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Topic: Pythagorean Perimeters
Replies: 1   Last Post: Sep 30, 2005 6:01 PM

 Charles Douglas Wehner Posts: 76 Registered: 12/8/04
Re: Pythagorean Perimeters
Posted: Sep 30, 2005 6:01 PM

"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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