As a follow-up to my previous Sunflower Study Part 1 post, here are more compositions along with a continued discussion about the amazing growth enabling mathematical structures consistently found in nature.
The foundation of this structure is the Fibonacci Numbers or Sequence. Start with 0 and 1 and then the sum them, drop the zero and you have the first two sequence numbers (1, 1). Then keep summing the the two numbers to build the sequence – 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and on. If you divide any number by the previous number you get 1.618 – the Golden Mean, or phi φ.
Next, consider a 360° circle. If you divide 360° by 1.618 you have an angle of 222.5°. The remaining angle in the circle is 137.5° – the Golden Angle (360° – 222.5°). If we build a model of square boxes, each with an adjacent box based on the Fibonacci Sequence, then and any two adjacent boxes will form a Golden Rectangle. Drawing an arch across the outer corners of each box forms the Fibonacci Spiral or Golden Spiral. Now we have the mathematics and geometry to reference what can be seen throughout the natural world.
The term phyllotaxis refers to the botanical study of phylla (leaves, seeds, flower pedals, etc.) on plants. A plant will typically set each phylla at a 137.5° “golden” angle from the previous one. This creates a “golden” spiral of leaves up the stem or seeds around the center of the flower. This enables not only the maximum possible exposure to the sun, but also the most efficient use of space. Fibonacci Spirals are a visual consequence or patterned observation based on this arrangement.
When observing the pattern of disc florets in the sunflower, it’s usually easy to see the both clockwise and counter clockwise spirals. On a typical medium size sunflower you can observe 34 spirals in one direction and 55 spirals in the other. Larger sunflowers can have spirals of 55 and 89. Did you notice these are Fibonacci numbers? John Edmark, a professor at John Hopkins University, is also an artist and inventor. Check out this wonderful video about his fascinating work based on Fibonacci spirals. Another great video about these amazing mathematical structures is Nature by Numbers, it shows a great animation on the distribution of sunflower disc florets.
Once you start looking for the golden spiral, you’ll find them all around you! And, if you’re like me, you’ll get a sense of awe and closeness with our Creator. Perhaps in the future, I can create some compositions to support a discussion around the Golden Mean. These compositions can be best viewed by clicking to see a high resolution version from my portfolio site. Thanks for taking time to visit my blog.