Understanding the Golden Proportion

This charming miniature toy grocery store was made during the depression from a kit and cigar boxes. We use it at Christmas with our own miniature additions. The original toy came with canned goods which were out of proportion with the store counters.

Decorating for energy involves utilizing the mathematics of nature. I generally just use the blueprint of a circle or an ellipse which is enough to bring energy into an environment. Sometimes you might hear of the Golden Proportion as an example of the mathematics of nature. It is good to know about this mystical number when we think about our world. In Understanding the Golden Proportion, I will bring some simplicity to its complexities.

It seems as the Golden Proportion has a lot of synonyms: the Golden Mean, Golden Ratio, Golden Cut, Golden Section. Sometime the word, Golden is substituted by the word, Divine. TPeople use them interchangeably but not all these words are the same. I want you to know the difference between a proportion and a ratio and understand the meaning of the Golden Cut and Golden section.

Making Ratios and Proportions Simple

When I was teaching math to Special Needs students, I found that the more I could put these abstract notions into pictures, the better the students understood the concepts. So it is best to do the same when learning ratios and proportions. One of the best examples to illustrate these concepts is the dollhouse my husband and I built for our granddaughter. We wanted to make everything in proportion unlike the folk art toy store above.

When we constructing it, we had to concentrate on the proportions to get everything to look real. Usually we could just buy the molding, baseboards and furniture already properly sized. Occasionally, however, we had to use some mathematical skills whenever I sculpted something from scratch.

Our Dollhouse Proportion

A typical doll house is 1/12 the size of a human house. [1] So for every foot in human size, an inch is substituted for the doll house. A six-foot human couch becomes six inches in a dollhouse.

The problem came in when I started working in with smaller items that were not in wholes numbers. One day I decided to make a pie. Now pies are usually baked in a 9″ pie pan. I could intuit that a 9 inch pie would be ¾ inch in doll house size.

The problem for us came when I decided to sculpt even smaller items in the human world like an orchid flower, a dish of spaghetti or a doughnut. .

To handle these conversions, we had to go back to some rudimentary high school math concerning ratios and proportions. In learning this process, you will be more apt to understand the golden proportion. As much as possible, keep real life examples of math problems as you work the numbers. Then you won’t run away screaming.

A Ratio

Before understanding the term proportion, it is necessary to understand the term, ratio. A ratio is a comparison of two numbers. Looking at the dollhouse, we see that 1 inch is compared to 1 foot or 12 inches. We can write that comparison in many ways:

1 inch is to 1 foot
1 inch:1 foot (which is read out loud as, 1 inch is to 1 foot)
1 inch/1 foot (which is read out loud as, 1 inch is to 1 foot)

1″ is to 12″
1″:12″ (which is read out loud as 1 inch is to 12 inches)
1″/12″ (which is read out loud as 1 inch is to 12 inches)

Proportions

A proportion is two ratios that are equal to each other. When we write them out, they are an equation. Think of the doll house. The scale is 1 inch to 12 inches. So a 6 foot human couch (72 inches) translated to dollhouse math would be 6 inches and would present itself in a proportion in many ways.

1 is to 12 as 6 is to 72
1/12 = 6/72 (which is read as “1 is to 12 as 6 is to 72)
1:12 = 6:72 (which is read as, “1 is to 12 as 6 is to 72)

1 inch is to 12 inches as 6 inches is to 72 inches
1″ is to 12″ as 6″ is to 72″
1″/12″ = 6″/72” (which is read as, “1 inch is to 12 inches as 6 inches is to 72 inches”)
1″:12″ = 6″:72″ (which is read as, 1″ inch is to 12 inches as 6 inches is to 72 inches’)

Proportion Problem Solving

When I was using Fimo clay to create an orchid plant for the dollhouse, I had to use an equation.. Each flower was 4 inches in diameter in human terms. How could I find the mini number to create the flower proportionate to the real world.

I wrote out the proportion which was:

1 inch is to 12 inches as the-unknown-number of inches is to 4 inches.

1:12 = ?:4 (Note that in mathematical terms, the outside numbers, 1 and 4 are called extremes and the inside numbers, 12 and the unknown number represented by x are called means. This is useful to understanding the golden mean.)

Then I had to change it to an equation to find the unknown number. In an equation, the unknown number is always written as x instead of a ?, a question mark. So your proportion to find the mini orchid blossom will look like this:

1 = x
12 4

I cross multiplied to solve the equation.

x times 12 = 1 times 4

12 times x = 4

x = 4/12 or 1/3 or.333…

When I solved the equation, the unknown number or x was .33 inch or between 1/4 and 1/2inch. That was the number I used in sculpting the orchid flower.

Using Our Knowledge of Proportion to Understand the Golden Proportion

The Golden Proportion has been found extensively in nature and has been used for centuries by artists. It can be expressed geometrically, as a fraction or as an approximation of a number. To find the number, we can take a line and cut it at a specific point.

That cut is at a point where the whole line is to the the larger cut of the line as the larger part of the line is to the smaller cut of the line. That cut is called a golden cut.

green is to blue as blue is to red

The larger segment is to the middle size segment as the middle segment is to the smaller segment.

.

Also written as a + b: a = a:b Read as a+ b is to a as a is to b.

Note that a is in the inner part of the equation. That makes it a mean.

Wikipedia offers various ways to solve for for the golden ratio as a number. The golden ratio is 1.61803……. The periods at the end means the number will go on forever, never repeating itself.

The line segment a is 1.61803 because the line segment b is 1 and can be expressed as:

1.61803… is to 1

1.61803…:1

1.61803… to 1

1.61803…/1

or simply, because it is fraction with a 1,

1.61803…

Just remember this is a ratio even though it is expressed as a number.

This number is also known as phi or the symbol Φ.

Just a Warning

If you google the golden proportion to learn more about how it was solved, you are going to get different diagrams with different letters than the example above. Sometimes the cut is given a letter.

A_______________________C_________B

Just keep in mind that you are comparing line segments no matter what letters are.

Decorating with the Golden Proportion

The golden proportion is not of this earth even though we see the geometry in galaxies, hurricanes, roses or the curl of an eye lash. That is because the golden number, like pi, is an irrational number, a number going on for infinity without repeating numbers..

For example:

1.6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788… You could go on for another 10,000 numbers and for infinity, never repeating the same number.

To understand the overwhelming idea of a number going on into infinity, think of this scenario: If each number or set of numbers were given a letter correspondence, (eg. a = 1, k = 11)) we would find in infinity everything that was ever written. You would find the Constitution and all its drafts, my Aunt Rose’s cookie recipe, every love letter ever sent, Julia Child’s cook books, the text to your ex you thought was deleted, your first grade report card, War and Peace with all its revisions, every political speech.

It would have every NY Times article, every essay written by every elementary child in the world, every Dear John letter, all the CIA’s non redacted files, every person registered to vote and every ballot – EVERYTHING ever written.

That is infinity.

Practical Uses

Therefore, we can see how difficult it would be to use the golden proportion/ratio in decorating as a number. The Golden proportion is from another world, Fortunately we have a system that can approximate it on an earthly plane. That is the Fibonacci series:

1, 1, 2, 3, 5, 8, 13, 21, 34…………

We can do proportions by using those numbers like 5 is to 8 as 8 is to 13. Somewhat useful if you are trying to figure out where place a chair on a 13 foot wall. Fun to play with. These numbers, however, are more useful in building a room rather than doing interior design. That is why I use an easier approach based on a circle to bring harmony to my decorating. See Let the Eyes have a Pleasurable Dance, Lesson VIII, Part A.

Musing on the Golden Proportion

Despite my use of the circle as a basis for decorating for harmony, I love musing on the Golden Number and reference it often in my blogs and book lessons. I can just kind of feel it as the mystical glue to the universe. I know it determines the proportions of my body right down to the geometry of my DNA. It is the heart of sacred geometry.

Plato and many of his peers believed that geometry existed before creation and when matter was created it simply filled in the geometric templates. The Golden Mean was major blueprint. It is a beautiful number which makes me understand why da Vinci’s friend, Luca Pacioli changed the word “golden” to the word “divine.”

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It shows Pacioli standing behind a table and wearing the habit of a member of the Franciscan order. He draws a construction on a board, the edge of which bears the name Euclides. His left hand rests upon a page of an open book. This book may be his Summa de Arithmetica, Geometria, Proportioni et Proportionalità or a copy of Euclid. Upon the table rest the instruments of a mathematician: a sponge, a protractor, a pen, a case, a piece of chalk, and compasses. In the right corner of the table there is a dodecahedron resting upon a book bearing Pacioli’s initials. An rhombicuboctahedron (a convex solid consisting of 18 squares and 8 triangles) suspends at the left of the painting. The identity of the young man at the right is uncertain, but one commentator recognizes the “eternal student” instructed by Pacioli. Some authors have also mentioned the possibility that the student is Dürer.

Footnote:

[1] A Barbie dollhouse scale is 1 foot to 6 inches.

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There are two parts to this website, The Lessons, which are more difficult in concept, and the blogs, which are lighter in nature. Blogs that you might enjoy with the same theme as Understanding The Golden Proportion are:

Lesson that relate to this blog are:

Please note that my website allows you to leave comments at the end of the blogs but not at the end of each lesson. If you have a comment or question about a lesson, you may email me at ruta@rutas-rules.com

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