Pythagorean Ratios
By Charles Douglas Wehner

It is possible to map all Pythagorean triples to rectangles of rational aspect ratios.

The method is as follows:

1. Choose a ratio
2. Double it
3. Add 1 First side
4. Square it
5. Subtract 1
6. Divide by two Second side
7. Add one Third side
8. Possibly adjust results 3, 6, 7 to integers.

This follows from the following:
Let the ratio be R. The first side is then 2R+1. That squared less one is 4R2+4R. The second side is therefore 2R2+2R and the third side is 2R2+2R+1.

The equation that brings it all together is:
 (2R2+2R+1)2-(2R2+2R)2=(2R+1)2

So we arrive at the following series for the Pythagorean base triples:

Integer Ratios

The Square

1. 1:1
2. 2:1
3. 3:1 3
4. 9:1
5. 8:1
6. 4:1 4
7. 5:1 5

2:1

1. 2:1
2. 4:1
3. 5:1 5
4. 25:1
5. 24:1
6. 12:1 12
7. 13:1 13

3:1

1. 3:1
2. 6:1
3. 7:1 7
4. 49:1
5. 48:1
6. 24:1 24
7. 25:1 25
And so on.

Halves

3:2

1. 3:2
2. 6:2
3. 8:2 4
4. 64:4
5. 60:4
6. 30:4 15:2
7. 34:4 17:2
8. 8 15 17

5:2

1. 5:2
2. 10:2
3. 12:2 6
4. 144:4
5. 140:4
6. 70:4 35:2
7. 74:4 37:2
8. 12 35 37
And so on.

Thirds

4:3

1. 4:3
2. 8:3
3. 11:3 11:3
4. 121:9
5. 112:9
6. 56:9 56:9
7. 65:9 65:9
8. 33 56 65

5:3

1. 5:3
2. 10:3
3. 13:3 13:3
4. 169:9
5. 160:9
6. 80:9 80:9
7. 89:9 89:9
8. 39 80 89
And so on.

It will be noticed that if the ratios are coprime, the triples will be the base triples - as 3 4 5 is the base of 6 8 10 and others.

It will be noticed that further, if the triples are base triples, their three numbers will be coprime.

This conversion of a coprime duple to a coprime triple may prove useful in the search for a formula for primes. However, as the coprimality of the starting ratio must first be established by human logic before the coprime triple is found, it is not of itself a stand-alone formula for coprimes.

ADDENDUM, 9 May 2007

When a coprime pair turn into a coprime triple, a new prime must be generated. Thus, the concept contains the idea of "neogenesis of primes". Further, if a b are turned into c d e one can start again with c d, with c e and with d e and their inverses.

It was hoped that the study of "neogenesis" would reveal where the primes are coming from. It was also hoped that ALL the primes would be found.

In an internet newsgroup, somebody who used the "handle" Quasi showed that if a b are a coprime duple, a b (a+b) are a coprime triple. As a corollary, the author showed that a b (a-b) is also a coprime triple. Accordingly, the neogenesis of primes is contained within the addition and subtraction functions in arithmetic, and is not unique to Pythagorean geometry.

The suggestion that one looks for a general formula for primes by this method is therefore retracted.

HOME

A new theorem on the decidability of primes emerged, which shows that the problem is not soluble arithmetically.

THEOREM

(C) 2005 Charles Douglas Wehner.
Use freely but do not plagiarise.