Golden Diagonals and Fibonacci Numbers
Introduction
Before we begin introducing the golden rectangle, we must first introduce the golden ratio ($\varphi$). An irrational number with a value of $\frac{1+\sqrt{5}}{2}$ or $1.61803$. It belongs to a family of irrational numbers called metallic means.
The golden rectangle is a rectangle with sides in proporation to the golden ratio ($\varphi$) but typically depicted with a length of $\varphi$ and a height of $1$. The diagonal of the golden rectangle is what is referred to as the golden diagonal.
The golden diagonal is $\sqrt{\varphi^2 +1}$ or $\sqrt{\varphi+2}$ and each consecutive rectangle is $\varphi$ times larger than the previous rectangle so the length of each consecutive diagonal is $\varphi^{n}\sqrt{\varphi+2}$.
For more information on the golden ratio and golden rectangle check out:
The Fibonacci Sequence and The Golden Ratio by David Henderson
Table
$nint()$ is the nearest integer function.
$F_n$ is a Fibonacci number.
% diff $ = (\frac{\text{diagonal}-F_n}{F_n}) \times 100 $
| Diagonal | Decimal | nint() | $F_n$ | % diff |
|---|---|---|---|---|
| $\varphi^{-3} \sqrt{\varphi+2}$ | $0.44902$ | $0$ | $0$ | |
| $\varphi^{-2} \sqrt{\varphi+2}$ | $0.72654$ | $1$ | $1$ | $-27.34574\%$ |
| $\varphi^{-1} \sqrt{\varphi+2}$ | $1.17557$ | $1$ | $1$ | $17.55705\%$ |
| $\varphi^0 \sqrt{\varphi+2}$ | $1.90211$ | $2$ | $2$ | $-4.89434\%$ |
| $\varphi^1 \sqrt{\varphi+2}$ | $3.07768$ | $3$ | $3$ | $2.58945\%$ |
| $\varphi^2 \sqrt{\varphi+2}$ | $4.97979$ | $5$ | $5$ | $-0.40406\%$ |
| $\varphi^3 \sqrt{\varphi+2}$ | $8.05748$ | $8$ | $8$ | $0.71850\%$ |
| $\varphi^4 \sqrt{\varphi+2}$ | $13.03727$ | $13$ | $13$ | $0.28674\%$ |
| $\varphi^5 \sqrt{\varphi+2}$ | $21.09475$ | $21$ | $21$ | $0.45122\%$ |
| $\varphi^6 \sqrt{\varphi+2}$ | $34.13203$ | $34$ | $34$ | $0.38833\%$ |
| $\varphi^7 \sqrt{\varphi+2}$ | $55.22679$ | $55$ | $55$ | $0.41234\%$ |
| $\varphi^8 \sqrt{\varphi+2}$ | $89.35882$ | $89$ | $89$ | $0.40317\%$ |
| $\varphi^9 \sqrt{\varphi+2}$ | $144.58561$ | $145$ | $144$ | $0.40667\%$ |
| $\varphi^{10} \sqrt{\varphi+2}$ | $233.94443$ | $234$ | $233$ | $0.40533\%$ |
| $\varphi^{11} \sqrt{\varphi+2}$ | $378.53005$ | $379$ | $377$ | $0.40584\%$ |
| $\varphi^{12} \sqrt{\varphi+2}$ | $612.47448$ | $612$ | $610$ | $0.40565\%$ |
| $\varphi^{13} \sqrt{\varphi+2}$ | $991.00454$ | $991$ | $987$ | $0.40572\%$ |
| $\varphi^{14} \sqrt{\varphi+2}$ | $1603.47903$ | $1603$ | $1597$ | $0.40570\%$ |
| $\varphi^{15} \sqrt{\varphi+2}$ | $2594.48357$ | $2594$ | $2584$ | $0.40571\%$ |
| $\varphi^{16} \sqrt{\varphi+2}$ | $4197.96260$ | $4198$ | $4181$ | $0.40570\%$ |
| $\varphi^{17} \sqrt{\varphi+2}$ | $6792.44617$ | $6792$ | $6765$ | $0.40570\%$ |
| $\varphi^{18} \sqrt{\varphi+2}$ | $10990.40877$ | $10990$ | $10946$ | $0.40570\%$ |
| $\varphi^{19} \sqrt{\varphi+2}$ | $17782.85494$ | $17783$ | $17711$ | $0.40570\%$ |
| $\varphi^{20} \sqrt{\varphi+2}$ | $28773.26371$ | $28773$ | $28657$ | $0.40570\%$ |