Editing Pythagorean triple (section)

===The Platonic sequence===
The case {{math|1=''n'' = 1}} of the more general construction of Pythagorean triples has been known for a long time. [[Proclus]], in his commentary to the [[Pythagorean theorem|47th Proposition]] of the first book of [[Euclid's Elements|Euclid's ''Elements'']], describes it as follows:

<blockquote>Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to [[Pythagoras]]. (The latter) starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse.<br />
...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which was obtained by the other method.</blockquote>

In equation form, this becomes:

{{math|''a''}} is odd (Pythagoras, c. 540 BC):

:<math>\text{side }a : \text{side }b = {a^2 - 1 \over 2} : \text{side }c = {a^2 + 1 \over 2}.</math>

{{math|''a''}} is even (Plato, c. 380 BC):

:<math>\text{side }a : \text{side }b = \left({a \over 2}\right)^2 - 1 : \text{side }c = \left({a \over 2}\right)^2 + 1</math>

It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence ({{math|''a''}}, {{math|(''a''{{sup|2}} − 1)/2}} and {{math|(''a''{{sup|2}} + 1)/2}}) by allowing {{math|''a''}} to take non-integer rational values. If {{math|''a''}} is replaced with the fraction {{math|''m''/''n''}} in the sequence, the result is equal to the 'standard' triple generator (2''mn'', {{math|''m''{{sup|2}} − ''n''{{sup|2}}}},{{math|''m''{{sup|2}} + ''n''{{sup|2}}}}) after rescaling. It follows that every triple has a corresponding rational {{math|''a''}} value which can be used to generate a [[similarity (geometry)|similar]] triangle (one with the same three angles and with sides in the same proportions as the original).  For example, the Platonic equivalent of {{math|1=(56, 33, 65)}} is generated by {{math|1=''a'' = ''m''/''n'' = 7/4}} as {{math|1=(''a'', (''a''{{sup|2}} –1)/2, (''a''{{sup|2}}+1)/2) = (56/32, 33/32, 65/32)}}. The Platonic sequence itself can be derived{{clarify|What does this mean?|date=April 2015}} by following the steps for 'splitting the square' described in [[Diophantus II.VIII]].