In Many Realms, the Shape of the Playing Field Counts

The last place you’d think to find insights from Einstein is the debate over Proposition 209, the ballot initiative seeking to ban “preferences” designed to give women and underrepresented minorities (in state and local government and public education) a helping hand. And yet, there’s a sense in which the whole thing is a fairly simple problem in geometry.

The underlying question here is: What’s the shape of the playing field?

If the playing field is tilted in favor of women and minorities, then obviously affirmative action is unneeded and unfair to white guys; if the playing field is tilted against women and minorities, then, initiative opponents say, affirmative action just as obviously is needed.

Alas, few people stop to think about the shape of the stage on which we play out our lives, mainly because it’s normally invisible. But ever since Einstein refashioned the way we think about space and time, it’s become a real factor in every physical equation.

To get a sense of how the geometry of the playing field can change things, consider the following riddle. You are at some unknown location on Earth. You walk one mile south, make a 90-degree turn and walk one mile east, then walk one mile north. You are back at your starting point. What color are the bears?

The answer is white, because essentially the only places on Earth you can make three right-angle turns and be back at your starting point are the poles.

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Of course, anyone who took geometry knows you can’t make three right-angle turns on a flat surface and get back to your starting point. But if the surface is curved--like the surface of the Earth--you can do all sorts of things your geometry teacher never taught you.

For example, most people learned in geometry that two parallel lines never meet. Again, this is true enough for flat, two-dimensional space like a piece of paper. But two lines of longitude that are parallel at the equator meet at the poles.

The shape of the background, in other words, makes a huge difference in how things work, whether we’re aware of it or not. And it affects a great deal more than lines and angles; it can also determine how physical forces act.

According to Einstein’s relativity, for example, gravity is really the result of the curvature of four-dimensional space-time (a joining of space and time that creates the backdrop for our universe).

Imagine space-time as a flat sheet--say, like a sheet on a water bed. Now, put something heavy on the sheet--like a bowling ball. The sheet sags. If you roll a small marble on the bed and it comes near the indentation made by the bowling ball, it falls right in.

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In Einstein’s gravity theory, the surface of the water bed is space-time. The bowling ball is the Earth, which warps space-time because of its mass. The marble “falling” is gravity pulling everything “down” toward the Earth.

In the same way, a bigger mass--the sun--warps space-time so drastically that nine planets and numerous asteroids fall into its grasp, rolling around the rim of a huge well in space-time like so many marbles rolling around inside the rim of a bowl.

The geometry of the unseen background, in other words, determines what rises and what falls on its face. So it’s important to measure how, or whether, the background curves.

How can you measure the shape of something you can’t even see? Surprisingly, perhaps, it is doable.

It turns out that if you can make any triangle whose three angles add up to more than 180 degrees, you know you’re on a surface that curves inward--like the surface of the Earth. And if you can make a triangle whose angles add up to less than 180 degrees, you know you’re on a surface that curves outward--like the inside of a spoon.

Scientists use these tools to figure out such unwieldy things as the size and shape of our universe--as well as its ultimate fate. If the universe curls in on itself, then it will eventually collapse into a single point; if the universe curves outward like a saddle, then it could keep expanding indefinitely.

To be sure, debating the merits of Proposition 209 will require measuring devices different from rulers and protractors. Still, if we can measure the shape of the universe at large, it shouldn’t be all that difficult to come up with a set of tools for figuring out the shape of the economic and social playing field here on Earth.