Day 143/365 - On phyllotaxis, Fibonacci and the Golden Ratio



Another fascinating concept that was introduced by Richard Mandelbaum in class was that of Sacred Geometry. It seems that all things in nature tend to grow in a particular pattern, which in the end resembles a spiral. Not only is a spiral representative of infinity and continuity, it’s also the optimal way for things to grow while conserving energy and maximizing vital flow.


The study of this phenomenon all started when people became interested to learn how plants grow in size, and why they arrange their branches, leaves and flower petals (i.e. phyllotaxis). After much study, it turned out that the pattern is pretty consistent and corresponds to the Fibonacci sequence, where each successive number is the sum of the previous two (0, 1, 2, 3, 5, 8, 13, 21...etc). These numbers show up consistently corresponding to plant growth, and can be converted to ratios where the numerator is the number of spirals it takes for leaves to arrive at the node directly above in an alternating growth pattern, and the denominator is the number of leaves it took to get there, starting with zero. Converted to fractal forms, as the numerators and denominators are each following the sequence, with the denominator lagging by one spot (½, ⅓, ⅖, ⅜, 5/13, 8/21...etc) - they approximate a consistent value as they extrapolate to infinity. The ratios tend to either 0.618 or 1.618, which is known as the Golden Ratio. Furthermore, taking the ratio of each subsequent number in the sequence to the previous number, is always equivalent to the ratio between the sum of the two numbers and the latter number (or a:b=[a+b]:a)! The golden ratio can be expressed as the angle within a circle that opens up to section “a” on the perimeter within a circle whose circumference is “a+b.” This ‘golden angle’ approximates 137.5 degrees, and is common to spherical and circular shapes everywhere in nature. Plants and trees grow in the circular pattern with this angle, and some tree trunks have striations that stretch out to show this to the naked eye.


We are naturally attracted to the symmetry and enigma of the spiral form, and it appears everywhere in the natural world. It is the shape of Nautilus mollusk shells, human DNA helices, hurricane patterns, and most galaxies. Those people that have studied this effect have tried to apply it to the consumer world as well, and it can be seen in art, architecture and even commercials. Famously, the Mona Lisa and the Parthenon exhibit the golden ratio spiral. Tons of watch commercials show the golden angle (the hour handle on 10 and the minute handle at 2) in advertisements to cater to the natural pull of the consumer toward this familiar and flattering shape. Try to pay attention and you may just be surprised at how much of this pattern you start to notice. Share below if I’ve missed anything!