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MatheMUSEments
Mental Math
By Ivars Peterson
Muse, April 2002, p. 44.
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.
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Volunteer 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.
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Volunteer 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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