Golden Diagonals and Fibonacci Numbers

Figure 1. The bold numbers on the diagonal are the nearest integer value of the length of the segment.

Just something interesting we noticed while studying the golden mean and rectangle.

Table of contents

Introduction
Table

Introduction

Before we begin introducing the golden rectangle, we must first introduce the golden ratio ( $φ$ ). 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 ( $φ$ ) but typically depicted with a length of $φ$ and a height of $1$ . The diagonal of the golden rectangle is what is referred to as the golden diagonal.

The golden diagonal is $\sqrt{φ^{2} + 1}$ or $\sqrt{φ + 2}$ and each consecutive rectangle is $φ$ times larger than the previous rectangle so the length of each consecutive diagonal is $φ^{n} \sqrt{φ + 2}$ .

For more information on the golden ratio and golden rectangle check out:
The Fibonacci Sequence and The Golden Ratio by David Henderson

Table

$n i n t ()$ is the nearest integer function.
$F_{n}$ is a Fibonacci number.
% diff $= (\frac{diagonal - F_{n}}{F_{n}}) \times 100$

Diagonal	Decimal	nint()	$F_{n}$	% diff
$φ^{- 3} \sqrt{φ + 2}$	$0.44902$	$0$	$0$
$φ^{- 2} \sqrt{φ + 2}$	$0.72654$	$1$	$1$	$- 27.34574 %$
$φ^{- 1} \sqrt{φ + 2}$	$1.17557$	$1$	$1$	$17.55705 %$
$φ^{0} \sqrt{φ + 2}$	$1.90211$	$2$	$2$	$- 4.89434 %$
$φ^{1} \sqrt{φ + 2}$	$3.07768$	$3$	$3$	$2.58945 %$
$φ^{2} \sqrt{φ + 2}$	$4.97979$	$5$	$5$	$- 0.40406 %$
$φ^{3} \sqrt{φ + 2}$	$8.05748$	$8$	$8$	$0.71850 %$
$φ^{4} \sqrt{φ + 2}$	$13.03727$	$13$	$13$	$0.28674 %$
$φ^{5} \sqrt{φ + 2}$	$21.09475$	$21$	$21$	$0.45122 %$
$φ^{6} \sqrt{φ + 2}$	$34.13203$	$34$	$34$	$0.38833 %$
$φ^{7} \sqrt{φ + 2}$	$55.22679$	$55$	$55$	$0.41234 %$
$φ^{8} \sqrt{φ + 2}$	$89.35882$	$89$	$89$	$0.40317 %$
$φ^{9} \sqrt{φ + 2}$	$144.58561$	$145$	$144$	$0.40667 %$
$φ^{10} \sqrt{φ + 2}$	$233.94443$	$234$	$233$	$0.40533 %$
$φ^{11} \sqrt{φ + 2}$	$378.53005$	$379$	$377$	$0.40584 %$
$φ^{12} \sqrt{φ + 2}$	$612.47448$	$612$	$610$	$0.40565 %$
$φ^{13} \sqrt{φ + 2}$	$991.00454$	$991$	$987$	$0.40572 %$
$φ^{14} \sqrt{φ + 2}$	$1603.47903$	$1603$	$1597$	$0.40570 %$
$φ^{15} \sqrt{φ + 2}$	$2594.48357$	$2594$	$2584$	$0.40571 %$
$φ^{16} \sqrt{φ + 2}$	$4197.96260$	$4198$	$4181$	$0.40570 %$
$φ^{17} \sqrt{φ + 2}$	$6792.44617$	$6792$	$6765$	$0.40570 %$
$φ^{18} \sqrt{φ + 2}$	$10990.40877$	$10990$	$10946$	$0.40570 %$
$φ^{19} \sqrt{φ + 2}$	$17782.85494$	$17783$	$17711$	$0.40570 %$
$φ^{20} \sqrt{φ + 2}$	$28773.26371$	$28773$	$28657$	$0.40570 %$

FibonacciGolden meanGolden ratioGolden rectangle