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Re: Pythagorean Perimeters
Posted:
Sep 30, 2005 6:01 PM
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"Guess who" wrote:
>
> The perimeter of a square drawn on the side is four times the length
> of that side?
A square has four identical sides.
>
> You'd be better perhaps to consider still areas: Areas of *any*
> similar figures drawn similarly on the sides of a right triangle bear
> the same Pythagorean relationship as do the squares; the sum of the
> two smaller add to the larger. Also, it is useful to know that in
> similar figures linear measures are in the same ratio as the square
> roots of the areas. That can be useful in so many ways.
This is off topic, and adds nothing new.
I do not wish to guess who wrote this and more.
WHG, however, was completely in touch with what I am describing.
Perhaps I should give a few more extensions to the Pythagorean
Perimeter theorem?
First, the "SQUARE TRIANGLE" is 3 4 5. That is because a square on side
3 matches the perimeter of the triangle.
Secondly, the "TWO-TO-ONE" triangle is 5 12 13 because the perimeter of
a 10:5 rectangle matches the perimeter of the 5 12 13 triangle.
INVERSIONS.
I shall call these "inversions" because it is a term in the mathematics
of music. An octave UP is 2:1, whilst an octave DOWN is 1:2. That is
musical inversion - the ratio is stood on its head.
To invert 2:1, we use the ratio 1:2.
As before, the system is Diophantine, and uses no algebra because
Pythagoras lived about 1300 years before al Kwarismi, who introduced
algebra.
1. Take the ratio 1:2
2. Double it 2:2
3. Increment it 4:2 First side
4. Square it 16:4
5. Decrement it 12:4
6. Halve it 6:4 Second side
7. Increment it 10:4 Third side
8. Eliminate common denominator 4 3 5
So we have the 3 4 5 triangle reused as a 1:2 triangle (4 3 5).
The 1:2 rectangle on side 4 will have sides 2 and 4, giving a perimeter
12. That matches that of a square on side 3.
We can see that two ratios will map to each Pythagorean triangle - but
the ratios will not be the reciprocals of one another.
EXCHANGES
Here I propose to exchange the rectangles on the non-hypotenuse sides.
We have seen how the 3 4 5 triangle gives a 1:1 ratio whilst the 4 3 5
triangle gives a 1:2 ratio.
This theme - the exchanging of the rectangle from one side of the
right-angle to the other, and so halving or doubling the ratio -
repeats.
HYPOTENUSE
Here I propose that Pythagorean triangles may equally easily be
catalogued by matching them to Diophantine rectangles on the
hypotenuse.
Lengthy studies of these things would be boring. But I have to mention
them to show that I have studied them.
Charles Douglas Wehner
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