Ugh, well links don't work... look at the first link!! copy + paste! ðŸ˜€

Notice on the left hand rectangle the diagonal travels up 3 in 8 across and then up 2 in 5 across (to and from the piece divide in the bottom half). Clearly these are not the same gradient and therefore they must not be exactly matching up with the gradient of the upper right triangle.

That's where your area difference comes from. Obvious really.

Originally posted by XanthosNZ Notice on the left hand rectangle the diagonal travels up 3 in 8 across and then up 2 in 5 across (to and from the piece divide in the bottom half). Clearly these are not the same gradient and therefore they must not be exactly matching up with the gradient of the upper right triangle.

That's where your area difference comes from. Obvious really.

Aren't those different gradients? ... I think they are... from what I counted they are the same.

I go to a bank, ask for a loan of 65 big ones arranged as a 5 by 13 rectangle.
Then I rearrange them as a 8 by 8 square and return theese 64 big ones and keep one big one for myself.
Bank doesn't lose anything (65=64 remember?) but I have a big on in my pocket.
Where does this come from?

Originally posted by FabianFnas I go to a bank, ask for a loan of 65 big ones arranged as a 5 by 13 rectangle.
Then I rearrange them as a 8 by 8 square and return theese 64 big ones and keep one big one for myself.
Bank doesn't lose anything (65=64 remember?) but I have a big on in my pocket.
Where does this come from?

Ok, XanthosNZ was saying that the two different shaped pieces - the triangle and the rectangular prism thing both have different slopes on their gradients which are supposed to line up and make a straight line in the 13x5 configuration... so it is kind of an illussion... the extra square is squeezed between the lines!

Oh, and i guess this kind of thing only works when the slopes are consecutive Fibonacci numbers like 5, 8, and 13 as in this example... it also works with 3, 5, and 8 or 8, 13, and 21... kind of weird how that is...

there are a bunch of other examples of this type of problem... but why does it only work with fibonacci numbers? The only reason we discovered the fibonacci sequence is because it is so prevalent in nature... why does nature swear by this sequence? another mystery of nature unsolved.

it is because if you square a fibonacci number it differs by only one from the neighbouring fibonaaci numbers multiplied together. This means that you are in effect stretching a square along the diagonal. It works well because a single square gap is difficult to spot with the naked eye.

(sometimes the single square will be an overlap 13 x 13 = 169 8 x 21 = 168 so in this case the rectangle will appear to have a smaller area than the square but it may be easier to spot an overlap)

Originally posted by XanthosNZ Notice on the left hand rectangle the diagonal travels up 3 in 8 across and then up 2 in 5 across (to and from the piece divide in the bottom half). Clearly these are not the same gradient and therefore they must not be exactly matching up with the gradient of the upper right triangle.

That's where your area difference comes from. Obvious really.

So then in the first link the large "triangle" really isn't one but is rather a quadrangle with one angle very close to 180 degrees.

Originally posted by AThousandYoung So then in the first link the large "triangle" really isn't one but is rather a quadrangle with one angle very close to 180 degrees.

Originally posted by FabianFnas Funny, funny. ðŸ˜•
Why would I go to the bank if I had my own big one?
I was referring to currency, what did you think, exactly?

Hehe, a little double entendre....Just pulling your chain!