Logarithmic Spirals Logarithmic spirals are spirals that often appear in nature. Logarithmic spirals were first described by Descartes but later on Jacob Bernoulli investigated extensively on the logarithmic spiral, Bernoulli called this spiral Spira mirabilis, translated to "the marvelous spiral." The logarithmic curve can be written as r=ae^b0, using polar coordinates(r,0). Logarithmic spirals can be found in nature such as a Romanesco brocoli which grows in a logarithmic spiral and nautilus shells to spiral galaxies ad the approach of a hawk to its prey. Golden Spiral, Golden Ratio/Rectangle and Fibonacci Sequence To understand the golden spiral you have to understand the basics. For example, everyone knows the famous Fibonacci sequence. Well Fibonacci's sequence has inspired artists and scientists for centuries. For those of you who need a refresher the Fibonacci sequence is 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, and so on forever. The pattern to this sequence is the sum of two numbers that follows behind. For example, 1+1=2, 1+2=3, 2+3=5, 3+5=8, and so on. This is a simple pattern but it has become one of the most influential built in numbering system that leads to the cosmos. Fibonacci sequence had a connection to phi which is the golden ratio. Also you need to understand the golden ratio and golden rectangle and its purpose. The golden ratio is 1/1.618 it is an irrational number and it is associated with the golden rectangle which is 1 for the width and 1.618 for the length of a rectangle. For example, a pentagram is a shape that has a ratio of 1.618 which is the golden ratio. The golden ratio is displayed in music, architecture, and even painting. Did you know, that Leonardo da Vinci's illustrations and paintings of Mona Lisa and Vitruvian Man use the golden ratio. Even some composers of music use the golden ratio to exhibit the main climax of a piece at the phi position. You can tell why the golden ratio is now called the golden ratio. The golden spiral is a logarithmic spiral whose growth factor is phi also called the golden ratio. As a golden spiral gets wider (further from the origin) by a factor of phi every quarter turn it makes. Now its time to discuss what things in nature have a golden ratio, Fibonacci sequence, or golden ratio. The petals of a flower have the golden ratio in it because each petal is placed at 0.618034 per turn. The seeds of a sunflower use the Fibonacci process and golden logarithmic spirals when looked with just logarithmic lines. Spiral Galaxies use the Fibonacci pattern and the golden spiral including the Milky Way. The Milky Way has a logarithmic spiral of about 12 degrees. Hurricanes also form a Fibonacci pattern and logarithmic spiral. Even human faces can depict the Golden Ratio. The golden sections of the mouth and nose are each positioned at golden section between the eyes and the bottom of the chin. The theory behind this is that the most attractive or the most beautiful smiles are those which follow 1.618 or the golden spiral. Why do many natural patterns use the Fibonacci sequence or golden spirals? Even scientist are trying to figure this out is this a coincidence or do these ratios exist because of evolution, natural selection, and a way nature forms. | |