What you are viewing in each of the above videos is a solid 3D printed sculpture spinning at 550 RPMs while being videotaped at 24 frames-per-second with a very fast shutter speed (1/2000 sec). The rotation speed is carefully synchronized to the camera’s frame rate so that one frame of video is captured every time the sculpture turns ~137.5º—the golden angle*. Each petal on the sculpture is placed at a unique distance from the top-center of the form. If you follow what appears to be a single petal as it works its way out and down the sculpture, what you are actually seeing is all the petals on the sculpture in the order of their respective distances from the top-center. Read on to learn more about how these were made, and why the golden angle is such an important angle.

*Note: the exact value for the golden angle is irrational. Here it is to five decimal places: 137.50776º

© edmark, 2014

Regarding the composite flowers and seed heads, Leonardo da Vinci was the first to suggest that the adaptive advantage of the Fibonacci pattern is to maximize exposure to dew. Current thinking supports this interpretation. Phyllotactic architecture optimizes access to moisture, rainfall and sunlight.

The most famous example is the sunflower head. This phyllotactic pattern creates an optical illusion of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and the number of clockwise spirals. These also turn out to be Fibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.

“Karl Blossfeldt (1865-1932) was a German artist and teacher who immersed himself in plant morphology by photographing nothing but plants for 35 years. He devised a self-made system to shoot close-up photographs of flowers, buds, seed pods, tendrils, and more–in order to study their form and design in detail. His photographs were originally seen as teaching material and only later presented as autonomous art works. Urformen der Kunst (Art forms in Nature) was published in 1928”

“Three photographs seen from above of typical phyllotactic patterns formed by ferrofuid drops for different values of the control parameter G [11, 12].  (a) For G ≈ 1 each new drop is repelled only by the previous one and a distichous mode is obtained, φ = 180º. (b) For G ≈ 0.7 the successive drops move away from each other with a divergence angle φ = 150º (between drop three and four).  Drops define an anti-clockwise spiral shown as a dashed line with parastichy numbers (1, 2).  © For smaller G values (G ≈ 0.1) higher order Fibonacci modes are obtained.  Here φ = 139º and parastichy numbers are (5, 8).”