Introduction

The golden ratio, also known as the divine proportion or phi, is a mathematical constant found in nature that has been used by artists and architects for centuries. It is said that the proportions of the golden ratio are aesthetically pleasing to the human eye, and can be found in everything from flowers to galaxies. In this blog post, we will explore Euclid's pentagon and how it is related to the golden ratio.

In order to construct the geometry of the pentagon, we will use the excellent luxor.jl package.

Golden Ratio - Euclid's Elements and the Pentagon

The discovery of the golden ratio is one that dates back to ancient Greece. It was first mentioned in works by Euclid, Ptolemy during their respective times who all had differing views on what it actually meant but were still able recognize its significance for mathematics.

The importance of the golden ratio is evident in many ancient cultures. In Greece, Euclid and his students were interested to figure out how best could constructoids like pentagons or icosahedrons since these shapes provide us with some insight into architectural designs such as temples where they had fivefold symmetry around their circular bases which led them down a path towards discovering this constant that still impacts mathematics today.

So we are going to show why this constant is important when constructing the pentagon.

The following code constructs the pentagon using the luxor.jl package:

#load dependancies

using Luxor;
using Base.MathConstants;
using SymPy;
using Images;

url = "https://upload.wikimedia.org/wikipedia/commons/c/ce/Scuola_di_atene_23.jpg";
download(url, "scuola.jpg");#download to local file

euclid = load("scuola.jpg")

"""
    guidelines()

Set axis guideline for orientation


"""
function guidelines()
    setdash("dot")
    background("antiquewhite");
    setcolor("darkblue");
    arrow(Point(-250,0),Point(250,0))
    arrow(Point(0,-250),Point(0,250))
    #circle with radious of 200
    circle(Point(0,0),200, :stroke)
    #box 400 x 400
    box(Point(0,0), 400, 400, :stroke)
    setdash("solid")
end

guidelines

@draw begin
    guidelines()
    

    setline(0.3);
    ngon(Point(0,0), 200, 5, -pi/2, action = :stroke)


   
    setdash("dot")

end

#get polygon vertices coordinates

@draw begin
    fontsize(14)
    guidelines()
    setline(0.3);
    setdash("solid")
    ngon(Point(0,0), 200, 5, -pi/2, action = :stroke)
    a = ngon(Point(0,0), 200, 5, -pi/2, action = :stroke, vertices=true) 
    setdash("solid")
    setline(0.5);
    l1 = line(a[3], a[5], :stroke)
    l2 = line(a[2], a[4], :stroke)

    f1,ip1 = intersectionlines(a[3], a[5], a[2], a[4])

    circle(ip1, 3, :fill)
    label("C", :W, ip1)
    label("A", :W, a[3])
    label("B", :W, a[5])

    label("D", :W, a[2])

end

When examining figure above,if we draw two diagonals from the center of the pentagon, we can see that the two diagonals are equal in length, forming 3 isoceles triangles. These diagonals are $AB$ and $BD$.

Using elementary geometry, we can show that according to Euclid's definition, point $C$ divides the line $AB$ precisely in a Golden Ratio $\phi$. Furthermore, the ratio AB to AD is also equal to $\phi$.

This fact illustrates that the ability to construct a line divided in a Golden Ratio provides a simple means to construct a pentagon. For this reason, the ancient Greeks were interested in $\phi$.

Thru simple measurements of line segments of figure, we can derive the ratios in question and show inevocably the presence of the golden ratio in the pentagon.

C = ip1;
A = a[3];
B = a[5];

Point(-3.6739403974420595e-14, -200.0)

AB = dimension(A,B)[1];
AC = dimension(A,C)[1];
BC = dimension(B,C)[1];

Ratio of Line $BC / AC$:

BC_AC = round(BC/AC, digits = 5)

1.61803

Ratio of Line $AB / BC$

AB_BC = round(AB / BC; digits=5)

1.61803

BC_AC == AB_BC

true

φ

φ = 1.6180339887498...

Deriving the Value of $\phi$

Lets look again at the line $AB$ and $AC$ from the figure above.

Let Line Segment $AC$ (shorter segment) be 1 unit long and the length of Line Segment $AB$ be $x$.

If the ratio of $x$ to 1 is the same as that of $x+1$ to $x$, then the line has been cut in a Golden Ratio $\phi$:

@syms x

(x,)

We can set up an equation to solve for the length x. The equation leads to a quadratic:

$\frac{x}{1} = \frac{x+1}{x} \implies x^2 - x -1 =0$

eq = ((x / 1) ~ (x+1) / x); #setting up the equation in SymPy

sols = solve(eq, x)

sols[2] # value of Phi

sols[2] |> float

1.618033988749894848204586834365638117720309179805762862135448622705260462818892