Like the Golden section, the Fibonacci numbers are used to understand the way trees branch, leaves occur, fruit ripens, etc. -- it is a set of numbers that explains nature's patterns.

Subdividing shapes has no effect on their ratio or relationship to Fibonacci.

Chapter 4 -- Root Rectangles- the idea of the root angel reduction allows the Golden section to become more vital in several aspects of modern life. Not only does this impact modern design of furniture, technology, and appliances; it has a far larger and more robust meaning as we begin to understand the roots of organic chemistry and the structure of living organisms. One of the most interesting aspects of this behavior is that of the human's innate ability to follow some of the principles of the Fibonacci and Golden section rules without even realizing it. In a similar manner, one need only listen to a composition by Bach to find it emotionally pleasant, calming, and that something about it "seems to fit." One need not analyze the chords or structures to understand that it, too, operates off the Golden section ratio.

Talking Points

Root rectangles (and therefore Golden sections) may be created using any shape or combination of shapes; circles, ovals, trapezoids, or combinations.

Snowflakes and ice crystals are examples of hexagonal constructions of the Golden section and form the basis for inorganic chemistry.

If one takes a look at most design, from movie posters to magazine art, one can easily find the Golden section and divide the document into coherent triangles.

Furniture, too, is based on the principle of the Golden ratio; the curve proportion, the way the back of the chair...

Designing using the Golden ratio looks and feels better to humans, trying to purposefully not adhere to a ratio causes the print or item to look askew, uneven, and unpleasant. The overall basis for this is the aesthetic nature of the Golden section. Perceptual studies show that the Golden mean has a direct impact on the way humans perceive and appreciate beauty; for the way music is put together, even modern music, based on the intervals between tones. Debussy's La Mer is a great example of this, the sequences are marked out as in the Fibonacci sequence, and the mood is both pleasant and relaxing.
Talking Points-

The ratio of the Golden mean is universal; it appears in plants, veins in leaves, skeletons, crystals, art, music, and seems to be one of the constants in universal physics -- at least on this quantum plane.

In the early 1990s, researchers proposed that there was a connection between DNA base pairs and the Fibonacci number system.

In 2003, a study showed that based on psychometric data, the Golden mean appears in the chronological cycle of brain waves. This was empirically confirmed in 2008.

In 2010, the journal Science reported that the golden ratio is present at the atomic level.

For a designer, the importance of the ratio cannot be overstated; from the Volkswagen Beetle to the Gutenberg Bible, the Golden ratio is everywhere; natural and man-made.