Pythagorean Perimeters

[charle...@hotmail.com]

It is not often that a new theorem of exquisite simplicity arrives -
least of all in the context of Pythagoras.

The world is vast, and time has aeons - so perhaps some other has made
the same discovery.

However, to me, and everyone I know this is a new discovery.

It begins with the 3 4 5 triangle.

You know - you create a square on side 3, add the square on side 4 and
get the square on side 5. This is all to do with AREAS.

However, what if you examine the PERIMETER of the square on side 3, and
compare it with the perimeter of the triangle?

What of the 5 12 13 triangle?

Create a 2:1 rectangle on side 5. That is a 10:5 rectangle. How do the
perimeters compare?

What of the 7 24 25 triangle?

Try a 3:1 rectangle - 21:7.

And so on.

These are the base triangles - the smallest.

Of course, for the square you can scale up the 3 4 5 to 6 8 10, to 9 12
15 and onward.

There are also 3:2 rectangles that convert to triangles. Indeed all
whole-number ratios relate to base triangles.

It is the kind of thing that can be taught even to children. Yet it is
new.

Go to http://wehner.org/pythag for the introduction, and follow the
link, or directly to http://wehner.org/pythag/ratios.htm for more
examples of this perimeter-classification of right triangles.

Charles Douglas Wehner

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[W H G]

<charle...@hotmail.com> wrote in message
news:1127826603.6...@g49g2000cwa.googlegroups.com...

>
> It is not often that a new theorem of exquisite simplicity arrives -
> least of all in the context of Pythagoras.
>
> The world is vast, and time has aeons - so perhaps some other has made
> the same discovery.
>
> However, to me, and everyone I know this is a new discovery.
>
> It begins with the 3 4 5 triangle.
>
> You know - you create a square on side 3, add the square on side 4 and
> get the square on side 5. This is all to do with AREAS.
>
> However, what if you examine the PERIMETER of the square on side 3, and
> compare it with the perimeter of the triangle?
>
> What of the 5 12 13 triangle?
>
> Create a 2:1 rectangle on side 5. That is a 10:5 rectangle. How do the
> perimeters compare?
>
> What of the 7 24 25 triangle?
>
> Try a 3:1 rectangle - 21:7.
>
> And so on.
>
> These are the base triangles - the smallest.
>

The OP considers right triangles with the hypotenuse one more than
the longer side. Let a < b < c and a, b and c are a Pythagarian triple,
such that c = b +1 then
a^2 + b^2 = c^2 or a^2 + b^2 = (b+1)^2,
clear the parantheses and you can show
b = (a^2 - 1) / 2
If you choose an odd number for a then b will be an integer.

The perimeter is p = a+b+c = a + (a^2-1)/2 + (a^2-1)/2 + 1
so p = a^2 +a = a(a+1)
The rectangle with same perimeter is a by, call it, s
p = 2a + 2s so s = (p - 2a)/2 = ( a(a+1) -2a ) /2
and s = a(a-1)/2

Interesting, eh? The perimeter of the rt triangle is the same as
that of a rectangle with the shortest side (and another
integer side)! Indeed the side is determined by a.

> Of course, for the square you can scale up the 3 4 5 to 6 8 10, to 9 12
> 15 and onward.
>
> There are also 3:2 rectangles that convert to triangles. Indeed all
> whole-number ratios relate to base triangles.
>

Indeed there are. Consider the 8 15 17 triangle. Perimeter is 40,
same as a 8 by 12 rectangle with ratio of sides 3/2.
As above we can show that b = (a/2)^2 -1, a must be chosen
as an even such that b is odd or else the triangle is double one
of the triplets with c = b+1.
I'll let someone else do the algebra to derive the length of the
rectangle side.

> It is the kind of thing that can be taught even to children. Yet it is
> new.
>

Why teach it to children? New? maybe.

[Guess who]

On Wed, 28 Sep 2005 05:01:12 GMT, charle...@hotmail.com wrote:

>You know - you create a square on side 3, add the square on side 4 and
>get the square on side 5. This is all to do with AREAS.
>
>However, what if you examine the PERIMETER of the square on side 3, and
>compare it with the perimeter of the triangle?

The perimeter of a square drawn on the side is four times the length
of that side?

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.

So, you can draw similar triangles, or semi-circles, or whatever [even
very odd-looking, but exactly similar closed shapes] on the sides of a
right triangle, and the sum of the areas of the smaller will add to
the larger. Then there will also be a relationship between the
perimeters since the sides of the triangle are each the same factor of
the total perimeter of that similar figure. For example, the side of
the triangle is a diameter of a semi-circle drawn on that side, and is
so 2:(2+Pi) of that shape's perimeter, large or small.

I'd be careful of what I'd teach to children and what I'd leave for
later examination in a fuller context. Children have at first enough
difficulty with the normal Pythagorean theorm as it is usually taught.
Some discoveries are exciting to you because you already have the
necessary background to compare and contrast. To them, it could be
something the teacher wants, so they will do it, but it might have
little or no meaning to them out of a larger context.

[charle...@hotmail.com]

W H G wrote:

> Why teach it to children? New? maybe.

1. Why teach it to children? I said EVEN to children. It is not good to
underestimate children, and those who can understand Pythagoras can
understand this perimeter classification.

2. New? New to my understanding, and to those of my friends and
associates. However, such is the vastness of time and space that we
cannot know whether it was discovered before, unless a student of
Diophantus, or of Euclid's "Elements", or of Chinese or Indian
mathematical history finds the prior art.

[charle...@hotmail.com]

"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