Thursday, February 07, 2008

Multi-Dimensional Magic

I've recently been watching a series of lectures on DVD (borrowed from Pa) called The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas. It's pretty well done, and I've been enjoying it. The lectures on the fourth dimension got me thinking, of course, about magic.

Most of what they covered was pretty much what you'd get from reading Flatland, which is worth doing if you haven't already. In particular, think about a two-dimensional world where a two-dimensional being is looking at a square. Maybe the square is his safe, and he thinks his money is securely locked up there. But we, as three-dimensional beings, can simply reach inside the square and remove the money without the two-dimensional being seeing anything. The disappearance of his money would seem magical from his perspective. Similarly, a four-dimensional being could perform analogous feats that seem impossible from our three-dimensional perspective.

Now, something else mentioned in the lecture was that any attribute can technically be called a "dimension" (it's just fun and tricky to try to think about an extra dimension of space). For example, you could call color your fourth dimension, and then use x, y, and z coordinates plus a color to uniquely identify any point in that particular view of the universe. That mention of color reminded me of something I learned way back in one of my only two linguistics classes, and that is how people's perception of color isn't a fixed, physical thing. People from cultures with fewer words for color actually perceive fewer colors than people with more words for them. (This was tested by finding some obscure jungle tribes somewhere, with only three or so color words, and asking them to tell whether certain swatches of colors were the same or different. Many colors or shades that we would perceive as different actually looked the same to them.) I think this is interesting because it's a mental limitation (or enhanced ability if you take it the other way), and therefore something you can learn, as opposed to something physical, like color blindness.

So connecting these two ideas of perception and dimensions, makes me wonder if there are other dimensions we can't perceive now but can learn to. A fourth dimension of space would be a good start, but other unusual attributes would be intriguing as well. Taking it to an extreme, perhaps, what if you could consider "reality" a dimension? Would someone who learns to perceive that dimension be able to see alternate universes?

Of course, simply perceiving a dimension or attribute doesn't necessarily mean you have power to change it. Just because we can perceive colors doesn't mean they change at our will, and just because we might learn to perceive four-dimensional space doesn't mean we could actually do anything in it. On the other hand, we don't have complete control over our standard dimensions either, but we get by alright. We can't change the location of a large building in space, but we can move our own bodies to a reasonable degree, and use them to move smaller objects. And if we have paint at our disposal, we can change the colors of things. So it's probably worth just figuring out what else we can perceive and then working from there.


Anonymous said...

We do live in a (macroscopically) four-dimensional world: 3 space dimensions and one time dimension, or 3x1. Similarly, Flatland has three dimensions: 2x1. The difference between time and space is that time only goes forward. Mathematicians who study the partial differential equations modeling nature talk about boundary-value problems -- space can be bounded on both sides; and initial-value problems -- time can be given a boundary only at the beginning.

Thinking about how 3D objects move in time can give you some idea about how to think in 4D; just pretend that the usual rules of time don't apply.

Take Flatland and the safe, for example. The safe is a square, and the coin is a disc in the safe. Now add the time dimension. The square becomes a rectangular prism (the prism dimension being time), and the coin becomes a cylinder. Now, wiggle the safe all you want: you can't get the coin (the cylinder) to leave the safe (the prism) without going through the temporal wall of the safe. But if you break the rules of time, you can get the coin out. One way is to suppose the coin blinks in and out of existence as time passes (but it's in the same place whenever it exists). The 3D picture for this is a cylinder with chunks taken out of it. When the coin blinks out of existence, move the safe. (Now the rectangular prism is no longer a prism: part of it is sheared.) When the coin blinks back into existence, it's outside the safe. A second method is to go back in time. The coin ended up in the safe at some earlier time, so go back there, move the safe to a different location, and the coin ends up outside it (at the expense of breaking causality).

Once you see the picture for these scenarios in Flatland, you can transfer them to our macroscopic 3x1 world. And now you have a way of thinking in 4D.

dancing dragon said...

Hi Graham,

That's so cool, I was just thinking about higher dimensions and Flatland a few days ago, particularly how a fourth dimension to us relates to belief in God or a spiritual world.

Movement in a dimension one higher than our own makes possible things like teleportation, telekinesis, telepathy, and invisibility. I was also thinking how people describe hearing God through a voice in the heart, and the Flatland idea makes it possible to think of that sort of literally.

William Wren said...

it seems to me that a 3 dimentional person would be unable to reach into a 2 dimentional space and pick up a 2 dimentional coin sice we can only phisically move objects whom are 3d

florin said...

Interesting post…and interesting comment by “dancing dragon”

I seems that we (humans in our 4 dimensional world) need to ask and answer coherently 4 basic questions:
1. questions of origin (who am I, where did I come from) – this relates not only to the 4th dimension, but to ALL dimensions
2. questions about meaning in life (why am I here, what is my purpose in this life, I want to do something meaningful),
3. questions about morality (what is the right thing, who can set the standard for right and wrong, what is justice), and
4. questions about our ultimate destiny (what happens when/after I die, is there an eternity). – like #1, this relates to all dimension

Thanks for listening said...

My favorite magic thing is for a magician to create something from nothing. Then you started talking multi dimensions...reminding of my favorite physics trick:

a2 + b2 = c2 descibes a triangle that might well be in flatland.

Similarly, a2 + b2 + c2 = d2 describes the diagonal distance in a box (front left top corner to the back right bottom corner) - a
3 dimensional object.

If a2+b2=c2 and a2+b2+c2=d2, then
a2+b2+c2+d2=e2, discribing the fifth dimension. Abracadabra! Magic.