Archive for April, 2010

Four Dimensions: Objection, Refutation, Objection, Refutation, Confusion

April 24, 2010 3 comments

This is a complication of a refutation of an objection to a refutation of an objection to my proof that all combinations of space and time can be represented in four dimensions. I apologize for not posting much recently, but I think that this behemoth of a post more than makes up for it.

A note on etymology: When I refer to “the universe”, I do not mean the actual physical universe that we live in. Rather, I refer to a more theoretical universe that has infinitely many points in space (it is argued by some that our universe is discrete, rather than continuous) and expands infinitely in all directions. Our physical universe is actually a subset of this much broader infinite universe. When we are talking about dimensions in a theoretical way, it makes more sense to refer to “the universe” as being infinitely large, because dimensions extend infinitely in either direction.

The objector concedes that any combination of space-time points is possible in four dimensional space. However, not every combination of space-times points is possible because the points are not all touching each other. In order to be able to move through every possible space-time, there must be a line connecting every combination of points. In mere four-dimensional space, there is no such line.

If we take space-time and drag it through a fourth spacial dimension, then each point of space will be capable of occupying infinitely many things. But this still limits each point of space to being able to move to the previous value or to the next value; it cannot move to just any value. So we need to drag space-time through a fifth spacial dimension to be able to represent every possible timeline in the universe — that is, a total of six dimensions.

If you are satisfied with this objection, please continue reading as I carry it to its absurd conclusion.

Unfortunately, six dimensions are not sufficient. If we drag 4d space-time through two more dimensions, it is still dragged as a whole unit. The different zero-dimensional points cannot move independently of each other. This may not be obvious, so allow me to use an analogy. Imagine that you have two red points. You can drag them through the first spatial dimension, and hit every possible shade of red. But what if you want the first point to darken twice as fast as the second point? Now you need a new direction. What if you want the first to darken three times faster? Now you need still another direction. It’s not hard to see that you need infinitely many directions. All these directions cannot be contained in one spatial dimension, but they can be contained in two.

But what if you want to add a third point? That one has to be able to change, too. So now you need three spatial dimensions.

What about a fourth point? You guessed it: four spatial dimensions.

Now let’s return to our six-dimensional space. For two points to be able to change brightness independently of each other, we need seven dimensions (one point can change along the sixth axis, while the other point changes along the seventh axis). For three points to be able to change independently of each other, we need eight dimensions. For four points, nine dimensions. So since there are more than infinite points, we need more than infinite dimensions. (If you’re confused, see this site for a fun explanation of how something can be bigger than infinity.)

But this itself is incorrect: it assumes that, to move through our dimensions, we must move in a straight line. Why can we not jump from point to point? We don’t jump from point to point in physical space, but there is no reason to assume that we cannot do it in other dimensions. Physically, that sort of jumping is impossible; but if we think of space not as something physical but rather as a model, then we can most certainly jump from point to point. In that case, to represent all possible brightnesses of the color red, we only need one additional dimension. Since we were just using color as an analogy, we might as well remove that dimension altogether — for now. (I’ll talk more about it later.) Additionally, the fifth and six dimensions were only added because we were assuming that we had to more in a line. Sure, a line is a convenient construct for some sorts of practical applications, but why limit ourselves to it? If we can jump around, then let’s remember that every possible point in space and time is represented within four dimensions. In that case, we can pick and choose which points we want, creating any possible timeline for any possible universe.

So now we’re back down to four dimensions. But I’ll bet I can add in some more.

You may remember that I said that color has a dimension, that saltiness has a dimension. Well, the universe contains color and saltiness. It also contains many other things. Any possible thing that can be measured independently could be considered another dimension, which means that then universe hardly requires four dimensions — it requires infinitely many. I addressed this point in my original critique.

But this objection can also be refuted, and considerably more easily. Remember that the four dimensions contain every possible point in space and time, but only space and time. Color or saltiness might be contained within space, but it’s not necessarily space. So if we are talking strictly about space and time, then there are only four dimensions.

Why, then, do we live in a universe with saltiness and color? Doesn’t that mean that we have to have more dimensions than just four? What happened here is that our model deviated from reality.

In the actual universe, color, saltiness, sound — every perceptible dimension — is really just a product of physical space. Saltiness is really just chemical reactions on your taste buds, which boils down to reactions between atoms, and even further, to reactions between quarks. Color is just the movement of light through space. Sound is just the vibration of particles in certain ways. According to the Standard Model, a point in space can only be one of so many things: one of the fundamental particles, or just empty.

But remember my previous argument about dragging color through space? Don’t we need infinitely many dimensions now?

If the universe is continuous, yes. But what if it is discrete — that is, the set of points in space is countable? In that case, it is easy to prove that we only need one more dimension.

I’m not sure how many fundamental particles (plus empty) the Standard Model predicts or how many particles there actually are, but if there are finitely many, then we can use logic similar to Cantor’s proof that rational numbers are countable to prove that we only need one additional dimension.

To make thing simpler, let’s match every particle (and the empty space) to a number. And, for the sake of convenience, let’s say that there are ten of them; the number doesn’t really matter. (If the number is one — that is, space is just space — then this proof doesn’t apply; we will still only be working with four dimensions.)

We will be moving through the fifth dimension by counting up through the numbers. If every possible combination of numbers is represented, then every possible universe can exist in five dimensions.

To start off with, let’s say that every point is set to 0. Now look at some point. Change it from a 0 to a 1. (We are now moving through an additional dimension, because this change cannot be done in space or in time.) Now change it from a 1 to a 2. Now 2 to 3. Keep counting up until you get to 9. Now you’ve covered every single combination of universes where every particle but that one is a zero.

Reset that particle to 0 and add 1 to the particle next to it. Now start counting up again with the original particle. The next time when you get to 9, add another 1 to the second particle, which now becomes 2. Do it again, and it becomes 3. Keep going until this second particle also gets to 9.

Now what do we do? Well, this time, reset the first two particles to zero and add 1 to a third particle. Keep adding numbers to the first particle again.

If we place these three particles in a row, we get a pattern that looks like this:



Gee, that sure looks familiar. We can continue to extrapolate this to every single point moving along a line, and we can represent every combination of particles using a number.

But wait: if we are moving along a line, we are missing all of the other particles that don’t fall on that line. How do we fix this problem? Well, it’s pretty simple. Instead of moving in a line, we move in a spiral. This way we hit every single particle in all of space and time. It will take a lot longer to reach the outer points, but it is impossible to name a point that won’t eventually be reached. By this process, every possible combination of particles — that is, every possible timeline of the universe — is assigned to a number.

Wait, did I say that we need five dimensions? Hardly. In this case, we can represent every possible timeline of every possible universe using a simple number line. And a number line isn’t called a line for nothing — it’s one-dimensional. That means that if our universe is discrete, then it can be represented using a one-dimensional line.

Notice that this isn’t always true. If the universe is continuous, then there is no way to enumerate all the possible points, which means that we can’t give a number to every single point in space and time. In this case, we need infinitely many dimensions to represent every possible combination of particles.

So let’s recap. If the universe is discrete, it can be represented with one dimension. If the universe is continuous but only represents space-time, then it can be represented with four dimensions. If the universe is continuous and also contains different particles, then it cannot be represented in any finite number of dimensions. If the universe is continuous and saltiness, color, etc are dimensions rather than physical constructs, then we need however many dimensions as there are measurable things — presumably infinite.

That was fun.

Categories: Math, Science

Four Dimensions: A Simple Proof

April 23, 2010 10 comments

This is a proof that four dimensions is sufficient to contain all possible spaces and times — ten are not necessary.

First, let us remember what a dimension is. Mathematically, a dimension is an axis across which something can change. Space has three dimensions, because you can change along length, width, or height. Time has a dimension because you can change through time. Saltiness also has a dimension, although it’s not particularly useful for most purposes; but even so, my science teacher informed me that oceanographers treat saltiness as a dimension.

One dimensions represents every possible value of some particular thing. A dimension could be represented with a line, which contains every possible location in one-dimensional space. It could be represented as moving through every possible shade of red, or every possible level of saltiness, or any other measurable thing that you can think of. You can also use this line to represent every possible combination of this thing: you can represent one shade of red along the redness dimension, or you can use two points to represent two shades of red, or three points to represent three shades of red, and so on.

What about when you introduce a second dimension? Now you can represent every combination of two things and every possible set of combinations. The most obvious example is length and width. On a spacial plane, you use a single point to represent length and width, and a set of points to represent many different lengths and widths.

You can also extend this into three dimensions, for example with three-dimensional space. Using the axes of length, width, and height, you can represent any point in space.

Next, bring it into the fourth dimension. If the first three dimensions are space and the fourth is time, then we are now able to represent every combination of space and time that can possibly exist. This four-dimensional construct contains all possible universes and all possible timelines.

I have an objection to this proof, as well as a refutation. It is much more complex, so I will be addressing it in an upcoming post. If you still have a problem with my proof, stick around.

Categories: Math, Science

A Critique of “Imagining the Tenth Dimension”

April 18, 2010 33 comments

Recently I saw a video called Imagining the Tenth Dimension (there is also a part 2). I suggest that you watch the video before reading this post. If you can get past the annoying sound effects, it’s actually pretty entertaining.

The beginning of the video is perfectly acceptable; it discusses some widely-known ideas popularized by Edwin Abbott’s Flatland.

The first problem is that this video assumes that time is the fourth dimension. Well… not really. Space and time are not actually dimensions, but rather are constructs that can be represented as dimensions. A dimension is simply a range through which something can change. In a square, you can either move up and down or you can move left and right; hence, two dimensions. Time is one-dimensional, as you can only move forward or backward (in practice you cannot move backward, but that’s irrelevant in this case). Dimensions can also be represented in other ways. Images of the Mandelbrot Set, for instance, often represent a third dimension using color rather than depth, because color is more convenient in that case. But it could just as easily be represented using depth, or time, or even something else.

A dimension is a mathematical construct. Space is not three dimensions. Space has three dimensions; that is, it’s three-dimensional. To say that time is the fourth dimension can be useful sometimes, but it is not inherently true. The only reason time is the fourth dimension is because we say it is. The fourth dimension could be saltiness, for instance. That may seem counter-intuitive, but think of it this way. A dimension is something that you can change though. Your geographical location can change, and we call this a dimension (actually, three dimensions). But your saltiness can also change. You can become more salty or less salty. If you place saltiness values on a line, you can even represent saltiness using a spatial dimension. In fact, we often represent other dimensions in two spatial dimensions by using graphs.

There is nothing wrong with representing the fourth dimension with time; but there is something wrong with assuming that time is the only way to represent the fourth dimension, which is exactly what this video does.

The next problem is a good deal more serious. After five minutes, the narrator states,

The long undulating snake that is us will feel like it is moving in a straight line in the fourth dimension, but there will actually be in the fifth dimension a multitude of paths that we could branch to at any given moment.

Wait, hold on. How did the fifth dimension get in there?

To be able to have a multitude of paths through time, we do not need five dimensions; we still only need four. Think about Flatland. A Flatlander can get a sense of us when we pass through the second dimension. But it can get a different sense of us if we pass through while walking sideways rather than forward. It can get a completely different sense if our friend passes through the Flatlander’s space. The Flatlander can see a multitude of paths branching out, but only needs three dimensions to do so. Similarly, we would not need five dimensions to see all the branching paths of time — only four.

At five minutes and thirty seconds, there is a reference to quantum physics which is used as some sort of completely unnecessary metaphor which only seems to serve to make the video seem more strange and interesting.

At 6:40, the narrator proposes yet another unnecessary dimension. For this one, remember the analogy of the Möbius strip. Your own timeline is like a two-dimensional strip of paper, except that it is in four dimensions. To visit your own past, you can wrap the paper around itself through the third dimension, forming a Möbius strip. To visit an alternate timeline, you do not need another dimension; you only need to attach your current strip of paper to a different strip, one that contains the timeline of preference. This means that we do not need six dimensions, nor do we even need five; we still are good enough at four.

If you are following along on YouTube right now, please go to the next video.

As soon as the second video starts, it attempts to compress three dimensions into a single point. You’re not allowed to do that. The fourth dimension does not join the Big Bang to one of the possible endings of our universe; rather, that is what our particular cross-section of the fourth dimension does. The fourth dimension is capable of containing all possible timelines; but we are not able to simultaneously perceive them all.

Now, as we enter the seventh dimension, we are about to imagine a line which treats the entire sixth dimension as if it were a single point. To do that, we have to imagine all the possible timelines which could have started from our Big Bang joined to all the possible endings for our universe (a concept which we often refer to as infinity) and treat them all as a single point. So for us, a point in the seventh dimension would be infinity.

All right, now the author of this video clearly has no idea what infinity is. Infinity “is an unbounded quantity that is greater than every real number.” The number of possible points along one dimension is not even infinity, but is aleph one. The number of points in two dimensions is also aleph one — they have the same cardinality. In fact, the number of points in any number of dimensions (greater than zero) will always be aleph one.

What the author is calling “infinity” really refers to everything that can possibly be perceived by humans. And, if I may point out, we’re still not in seven dimensions. We’re only in four.

But if you think about it, the first sentence of my last paragraph is actually not true. We are capable (in theory) of perceiving infinitely many degrees of saltiness; these cannot all be represented in four dimensions. It would require adding a fifth. The same logic applies to the other four tastes, as well as to the three colors of light, every volume and pitch of sound, and every other thing that we are capable of perceiving. By the time you’ve added these all up, there are more than a dozen major dimensions, as well as at least a hundred less noticeable ones.

The video goes on to propose that the set of possibilities stemming from our Big Bang is only one infinity. There are other infinities resulting from other initial conditions. The problem with this, though is that having these two infinities does not necessitate another dimension because adding two infinities results in the same infinity.

At this point, you should watch up to 2:00.

If we’re really talking about adding other initial conditions, though, then we’d end up with a lot more than eight dimensions. We would need one axis to cover all the universes with different speeds of light; an axis for all the universes with different gravitational constants; all the universes with different fundamental forces, different types of particles, the list goes on and on. So either we are in four dimensions, or way too many to count, depending on how you look at it. But we are certainly not in eight.

The tenth dimension, which is introduced at around 3:30, is just absolutely ridiculous. The author keeps extending dimensions further and further when all we really need is four. He claims that there is “no place left to go” after 10 dimensions; hmm, according to this author’s perverse logic, wasn’t that true after seven?

The reference to string theory near the end is essentially the same as the reference to quantum mechanics: only in there to sound cool, and without any relation to the actual content of the video.

Before I wrap up, allow me to get in one final point. Mathematics has no problem with defining however many dimensions it wants. These dimensions are not spatial; space is merely a way to represent these dimensions, albeit only three of them. There is absolutely no reason to be limited to ten dimensions (or rather, four, as I have demonstrated). But remember, no matter how many dimensions you are using, you can’t cover any more ground. You’re still looking at aleph one points.

Maybe you watched that video and thought it was amazingly cool, and I have just crushed your dreams. But there are plenty of mind-blowingly cool things out there that don’t rely on lies and pseudoscience. If your mind is blown by a lie, is it really worth it?

Categories: Science

The Interpreter, Chapter 2

April 7, 2010 1 comment

The continuing story of one man’s quest to write an interpreter.

Jackson knew that all great languages needed string handling. So he began to add support for strings. What he did not know was the difficulty that strings would bring. Unlike anything he had created before, they required that the lexical analyzer turn a new eye on the input given. It must understand when text is enclosed in quotes. If not, it is read as usual; but if so, it must be treated as a single block of text. The analyzer had to be smarter.

The next great challenge was to implement arrays. In standard convention, they are much like parentheses: a pair of brackets, with some number of commas in between. Jackson found that a very similar definition could be used for both brackets and parentheses. But he changed the way arrays worked to be cleaner, to better fit the calculator-like syntax of his language. It was more Lisp-like: something like (array 1 2 3) instead of [1, 2, 3]. It required no special new syntax. And it allowed for a native implementation of both arrays and linked lists.

Yes, Jackson wanted linked lists. So many languages have either arrays or linked lists, but Jackson wanted both. To keep things simple, he defined a similar syntax for both: lists were defined to be written as (list 1 2 3).

There was some trouble with the implementation of strings, especially when putting strings inside parentheses. The nesting was confusing the interpreter and creating some very unusual bugs. But then the most amazing thing happened. Jackson saw one little line of code, the line that was reassessing the array of tokens, and saw a way to fix it. He had been trying to improve that line for weeks, but at this moment he had a sudden insight. He realized a way to vastly improve the line, making it much simpler. And, lo and behold, it worked perfectly.

Chapter 3

Categories: The Interpreter


April 2, 2010 1 comment

Video. A very entertaining one. About QWERTY. Unfortunately, it doesn’t look like this guy has heard of Colemak.

Categories: Humor, Keyboards