http://irradiate.net/fya/images/triangles.gif
5 bucks to whoever explains this :)

um... sure

call me dumb, but that still doesn't explain it..... :eek: :(

In math class a couple years back, we had that same problem.

But, I can't explain it. Been too long, and it's quite complex.

IIRC it involves the pythagorean theorem.

Here is the explanation. It's quite easy. All you need to know is just basic geometry:

Look at picture #1.
The angle of the red triangle should be the same as the angle of the dark green triangle (do you know why?). In other words, the tangent of the angles of the two triangles should be the same. But it's not! The tangent of the dark green triangle is 2/5 (2 blocks vertical, 5 blocks horizontal). The tangent of the red triangle is 3/8 (3 blocks up and 8 blocks horizontal). Now, the tangent of the whole triangle is 5/13 (5 blocks up and 13 blocks horizontal). 2/5 is not equal to 3/8 and it's not equal to 5/14. Can't have this in a triangle in this configuration. You can actually show this by actually take up a ruler and try to line up the hypothenuse of the two triangles . The lines bulge down ... It's not a straight line as it's supposed to be.

In picutre #2, the line bulge up ...

How do we get the extra hole? The optical illusion that causes the two lines to bulge down and up will create an area difference exactly of one hole. I should be able to prove this later.

cheers,
:beer:

Let's do some simple calculations:

Picture #1.
The area of the dark green triangle: 0.5x2x5 = 5.0
The area of the red triangle: 0.5x3x8 = 12.0
The area of the L-shaped (together): 5x3 = 15.0
--------------------------------------------------------------------------- +
Total Area for all shape: 32.0

Now, lets take all shape to be one object, a big triangle. Then the are of this big triangle is: 0.5x5x13 = 32.5

Difference in area: + 0.5. The sign (positive) here is important since it tells you that there is a net positive area due to this optical illusion.

Same goes for picture #2. You will get a difference in area of + 0.5.

Hence, the total "extra" area you will get is: (+0.5) + (+0.5) = +1.0, exactly the area of one hole.

No suprises ... :)

cheers,
:beer:

PS: you can paypal me the money :D

Originally posted by dektong
the tangent of the angles of the two triangles should be the same. But it's not! The tangent of the dark green triangle is 2/5 (2 blocks vertical, 5 blocks horizontal). The tangent of the red triangle is 3/8 (3 blocks up and 8 blocks horizontal). Now, the tangent of the whole triangle is 5/13 (5 blocks up and 13 blocks horizontal). 2/5 is not equal to 3/8 and it's not equal to 5/14. Can't have this in a triangle in this configuration.Ah.. you're absolutely right. It's one of those optical illusion thingys. :D The way they are drawn here, it looks like the large triangles are congruent. However when you compare various points along the hypotenuse, they don't match up with eachother.

Hey Matt, glad to see you are satisfied with my explanation. Does it mean I will get free colo this month from you? :D :D

cheers,
:beer: