Free Video – Golden Section Ratio

Discovering the Golden Section Ratio

Drawing Lesson 24, Part 1

Video Lesson Description

In this video lesson, you will discover the Golden Section Ratio, also known as Golden Proportion, Golden Section, Divine Proportion, Golden Mean, or Golden Ratio, which is a universal principle that is present in nature, science, and art.

Bonus – Golden Ratio Face Mask:

Golden Section Ratio

The Golden Proportion is mathematically determined by nature, and used by fine artists, sculptors, and architects as a guideline for beauty and harmony.

It’s time to examine the golden proportion, to understand how it works. Let’s compare two ratios as described by a famous Greek mathematician, Euclid of Alexandria, about 2,300 years ago.

In his book, “Elements”, Euclid drew a line and divided it into a ratio that was named by him as, “extreme and mean ratio”.

The line “A” is divided into two parts “B” and “C” so that the ratio of the line “A” to the larger part “B” is the same as the part “B” is to the smaller part “C”.

This Golden Mean Phi can be calculated as the sum 1, added to the square root of 5, divided by 2, and equals approximately 1.61803 to 1.

Using Euclid’s proportions, we can draw a pentagram in which the ratios of lines “A” is to “B” as “B” is to “C”, and “B” is to “C” as “C” is to “D”. All pentagram segments, in order of decreasing lengths, are in a ratio of Phi that is 1.816.

Nature uses the same principles of construction. Let’s cut an apple to check the pentagram sections in the centre. This shape is a five-pointed star and is also called the “Pentad”; for centuries, it was regarded as the symbol of power. The first written description of this symbol can be found in the book by Luca Pacioli, Leonardo da Vinci’s teacher, who revealed the method of the Pentad construction, and its unique geometrical properties, to the World.

The pentad geometry can also be found in seeds, leaves, flowers, star fish, and many other plants and living creatures.

Divine proportions can be found in nature in the form of spirals. Equiangular Spiral, also known as “Spira Mirabilis”, is the logarithmic spiral. The distances between the arms of this spiral increase in geometric progression. It can be geometrically constructed using golden triangles or golden rectangles.

Here’s the method of the triangles. The golden triangle is taken from the Pentad.

This triangle is duplicated and scaled to fit a previous triangle.

A spiral curve goes around the triangles’ corners.

The method of the golden rectangles starts with a square, and continues with additional squares that correspond to the golden ratio.

A spiral goes from a corner to the opposite corner of each square.

There are endless examples of golden spirals in nature. Its geometry even explains why the eye of the storm is calm, while the hurricane can reach as much as a hundred miles per hour in wind speed. The golden spiral’s eye is called Asymptote. This is the place that the spiral is approaching, but never reaches; therefore, the wind forces are in balance in the eye of the storm.

The golden proportion spiral works on a cosmic scale, as well. Spiral galaxies are formed around a centre of gravity, but they follow the same divine rules.

Numerous spirals can be found in our bodies. Examples of these spiral shapes include the human ear, our fists, the human embryo, and the structure of our DNA.

The human body and facial proportions follow the divine formula.

To illustrate golden proportions in a human face, let’s create the “golden mask”. As the base, we will take the Pentad that consists of golden triangles. On top of one Pentad, we place another one, which is rotated 180 degrees – upside-down. Now we connect all 10 outer points with blue lines. This grid is used for the facemask construction. Note how red lines follow green and blue lines. The mask is complete, now let’s put it into practice.

The harmony of the Golden Proportion is present in the human body.

Leonardo da Vinci’s famous drawing, Vitruvian Man, dating back to 1509, depicts a man in two superimposed positions, with his arms, and legs inscribed in a square and circle. A circle centred on his centre of gravity, navel, and a square is centred on the root of his penis. The man’s knees, the root of the penis, and the middle of the chest, divide the figure into 4 equal parts.

This drawing is named in honour of Marcus Vitruvius Pollio, a Roman architect, writer, and engineer, who wrote about 2040 years ago:

• The length of the outspread arms is equal to the height of a man

• For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height

• The open hand from the wrist to the tip of the middle finger is just the same

• The head from the chin to the crown, an eighth

• From above the chest to the hairline is one-seventh of the height of a man

• And with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth

• From the middle of the breast to the summit of the crown is a forth

• The maximum width of the shoulders is a quarter of the height of a man

• The distance from the elbow to the tip of the hand is a quarter of the height of a man

• The distance from the elbow to the armpit is one-eighth of the height of a man

• The root of the penis is at half the height of a man

• The foot is one-seventh of the height of a man

• The distances from below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one third of the face

The proportions of the human body correlate with the divine ratio Phi.

• The golden proportion can be found in the ratio of the distance between the finger tip and the elbow, to the distance between the wrist and the elbow

• The ratio of the distance between the navel and the end of the foot, to the distance between the navel and knee

• The ratio of the distance between the navel and the top of the head, to the distance between the navel and the shoulder line

• The ratio of the distance between the navel and the top of the head, to the distance between the top of the head and the middle of the breast

For centuries, the golden proportions of a human body have fascinated fine artists. Albrecht Dürer, a German painter, graphic artist, and humanist, wrote, illustrated, and edited the book on human body proportions. This book was designed to apply the science of human anatomical proportions to aesthetics.

The same divine proportions can be found in Albrecht Dürer’s illustrations.

Here is the solution to a practical question – how to divide a line according to the golden proportion.

• To divide any line at the point of the golden ratio, first find the middle of that line. You can use a ruler or a drawing compass

• Draw the perpendicular line “BC” which equals one half of the first line

• Join the end points of two lines “A” and “C”, making the right angle triangle

• Draw an arc with a radius “CB”; it cuts the longest side of the triangle in the point “D”

• Draw another arc with the centre “A” and radius “AD”

• The point “E” where this arc intersects the line, divides this line into its golden ratio

This task can be done much faster, without any geometrical drawings, by simply using the Golden Gauge tool. In the next part of this video lesson, you will discover how to make such a tool.

Great art starts with a great drawing; all great drawings are based on golden proportions.

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