Wednesday, April 7, 2010

Wednesday Math, Vol. 114: Vectors and dimension

An ordered list of numbers is called a vector. How many numbers are on the list is called the dimension of the vector. Let's take, for example, a list of numerical information about a pitcher in baseball. Among the relevant numbers that might be listed are height, weight, age, wins, losses, earned run average (ERA), innings pitched, runs allowed, hits allowed, the number of strikeouts, walks, complete games and saves. Since there are thirteen categories in which we can put numbers, this would be a 13-dimensional vector.

In common parlance, "dimension" deals with three directions which are perpendicular to one another, let's call them length, width and height, usually written in a math context as x, y and z. If I have a physical object like a book, I can say it's 11.5" tall, 8" wide and 2" thick, which describes it when it's standing up. If I lay it flat, now it's 2" tall, 8" wide and 11.5" thick, which is to say the x and z dimensions have switched places. It's still the same book and the important measurements haven't changed. The surface area of the front cover is still 11.5 × 8 square inches, or 92 in.² The volume is still 11.5 × 8 × 2 cubic inches, or 184 in.³. Properties like this that don't change even when the object is moved are called invariants, and any time a mathematical model has an invariant, there's a very good chance it is connected to some physical property in an important way.

It's been more than 100 years since Einstein popularized the idea that time is the fourth dimension, but it doesn't exactly work the way the other dimensions do. We are allowed to move objects in the three dimensions, changing the height for the length or transformations with even trickier math, but we can't interchange time with the other three. If we compare it to the vector of information about the pitcher we started with, we aren't allowed to switch wins for losses or number of strikeouts with innings pitched and say that we are still talking about the same pitcher's stats. Given that we are mortals who live our lives at speeds nowhere near the speed of light, time is the dimension with the least amount of freedom. We move forward in time not backward, and we can do almost nothing to change the pace at which time moves. The ideas of string theory say there may be other physical dimensions, but they are ultra-microscopic loops also can't be swapped with the three dimensions our senses can actually detect.

Multidimensional math is a "real thing", in that it solves problems that can be stated and that correspond to things in the real world. Linear programming problems can have as many dimensions as are needed, which is to say vectors with many entries, not unlike our pitcher's stats. But physical problems where we actually get to swap dimensions like height, width and length and keep some important information invariant are much rarer, and the vast majority of those problems don't stretch beyond the three physical dimensions we can experience.


Tengrain said...

Nicely explained, I really like it when difficult concepts can be expressed simply like this. But I also like to read simplified science and physics books for fun, too.

Concepts are important for me. If I don't get an "a-ha " moment, I know I did not understand the concept, so then I go back and try to find a simplified explanation and I rework my understanding.

Great post, and thanks for doing this. I feel smarter already!



Matty Boy said...

Thanks, Tengrain. I have set a goal of getting to 128 volumes of math, which is two to the seventh power. I often wonder if anyone is reading this stuff at all. It's nice to see that people are still getting to the end and it's making some sense.