less thought has gone into it. There's one example of a possibility that I find easy to conceptualize. Imagine a two dimensional being - a resident of Abbot's flatland. To Mr. Square, the world is two dimensional. But the actual two dimensional surface that is his world is the surface of a sphere. So there are a bunch of bizzare things that he can observe that indicate a third dimension; things like gravity from objects on the opposite side of the sphere being stronger than its apparent distance along the surface of the sphere. Now scale that up: think of us, in the three dimensional equivalent of that surface in a universe with four spatial dimensions.

    By Blogger MarkCC, at 5:36 PM  

  • Great post. I love your blog.

    Do you think you could go into topology a bit? When I got my BS in math, several of my classes touched on it, or waded in, but didn't get much further.

    By Anonymous Patrick, at 5:38 PM  

  • jp:

    It isn't just the popularized versions of string theory that assumed the tiny O(planck length) dimensions; that was a fundamental assumption of most string theorists intil relatively recently. There was some pretty good math to support the idea; until Randall, I don't think that anyone had made the math of multiple larger dimensions work with string theory.

    By Blogger MarkCC, at 5:40 PM  

  • patrick:

    I'll probably do a little bit of topology at some point. For now, I have admit that my topology is quite weak; I don't get it well enough to be able to explain it to someone else. I need to do some reading before I can do an adequate job.

    By Blogger MarkCC, at 5:42 PM  

  • There are actually five taste centers. The fifth is a sort of fat detector that gives food a sort of "full" flavor upon detecting things like MSG.

    By Anonymous Anonymous, at 7:14 PM  

  • big c:

    I was trying to just talk about the dimensions of color of pixels; but there is a way of looking at points in an "image space" where you need five dimensions - three color dimensions and two spacial dimensions. In fact, I think that's a great example of different kinds of dimensionality.

    By Blogger MarkCC, at 9:01 PM  

  • Mark:
    Try thinking about the flavors of food. It's commonly asserted that we humans have the ability to perceive sweet, salty, bitter, and sour flavors. Every flavor that we can experience is some combination of sweet, salty, bitter, and sour. Therefore, flavor has four dimensions.

    You forgot umami.

    Yes, I'm nitpicky.

    Great post, all in all. Keep up the good work.

    By Blogger The Neurophile, at 9:48 PM  

  • Also, in reference to your previous post, it took me a half dozen tries to post the previous comment.

    I'm pretty sure I'm not illiterate enough to have THAT much of a problem reading the letters for the verification.

    Perhaps David Berlinski has cursed you with his voodou powers.

    By Blogger The Neurophile, at 9:49 PM  

  • quote big c:
    "With the pixel example you used, I'd actually consider the pixels to be represented in 5 dimensions:

    Red brightness
    Blue brightness
    Green brightness
    *AND*
    x position on the screen
    y position on the screen"

    Additionally, contemporary image formats now also support an alpha channel for transparency.
    So that adds one more dimension for a total of 6 to your color concept.

    By Blogger Broadside, at 12:55 AM  

  • In a programming language I commonly work in, the available variables to the user workspace are called dimensions. A limitless number of dimensions may be added to the work environment. Since I was introduced to this concept, I have had little problem with the concepts of string theory or brane theory.

    By Blogger Broadside, at 1:01 AM  

  • A couple of other definitions of dimension:

    1) Topologically, you can define dimension inductively - an n-dimensional space is one such that an (n-1)-dimensional subset is required to "trap" a point. For example, in a 2-dimensional space such as a piece of paper, a single point can move in or out of any number of other points (1-dimensional objects), but a single circle (a 2-dimensional object) can massively restrict its motion. In a 3D space, a 2-dimensional object such as the surface of a sphere is required - circles won't cut it.

    2) In geometry, dimension is defined in terms of scaling up objects - roughly speaking it's the log of the ratio of size increase to scaling factor. So, for example, to scale a cube up by a factor of two you need eight copies of the original cube. Hence the dimension of a cube is log(8)/log(2) = 3. This is called the Hausdorff dimension of a shape

    With the Cantor ternary set, you need two copies of the set if you wish to scale it up by a factor of three. Hence the Hausdorff dimension is log(2)/log(3) ~= 0.631. This is what's known as a fractal dimension - a dimensionality that's not a whole number.

    By Blogger Lifewish, at 8:20 AM  

  • If the blog doesn't make sense, I say make folks read a Wrinkle in Time, that's where I finally got a basic understanding of dimensions. Everything's easier from a sixth grade fiction novel.
    -m

    By Blogger barrett 'n megan, at 10:04 AM  

  • Lifewish:

    Not that happy about the topological definition of dimension you gave, since you are assuming you have homogenous space in some sense, any singularities would be in some sense "of lower dimension".

    Your definition might be usable for real manifolds, but then it is pretty much tautological.

    p.s. I know what I wrote is aimed for the professional mathematician, will be happy to explain further.

    By Anonymous ParanoidMarvin, at 10:39 AM  

  • That confusion between "dimension" and "parallel universe" probably dates to some of the earliest science-fictional discussions of the latter idea, which often described the multiple worlds as, e.g. "like a deck of cards stacked into an extra dimension". Someone who didn't quite get the idea, might well get confused about the terminology.

    Also, with regard to "N coordinates for N dimensions", it's worth noting that there's usually a choice of several N-coordinate systems available, which ought to be interconvertible.

    Some examples: Color can use RGB or Hue, Saturation, Brightness; Flat planes can use rectangular or polar; In three-D space, we can use rectangular, spherical, or cylindrical coordinates. The options for 4-D space are left as an exercise for future commenters.... :-)

    By Anonymous David Harmon, at 12:21 PM  

  • I don't where I first heard it, but I've always thought this explanation was rather good for describing dimensions to the layman —

    Imagine a person on a tight-rope. They can walk forwards or backwards. Thus, they can move in one dimension, even in our three dimensional world.

    A flea on the same tight-rope, on the other hand, can move in two dimensions. It can travel the length of the rope, but it can also travel around the rope, so it is directly underneath and hanging upside down.

    The flea has two possible ways of moving to the person's one. That same dimension is still there for the person, but is too small to be regarded as viable. It's not a great leap of the imagination to assume a person so big that they can't even see the flea moving around circumference of the rope.

    By Anonymous Ithika, at 4:25 PM  

  • Ithika:

    That's almost the same as the example that Randall uses in her book for illustrating the same point. (She used an Alice-in-Wonderland metaphor, where Alice shrinks until she can see the second dimension.)

    By Blogger MarkCC, at 8:27 PM  

  • Not that happy about the topological definition of dimension you gave, since you are assuming you have homogenous space in some sense, any singularities would be in some sense "of lower dimension".

    Your definition might be usable for real manifolds, but then it is pretty much tautological.


    Well, this is based on vague recollections of the first chapter or so of Hatcher - I'm no topologist. Would be interested to know what "homogenous" and "singularity" mean in this context.

    By Blogger Lifewish, at 3:15 PM  

  • I've encountered the same problem trying to study dimensions from a philosophic angle. I'm usually reading or studying something about chaos theory, as well--I think there is a signifiant link between the two. But if I try to describe a "dimension of chaos", people think cheesy sci-fi movies, rather than a complex, aperiodic aspect of an object. Even a "dimension of time" is difficult for most people to swallow... It's easier just to think about "space"...and from there, why not assume they mean "outer space" or multiple universes?

    By Anonymous Karmen, at 1:07 PM  

  • This is a great stab at a layman's introduction. It leaves me wondering, though: what's the distinction between measurable characteristics and dimensions? Is quantum spin a "position" in a dimension? What about particle density?

    By Blogger Hans Gerwitz, at 4:49 PM  

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