Enter your email address:


August 5, 2009

64 = 65 Geometry Paradox





Comparison of the areas of the 2 shapes creates a paradox: 64 = 65

How is this possible?

8 comments:

Secret Squïrrel said...

It isn't. The longest side of each of the golden triangles in the "65" diagram are not straight; ie the pieces do not actually fit together exactly. In reality there is a very thin diamond-shaped gap, running from the top-right to the bottom-left of the rectangle and equal to 1 unit of area. Hence 5 * 13 - 1 = 64.

However, I can prove that 0=1 which probably explains why computers sometimes crash!

                                -20 = -20
                           16 - 36 = 25 - 45   equivalent
                    4^2 - (4)(9) = 5^2 - (5)(9)   factorise
     4^2 - (4)(9) + (81/4) = 5^2 - (5)(9) + (81/4)   + 81/4
4^2 - 2(9/2)(4) + (9/2)^2 = 5^2 - 2(9/2)(5) + (9/2)^2   rearrange as polynomial
                     (4 - 9/2)^2 = (5 - 9/2)^2   find the roots
                        (4 - 9/2) = (5 - 9/2)   sqrt of each
                           4 - 8/2 = 5 - 8/2   add 1/2 to both
                              4 - 4 = 5 - 4   reduce fractions
                                   0 = 1   Uh-oh

Anonymous said...

In the above proof, you missed a point.When squares are equated, the term on left hand side will be equal to either + or - of teh right hand term. In this case its is -ve. Your proof came from teh assumption that both terms are positive which is not true always.

Secret Squïrrel said...

You may surmise that I "missed the point" intentionally. That is the flaw in my "proof".

It's along similar lines to the following which also proves that 0=1 (and hence all integers are equal to 0).

Let: a = b = 1
So: a = b
Mult by a: a^2 = ab
Subtr b^2: a^2 - b^2 = ab - b^2
Factorise: (a+b)(a-b) = b(a-b)
Div by (a-b): (a+b) = b
Subtr b: a = 0
a=1 so: 1 = 0

Anonymous said...

hey secret squirrel.. are you working or are you still studying? working as a wat? studying at where?

Anonymous said...

That second one breaks at "divide by (a-b)" because that is a division by zero.

Nir...... said...

Hey Squirrel
The term writte as factorise results in zero itself
a-b=0 cuz a=b=1
still if you like to proceed this way its fine but ultimately you are deviding by zero which is complete NO NO.

Its a nice n tricky approach but there are always holes in the loop.

Anonymous said...

Your blog keeps getting better and better! Your older articles are not as good as newer ones you have a lot more creativity and originality now. Keep it up!
And according to this article, I totally agree with your opinion, but only this time! :)

Anonymous said...

Your blog keeps getting better and better! Your older articles are not as good as newer ones you have a lot more creativity and originality now. Keep it up!
And according to this article, I totally agree with your opinion, but only this time! :)