Mental Math

Want to impress your friends? Tell them you can do trigonometry just like that—without even thinking about it. You don't have to tell your friends your brain does the calculations without any help from you.

Trigonometry is a branch of mathematics that has to do with using angles to figure out distances; it is the basis for all calculations a surveyor does to make a map.

When a surveyor wants to know the distance to a building, he peers through a telescope called a surveyor's level, moving it until it is lined up with the building. He can then read out the angle between this line and his line of sight to the distant horizon. Knowing this angle and the height of the telescope, he can calculate the distance to the building.

The cool thing is you do this yourself all the time. As you look around, you constantly make decisions about how far away things are—whether it's your pal down the street or a tree in the distance. Of course you don't really do trig. That's great for drawing a map or locating a building, but it's more work than you want to do for an on-the-fly estimate.

Instead you guess how far away the object is by where it falls in your field of view. If you have to look down toward your feet, you can assume that it's close to you. If you have to peer toward the horizon to see it, it's far away. So, the farther away an object is on the ground, the higher it is in your field of view. This is actually more or less the same calculation the surveyor does. You're using an angle to figure out a distance.

Scientists now have evidence confirming that people really do use this angle to decide how far away things are. Ten Leng Ooi of the Southern College of Optometry in Memphis, Tennessee, and her coworkers asked volunteers to wear prism goggles. The prism changed the direction in which light travels, making objects appear lower in the field of view than they really were. The volunteers missed when they tried to walk to the objects blindfolded or throw beanbags at them.


Volunteer (above) viewing a distant object without crazy prism goggles.

When people were allowed to get used to the goggles beforehand, they judged distances correctly. (This proves the brain has an amazing ability to adapt to weird input.) Then, when they took off the goggles, they temporarily went to the opposite extreme, overestimating distances and overshooting objects when they tried to walk to them.


Volunteer (above) viewing a distant object with crazy prism goggles.

So, even if you haven't studied trigonometry yet, your brain is using it all the time. But beware of optometrists looking for volunteers. They are apt to make you look like a total idiot.

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Muse, April 2002, p. 44.