Worlds of 1 Spatial Dimension

Matt Strassler 11/07/11

A one-dimensional world is a lot simpler than a three-dimensional world, but already there is a lot to talk about. For instance, there are several different types of one-dimensional worlds to discuss. They share certain properties, but their differences are also really interesting.

For a first example of a dimension, let’s talk not about dimensions of physical space but more general ones.  That’s helpful for many reasons, especially shaking your intuition away from your natural biases about what dimensions are and how they work. Let’s talk about yearly income — how much money a person makes in a given year.  It’s as good a dimension as any.

The Income Dimension

How much money you made last year is a certain number in your home currency. It could have been positive or negative; it could have been small or large; it can be represented as a point on a line, as in Figure 1, which we’ll call the “income line”. Every point on the line represents a possible income.

Fig. 1: The Income Line is a one-dimensional "world" of infinite extent, extending to infinity in both the positive and negative directions. The line represents all possible incomes; a point on the line represents a particular income.

What makes yearly income a one-dimensional world is (very roughly speaking) the following obvious property:

  • A location within this space is specified by one piece of information: in particular, an income. 

Note also it is continuous (or close to continuous for all practical purposes) — if two people have different incomes A and B, we can presumably find a third person who has an income that lies between A and B.

These two facts together imply that an income can only change continuously within the income line by moving to the right or away from the right (i.e. to the left) — either to higher income or to lower.  There are no other options.

Of course the income line has nothing to do with any physical space that you and I could walk around in, but it is still a dimension. And (at least in principle) it doesn’t have any end in either direction: there’s no limit (in principle) to how much money could be made or lost in a given year. This one-dimensional world is not a very lush one, but still we can ask simple questions that make sense there, such as:

  • How are yearly incomes in the United States distributed?
  • What is the average yearly income in Japan?
  • How do the answers to these queries change with time?

These questions take on meaning in the one-dimensional world of the income line.

The Rainbow-Color Dimension

Here’s another, rather different, one-dimensional world. Colors of the rainbow form one dimension, from red through orange to yellow, and from there to green, blue, and finally violet. Color, from this point of view, forms a one-dimensional world of finite extent.   (There are invisible forms of light beyond red and beyond violet, but as far as your eyes are concerned, the dimension stops there.) It’s represented not as an infinite line but a line segment — the “rainbow line” — as in Figure 2. (This is to be distinguished from the color wheel! Note a rainbow itself shows no sign of the color wheel; it starts at red and ends at violet — period!) Again, a location on the rainbow line is specified by one piece of information (a color) and it’s continuous.

Fig. 2: The Rainbow Line represents all the colors of the rainbow, from red to violet. It is represented as a line segment, since there is nothing to be seen to the left of violet or the right of red. We may also represent it by a finite interval of numbers, representing the wavelength of the photons (particles of light, which make up light waves.) A nanometer is a billionth of a meter, or about 30 millionths of an inch. Photons with larger or smaller wavelengths exist, but are ignored by our eyes.

Obviously this isn’t a dimension of physical space either! You can throw a ball from your house to your neighbor’s, but you can’t imagine tossing a ball from green to orange — that’s a meaningless phrase.  But this is still most certainly a dimension. We can ask lots of questions that make sense there: as an apple turns from green to red, how does its color move along the rainbow line?  How much of the sun’s light is emitted in each color?  If an orange star turns red, will it first become yellow?

The Wind-Direction Dimension

And here’s a third, different yet again. If you listen to a weather forecast, it will tell you the wind will soon be blowing from the north, or from the northwest, or from the southwest. The possible directions that the wind can blow from form a dimension too. Notice this is not at all a dimension of space! You can’t throw a ball into this dimension, the way you can throw a ball up, or left, or forward.  This is a dimension’s-worth of directions pointing in space!

Fig.3: Many representations of the Aeolian Line. This line of all possible wind directions is of finite extent, like the Rainbow Line, but it closes on itself. If we represent it as a line segment (top) a changing wind can move off the left end and onto the right end of the line, or vice versa. We can represent it the line as starting at north (top), or east (below top), or anywhere in fact. Or we can represent it as a circle (bottom left). The latter technique is used in compasses and in wind vanes (bottom right).

How do we represent this dimension? Well, there are at least two natural ways, shown in Figure 3. One is using a line segment again — the “aeolian line[Aeolus being god of the winds] — but the aeolian line differs from the rainbow line in that it is periodic — the wind direction can move from north to east to south to west and back to north continuously. In fact, in making our representation we can cut the line anywhere; compare the two lines at the top of Figure 3, which are both equally good representations of the aeolian line.  The point is that the wind can wander off the right end of the line and come back immediately on the left end, and vice versa, so it doesn’t matter where we break the line. Or perhaps it is easiest to represent this periodic line as a circle. Which is what we do with a compass, or a wind vane!

Three different one-dimensional worlds

There you have it: three one-dimensional worlds. And look at how rich they are in details! Different shapes, with different properties. On the income line, an income can grow (or shrink) forever. On the rainbow line, your eyes can only travel as far as violet, and then beyond that point the space of visible color ends, leaving them no choice but to stop there, or retrace their steps, which they can do only as far as red. And on the aeolian line, the wind may go all the way round as many times as it pleases — but in doing so it will blow from the south again and again.

These types of one-dimensional worlds — worlds that are infinite, finite with end-walls, and finite but periodic, and represented by the infinite line, the line segment, and the circle — are the basic ingredients for understanding worlds with more dimensions. I’ll come back to them many times. See Figure 4, which also includes a fourth type that extends infinitely in one direction but hits a wall in the other. (As an example, temperature is such a dimension: it can be as high as you like, but there is a lowest-possible temperature — absolute zero, it is called — so temperature forms a line that starts at absolute zero and only goes up from there, not down.)

Fig. 4: Four types of one-dimensional worlds: an infinite line, a line segment with two end-walls, a circle and (not discussed in detail here) a half line with one end-wall. A location in one of these worlds can be specified using one number, and motion (back and forth) within each of the worlds can occur continuously in only one direction.

Representing Dimensions, Spatial and Otherwise

Along the way I’ve mentioned or used several different methods for representing dimensions. We can represent income by a number, or by an infinite line. We can represent visible rainbow color as a line segment, or by a color itself, and we could also use a number — the wavelength of the photons (particles of light) that correspond to a particular color would do nicely. We can represent the directions of the wind by a circle, or by a line segment whose left end connects to its right end — or by the words north-east-south-west — or even by a number that states the direction of the wind in degrees, running from 0 to 360 and back to 0 degrees again. The fact that we can represent a single dimension in many different ways offers us great flexibility in building intuition about extra dimensions, and we will put them to considerable use!

Now the reason I chose to illustrate these types of dimensions using ideas that have nothing to do with physical space — income, rainbow color, and wind direction — is that the abstraction that spatial dimensions are specific examples of the more general concept of dimensions makes representing worlds with more than three spatial dimensions a lot easier. Remember the two parts of learning how to think about extra dimensions, mentioned in the parent article to this one? First, learn to represent them; second, learn how things work in them. Spatial dimensions do have some special properties as far as how things work there, but not in how they are represented.

Effectively-One-Dimensional Spatial Worlds

That said, there are one-dimensional spatial worlds that we encounter regularly. Or, more precisely, there are particular situations in which some aspect of the world behaves as though there is only one dimension of space. What we say is that the world, for some actors or objects, becomes effectively one-dimensional.

Fig. 5: A person walking on a high-wire can only walk safely in an effectively one-dimensional world. The world shown has finite length and two ends.

Think of a tight-rope walker balancing on a high wire. What makes the tight-rope walker’s world effectively one-dimensional (though of course it remains truly three dimensional) is that he or she cannot safely move in any direction other than left or right. This world is like the rainbow world: it is finite in length, and when the walker reaches the end he or she must turn around and walk back the other way [or leave the wire, ending the situation in which the world is effectively one-dimensional.] What else is true about it?  Position along the wire can be specified using one piece of information (such as a distance from the left end to the walker.)  Two walkers on the line can meet, but they cannot pass each other.

We could make the high-wire world more like the aeolian line if the wire were bent around into a circle (Figure 6). It would still be true that two walkers could not pass each other — that’s a general property of one-dimensional worlds. And it would still be a dimension of finite extent.   But it would now be possible for a walker to continue around and around forever without stopping.

Fig. 6: As in Figure 5, but now with a circular wire, which eliminates the end-walls from this effectively one-dimensional world but keeps it finite in extent.

Other examples of one-dimensional spatial worlds we know:

  • a narrow one-lane road is a one-dimensional world for cars;
  • a narrow knife-edge ridge, for a hiker climbing a mountain;
  • the floors of a tall building, for an elevator.

The world as a whole remains three-dimensional, but for describing the cars, the hiker, the elevator, you only need to think about one dimension for the moment.

So for later purposes, remember this lesson: we live in an apparently three-dimensional world, and everything we encounter appears to us three-dimensional. But sometimes our three-dimensional world (or rather, some part of it) can behave as though it is effectively one-dimensional, or two-dimensional (can you think of examples?) or even zero-dimensional! (Anyone who has had the misfortune to be stuck in motionless traffic for hours knows what an effectively zero-dimensional world is like!) We’ll find this intuition very useful later.

18 responses to “Worlds of 1 Spatial Dimension

  1. Wouldn’t it have been clearer if the term “degrees of freedom” had been used from the start?

    • Hmm. In my experience, not to a typical layperson. My mom doesn’t know what “degree of freedom” means. And besides, two particles moving in one dimension have two degrees of freedom, not one; and a field in one dimension has an infinite number. So…a Pandora’s box, it seems to me. What did you have in mind?

  2. Using minimum length, (size), could you describe a one dimension and what motions would be allowed for a point of minimum size.
    There is a presumption of a time dimension when we assume motion, therefore, if we look at minimums, ( l,w,h). then shouldn’t there be a consideration of minimum time?

    Would our definitions of packing and density still apply at a small scale?

  3. As a high school student in Russia in seventies I studied geometries (spaces) of finite number of points. For example, a geometry of a space containing four points could have two parallel lines ( no common points ) while there point space has none. Although I never had to use it in my work, I always liked simplicity and elegance of it. Could you please tell if it is being used in any form in today’s particle physics?

  4. Ah sucks! Any way to pick up the pace Doc? Little bit rudimentary. I suspect the average interested readers are familiar with Hilbert spaces, eigenstates and branes, not to mention the imaginary 1-d world is riddled with inconsistencies, a prior i assumptions about causality and the constancy of space/time that result in false dilemmas. What about discontinuous 1-d space lines? Impossible? I don’t know.

  5. As a layman who cannot get his hands on enough books to feed my physics hunger, this site and it’s postings are without a doubt the best gift to me, and no doubt countless other people. The information is presented in a most understandable fashion, and there is no time to come up for air before the next wonderful tidbit arrives. My only problem is to find the time between my day job (should perhaps have been a physicist, but 40 years ago that never happened :-)) and family to digest all this great material. Forgetting about my time, I don’t know where Matt finds the time to serve this up, but keep it coming! Without a doubt this is the best repository of up-to-date, cross-referenced, detailed yet understandable information on the latest theories in physics in the entire internet. Thank you for this site !!!

  6. Of course, “wind direction” is actually a vector in a three-dimensional space, which is usually close enough to parallel with the earth’s surface that it doesn’t change much on orthographic projection to a two-dimensional space, and since it is usually reported using the dimensions of direction and magnitude (instead of using, for example, the north and east components), we can ignore the magnitude (when it is non-zero—which not always a safe assumption) to represent it as a one-dimensional manifold of unit vectors in the plane, or a circle as pictured.

  7. Matt,

    When you switch from a single “dimensional” line that represents some positive or negative item of information to representations of spatial dimensions you and everyone else who does this is making a mistake.

    Such drawing representations as a figure moving in a “one-dimensional” space is not that at all. It is actually a representation of two spatial dimensions and a frozen time dimension. The two spatial dimensions are of course width and height.

    That is something our brain can process only because when we don’t see the depth of something we nevertheless assume it is there. When you draw such things you are using a three-dimensional pencil and three-dimensional lead particles and three-dimensional paper. I challenge you to try doing that representation without using three-dimensional materials.

    All real things are of three spatial dimensions and no mind trick of that sort is going to help understand the nature of the real universe, which is only three-dimensional plus motion (time).

    • I’m sorry, but when you go around lecturing a physicist who’s been in the business for 25 years about these completely obvious things, you just make yourself look silly.

      I happen to trust that most of my readers understand the notion of “abstraction”.

  8. Vincent,
    I think you gave a good example of confusing ” world(s)” with their abstract presentations that Matt is talking here about.

    As an aside “stupid” Q, what’s the maximum possible T (in degrees of Kelvin), that “was” in the Big Bang, I guess ?!
    What consideration could be given to assess its order of magnitude ?

    Thanks, bob 8/29/12

    • Maximum possible may well be the Planck temperature, about 10^(32) degrees Kelvin. At temperatures that high, space itself may start to become hot, due to quantum gravitational effects, and probably you can’t make things any hotter than that even in principle. But our understanding of quantum gravity is imperfect and that argument might be wrong.

      Maximum actual may well have been much lower than this; we don’t yet know enough about the cosmos to guess. It almost certainly had to have been large enough to produce protons and neutrons and their antiparticles easily; that’s about 10^(14) degrees Kelvin. It also had to be large enough to allow an asymmetry between the number of protons and electrons and the number of anti-protons and anti-electrons (i.e. what is often called the matter-antimatter asymmetry) to arise but since we don’t know how that asymmetry came about we’re not sure what temperature was required. It might be as low as 10^(14-15) degrees Kelvin but is possibly much, much larger than this.

      • Pedantic point … I think Kelvin is an S.I. unit in it’s own right and has no “degrees” association.

      • First, hello, new fan here. I love your style, and your explanations of the concepts resonate for me. Regarding this topic…

        I wonder if, in an infinite universe, the asymmetry we see between the number of protons and electrons might not be a local asymetry, specific to our own region – or if inflation is localized as well, to our own pocket? In other words does it make sense that perhaps, if you grab up a portion of particles from the universe at random, there might be an asymetry of the scale that we observe?

        – John

  9. Professor,
    You are “killing” me by your knowledge.
    Thanks, bob

  10. To Dr. Matt Strassler: Q1)mechanical waves (like sound waves) seem to pass each other without collision, can any wave pass each other without collision “in a true(effectively?) one-dimensional world (circle)”? Why can they pass through each other without collision? Are these waves are “not” in true one-dimensional world but actually in more than one dimension? and is this why they can pass each other? do they actually dodge(take two different paths) each other when they pass each other even though they effectively look like superimposed or sit/move within the same (one?) dimension? Q2)I am not sure if “photons” can exist in one-dimensional world but can they pass each other without collision in (true) one-dimensional world? OR they MUST ALWAYS be in 3-dimension to exist and propagate? If so why?

    • Two waves can indeed pass each other, in one dimension or in any other, under some conditions. In fact you can see this in a bathtub: just make two ripples and watch them go right through. The issue is whether the equations for the waves are like those of water waves and light (which are “linear”) or whether the equations are more complicated (“non-linear”) in which case the waves will scatter, to a degree anyway.

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