Question

5. Find at least three interesting and fascinating facts or properties about the Fibonacci num- bers and/or the Golden Ratio that do not appear in the class presentation. You do not need to provide proofs. Support your findings with pointers to their resources.

Answer 1

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FACT 1: The unique properties of the Golden Rectangle provides an example. This shape, a rectangle in which the ratio of the sides a/b is equal to the golden mean (phi), can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It's called the logarithmic spiral, and it abounds in nature.

Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats, and the shape of certain spider's webs.

FACT 2: Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees. As an interesting aside, spiral galaxies appear to defy Newtonian physics. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate. Subsequently, after a few rotations, spiral arms should start to wind around a galaxy. But they don't — hence the so-called winding problem. The stars on the outside, it would seem, move at a velocity higher than expected — a unique trait of the cosmos that helps preserve its shape.

FACT 3:

Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can be seen from the side, and even the eye and ear itself (which follows along a spiral).

It's worth noting that every person's body is different, but that averages across populations tend towards Phi. It has also been said that the more closely our proportions adhere to phi, the more "attractive" those traits are perceived. As an example, the most "beautiful" smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It's quite possible that, from an Evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.

FACT 4:

Phidias, the Greek sculptor, used the Golden Ratio in his work, especially when he began to work with the bands he sculpted just above the Parthenon columns. It's also important to note that the numeric value assigned to the Golden Ratio, Phi, was named in his honor.

If you measure the dimensions of the Parthenon's exterior, you'll discover that it not only forms a Golden Rectangle but that there are also many Golden Rectangles between the columns. The use of the Golden Ratio accounts for the genius and beauty of this example of sacred architecture.

If there are any more facts you would like to know, let me know in the comment section. Thank you.