When creating things that have form, the process typically does not depend on a singular methodology when generating, perceiving and manipulating a shape. Among these methods are the intuitive sense, a behavioral response, observation of patterns, mathematics, and many more in many combinations. The observation of natural phenomena is commonly used as base for modeling form, but when observing the universe, one finds an enormous variety of phenomena to chose from! The beauty, order and variety in nature is readily visible. The golden proportion, or golden ratio is a classic pattern you've most likely seen that occurs in nature and demonstrates certain commonalities and relationships between various areas of physical, biological worlds. At the simplest level, it is a mathematical relationship between two quantities that is intrinsic to various fundamental geometries and forms. By definition, two quantities are in the golden proportion (aka, the golden mean, golden section, golden number) if their ratio is the same as the ratio of their sum to the larger of the two quantities.

a+b is to a as a is to b

In math, the ratio is commonly referred to as the greek letter Phi , As a decimal, it is represented as 1.61803398... It is an irrational number, meaning it has an infinite number of decimal places and it never repeats itself, like e or Pi. Phi is curious because in order to square it, you can just add 1. Incidentally, phi denoted in lowercase means just the numbers after the decimal.

Some proofs that show the uniqueness of the number.

Phi = 1.6180339

phi = 0.6180339

Phi * phi = 1

Phi - phi = 1

Phi = 1 + phi

phi = Phi – 1

Phi = 1 / phi

phi = 1 / Phi

Phi * 2 = Phi + 1

phi * 2 = 1 – phi

(–phi) * 2 = –phi + 1

Phi ^ 2 = Phi + 1

Phi ^ 2 – Phi – 1 = 0

Phi + phi = √5

Phi = (√5 + 1) / 2

phi = (√5 – 1) / 2

Ok, now that we've seen the math of the number, let's see how to find it with geometry. The proportion becomes prevalent when describing symmetry and descriptions of polyhedra and polygons, as the ratio appears in many forms as as a fundamental geometric building block.

Mathematicians, designers and artists throughout history have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and the golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio (as in the image in the title). It's a classic, right up there with Pi.

So, this article will demonstrate some distinct methods I've found for finding the proof of Phi with geometry. If you were doing this by hand, the tools you need are a straightedge and a compass. With a computer, you can use any drawing program that can do lines and arcs, and point-snapping is a big help too.

Step 1: The Golden Rectangle

Possibly the most common and widely used method to find the golden ratio is within the golden rectangle. The ratio of the short side of the rectangle to the long side is Phi. This is commonly seen when describing the Fibonacci sequence as a series of golden rectangles embedded within each other, such as depicting in this articles title.

Draw a square that has sides that are 1 unit long
Place a compass point at 1/2 of the bottom side of the square
Rotate the compass clockwise it is on the same horizontal plane as the base of the square.
Where there compass arc and the base of the rectangle meet is Phi

Step 2: Single Known Length

The simplest way to find Phi that I know of is not the traditional Golden Rectangle method. We can actually derive Phi from a single known length. In this method, we have a line that has a known length, in this case it has a length of 1.

Align the known length vertically (A)
Align another line of the same length to one end to the midpoint of the vertical length, and the other end to the horizontal plane. (B)
With a third duplicate length, align one end to the midpoint of the second length, and then other end to the horizontal plane. (G)
The ratio of (AB) to (AG) is Phi

Step 3: Triangle in a Circle

We can find the golden ratio when inscribing a triangle inside a circle. This step assumes you can draw a perfect equilateral triangle within a circle. There are a few ways to draw a triangle without a computer or other devices that I listed in the requirements, such as this article on drawing an Equilateral Triangle

Draw a circle.
Align an equilateral triangle so it fits within the circle bounds.
Draw a horizontal line that starts from the 1/2 point of the left edge of the triangle (A)
Continue the line to the other side of the triangle (B) until it intersects with the edge of the circle (G)

Step 4: Square in a Circle

The square-in-circle method is similar to the triangle, and related. It's interesting to see the pattern and relationships between these shapes emerge.

Draw a circle
Divide the circle in half with a horizontal line
Draw a square that rests on the horizontal line, and is scaled vertically so its two top corners intersect the edge of the circle.
The length one side of the square (AB) to the distance to the right or left edge of the circumference of the circle (AG) is Phi

Step 5: Pentagon in a Circle

The pentagon in a circle method is similar to triangle and square.

Draw a circle
Within the bounds of the circle, draw a pentagon, and intersect the vertices with the edge of the circle.
Draw two lines that emanate from the bottom vertices of the pentagon, and intersect at the top vertex of the pentagon.
Draw a horizontal line from the leftmost vertex (A) of the pentagon to the rightmost vertex (B) of the pentagon.
The ratio of (AB) to (AG) is Phi

Step 6: Pentagon in a Circle 2

If you've already inscribed a pentagon in a circle, the ratio actually exists in another measurable place.

Connect all the vertices into a 5 pointed star.
If the edge of the pentagon is 1, then the length of the connecting star lines is Phi
The length of the lines that create the points of the are 1/Phi, or phi

Step 7: 1/2 Square

The 1/2 square method

Draw a side of a rectangle with length 1 on the horizontal, and a height of 1/2 on the vertical.
Draw a diagonal line within the rectangle (xy)
Place your compass point at (y) and set it to length (yz)
Inscribe an arc until it intersects with the diagonal line. (w)
Place your compass point at (x) and set it to length (xw)
Inscribe an arc that starts at (w), and intersects with the bottom of the rectangle (u)
(xu) is 1/Phi, or phi

Step 8: Conclusion

There are a myriad of ways to find and utilize this unique numeric ratio with geometry. I've included the following reference files for all methods as AI, DXF, Solidworks, and PDF.

Additionally, in Processing.js: Phi in Processing JS

// Phi / Golden Ratio example written in Processing.

// online at http://studio.sketchpad.cc/7e3hFRhlhV

size(600, 400); smooth(); strokeWeight(2); background(255); stroke(0); noFill();

float phi = (sqrt(5) + 1) / 2;

float scale = width / phi / phi;

translate(scale / 2, scale / phi / 2);

// draw guides

stroke(50);

// golden rectangle

rect(0, 0, scale * phi, scale);

// guides