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New Blog and Hyper-geometry

Fri: March 18th, 2011 2 comments

Well, here I go with a new blog. I used to blog regularly on Myspace, but now that I use Facebook, I’ve gotten out of the habit. I found it was good exercise for my writing brain. It helped minimize the feeling of dread that I often get when contemplating the start of a new writing project. It also kept me in practice, translating thoughts into text, and presenting them in a readable manner.

I wanted to talk about hyperspaces. Often, the more I learn, the more I realize that I don’t know. Other times, the curtain is lifted and things at which I used to marvel, are revealed to be rather pedestrian. You may have heard of a figure called a hypercube.  Below is a 3d representation of what is supposedly a 5-d object. Stuff like this used to blow my mind. I still wouldn’t be able to make a video like this – project a n-dimensional figure on 3 dimensions – but it is displays like this that totally miss the point of n-dimensions, and prompt popular media to draw wild conclusions.

Hypercube in Hypercolor!

Here is what a scientist means when (s)he says “multiple dimensions”.

  1. One dimension. Popularly known as “numbers”. In one dimension the coordinates are valued like (2) and (3) . The distance between points (2) and (3) is |2-3| . This is a space often referred to as \Re^1 or just \Re . It includes any number you can think of, as well as all of those that you can not.
  2. Two dimensions. Often called the x,y plane. You use 2-dimensional math when you analyze a graph. Points in 2-space are often denoted by ordered pairs like (2,3) and (8,1) . The first number refers to dimension 1 and the second to dimension 2. The distance between (2,3) and (8,1) gets slightly more complicated, but many of you know the Pythagorean formula a^2 + b^2 = c^2 . Many may also know that the distance between two 2-space points is c=\sqrt{a^2 + b^2} where a is the distance between two points on the x axis and b is the distance between two points on the y axis. (On a side note, here’s a nifty visual explanation of how the Pythagorean identity works.
  3. Three dimensions. Now, it gets a little more interesting, but it’s more of the same. We now add a z-axis, and points become (3,4,5) and (1,5,2) . We can still visualize a meaningful concept of “distance” in 3-space. If we have two points, and we label their coordinates (x_1, y_1, z_1) and (x_2, y_2, z_2) , then the distance between the two |a-b|=\sqrt{(x_1-x_2)^2+ (y_1-y_2)^2+ (z_1-z_2)^2}.
  4. Now, let’s skip to n-dimensions. With n-space we usually drop the x, y, z (we don’t want to be limited to 26 dimensions) and label points like so, point a= (x_1, x_2, x_3, x_4, \cdots, x_n) and point b=(y_1, y_2, y_3, y_4, \cdots, y_n). Once we’re above three dimensions, distance has no real non-mathematical meaning. We do, however extend Pythagoras’ formula into n-space in order to give a measure of magnitude which we call “distance”, \|a-b\|=\sqrt{ (x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2 + \cdot + (x_n-y_n)^2} . (The double bars \| are a way to denote magnitude.)

Now, let’s do some n-dimensional math! Suppose I want to go to the grocery store. I have $100 to spend. Lettuce costs $2, (b)read 3$, bolo(g)na 1$, (m)ayo $2, m(u)stard $2, and (c)heese $3. I can choose how many of each to buy, as long as the total is no more than $100. The set of all of my possible mix of goods lies in 5-space. There, I’ve gone into five dimensional space and I didn’t even need my special time shorts! This “space” would be a purely mathematical construct, the set of all points (l, b, g, m, u, c) such that (2\times l)+(3\times b)+(1\times g)+(2\times m)+(2\times u)+(3\times c)<100. Isn’t that boring?

So the next time you see an article trying to explain to you how String Theory predicts 10 dimensions, you can scoff. “Hah, I’ve got more than 15 or 20 just in my shopping cart!”

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