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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$