The Platonic Sequence

From The Chapterhouse Codex

1 History
2 Generalizations
3 See Also
4 References
5 External Links

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History

"Odd" Variation

"Even" Variation

The Platonic sequence is one of the first rational arithmetic triangle theorems of the ancient world. It began as a popular investigation into indeterminate analysis by ancient Greek mathematicians. Although it bears only the name of Plato today, both Plato and Pythagoras are presumed to have contributed to it's development. Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows:

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.
...
But the method of Plato argues from even numbers. For 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 as that which was obtained by the other method.

Using this sequence, every integer greater than 2 produces the corresponding pairs necessary for the creation of a right triangle. This, and other interesting characteristics of the sequence, continue to make it an object of study for mathematicians today.

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Generalizations

One modern generalization the sequence is the Hayes formula, which was discovered by the amateur American mathematician Gregory S. Hayes in 2003. This formula is a result of using recent number theory to unify the previously disconnected pairs. Every integer a in the Hayes formula will produce the corresponding pairs of the even or odd variations of the Platonic sequence. The formula is given as follows:

$(b, c) = \left(\frac{a}{2}\right)^2(1+a\bmod2)\mp\frac{1}{1+a\bmod2}$