LS: Speaking of Science Column April 05, 2009
Math makes sense of a wacky, weird world
Every day, most of us use mathematics in some way.
We use it to make sure we get the right amount of change when we buy a cheeseburger.
Carpenters calculate areas of structures they plan to build. And we want to make sure we have enough money in our checking accounts to keep checks from bouncing. (Hopefully.)
As we get more familiar with math, we feel like we know how it operates. Even for more complicated activities than we normally take part in, we expect that mathematics still involves adding, subtracting, multiplying, dividing and maybe a few other twists.
But sometimes, our simple outlook is not quite the way the real world works.
Let’s say I can throw a baseball at 50 mph. (This will not result in a multi-million-dollar contract as a professional pitcher — I don’t even need math to figure that out.)
Now imagine that I am standing on a flat-bed train car, and you are standing on the ground beside the train. If I throw my “fastball” while the train is still, you will see the same speed for the ball as I do — 50 mph. But if the train is moving at, say, 50 mph when I throw the ball, what will you see as an observer on the ground?
If I throw in the same direction the train is moving, you will see the ball moving at 100 miles per hour — the speed of my pitch, plus the speed of the train. And if I throw the opposite direction of the train’s movement, you will see the ball stand still as it leaves my hand — zero mph.
So your view (from the ground) of the velocity of the ball is obtained just by adding, or subtracting, the speed of my pitch and the velocity of the train, depending on which direction I’m throwing. Simple, right?
Now let’s speed things up. Let’s say the train is moving at 90 percent of the speed of light.
And now I am able to pitch at 90 percent of the speed of light (hopefully without the use of steroids).
Simple addition would say that you as an observer on the ground would see the ball moving at 90 percent plus 90 percent, or 180 percent of the speed of light.
But the real world isn’t that simple. What you on the ground would actually see for the combined velocities would still be less than the speed of light (just a little more than 90 percent), not almost twice that speed, which you would expect from simple addition.
This is a consequence of the theory of relativity, in which Albert Einstein postulated that nothing can move faster than the speed of light. (To give proper credit, however, it should be pointed out that the mathematics involved were actually developed by Hendrik Lorentz and George Francis FitzGerald in the late 1800s, and built upon by Einstein.)
Other strange things happen when we push the limits of light speed — objects get “flatter” and they get more massive. And time passes more slowly for someone moving near the speed of light.
This is definitely not in keeping with our everyday mathematics.
Let’s imagine another scenario. Pretend you are an ant, living on a huge, smooth sphere, the size of the Earth. Imagine further that there is no “up” (away from the sphere) or “down” (toward the interior of the sphere). Your entire universe is the surface.
What happens when you try to draw parallel lines in your spherical universe? At first, the lines will look parallel, just like we imagine them in our three-dimensional world. But if you continue following them farther and farther, you will notice something strange — the lines approach each other, and finally intersect (just like longitude lines eventually meet on the earth at the north and south poles).
The odd conclusion we reach in the ant’s “sphere universe” is that there are no such things as parallel lines. Why not? Because you cannot draw two straight lines in that world that don’t eventually meet.
And believe it or not, the weird geometry in the ant’s “sphere universe” may be a more accurate description of our actual universe than the straight line version we visualize.
The world is stranger than we think.
Vincent King is a certified health physicist who has been involved in radiological sciences for more than 30 years. He is a volunteer at the Western Colorado Math & Science Center.