Kepler's Triangle¶
Kepler's Right Triangle is a right triangle with sides in geometric progression 1 : √Φ : Φ. It is named after Johannes Kepler, who first noted this remarkable relationship.
The Triangle¶
A right triangle with sides \(a\), \(b\), and hypotenuse \(c\) satisfies the geometric progression:
\[a : b : c = 1 : \sqrt{\Phi} : \Phi\]
This means: - \(a = 1\) - \(b = \sqrt{\Phi}\) - \(c = \Phi\)
Verification¶
\[a^2 + b^2 = 1^2 + (\sqrt{\Phi})^2 = 1 + \Phi = \Phi^2 = c^2\]
The Pythagorean theorem holds because \(\Phi^2 = \Phi + 1\).
Golden Pi Connection¶
The Kepler triangle provides the direct geometric link between Φ and Golden Pi:
\[\pi = \frac{4}{\sqrt{\Phi}} = 3.144605511029693\ldots\]