## August 5, 2009

### 64 = 65 Geometry Paradox

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

How is this possible?

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...

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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! :)