Difference between Terms of Wythoff Pair

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Theorem

Let $\tuple {\floor {k \phi}, \floor {k \phi^2} }$ be a Wythoff pair.

The difference between the coordinates of this Wythoff pair is $k$.


That is:

$\floor {k \phi^2} - \floor {k \phi} = k$


Proof

\(\ds \floor {k \phi^2}\) \(=\) \(\ds \floor {k \paren {1 + \phi} }\) Square of Golden Mean equals One plus Golden Mean
\(\ds \) \(=\) \(\ds \floor {k + k \phi}\)
\(\ds \) \(=\) \(\ds k + \floor {k \phi}\)

$\blacksquare$


Sources