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Phyllotaxis

2. F1000.com - Daisy
3. cs.berkeley.edu - CS285 Gear Generator
4. dgleahy.com - Botanical Units
5. hot.ee - Surface of Revolution
6. coroflot.com - Objects

“Phyllotactic Portrait of Fibonacci” by Robert Bosch

Mathematical artist Robert Bosch created this picture by adapting a well-known portrait of the Italian mathematician Leonardo Pisano Bigollo (c. 1170—1250), who was better known as Fibonacci.

Fibonacci described the sequence that bears his name in his 1202 book Liber Abaci, although the sequence was known to Indian mathematicians as early as the 6th century. The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, the key property being that each of the terms from the third term onwards is the sum of the preceding two terms.

Fibonacci used his sequence to study the growth of a population of rabbits, under idealising assumptions. The sequence can be used to model various biological phenomena, including the arrangement of leaves on a stem, which is known as phyllotaxis. Robert Bosch used a model of phyllotaxis to produce this picture. He explains:

Using a simple model of phyllotaxis (the process by which plant leaves or seeds are arranged on their stem), I positioned dots on a square canvas. By varying the radii of the dots, I made them resemble Fibonacci. Incidentally, the number of dots, 6765, is a Fibonacci number. So are the number of clockwise spirals (144) and counterclockwise spirals (233) formed by the dots.

A framed version of this picture is currently being exhibited at the Bridges Exhibition at Gwacheon National Science Museum, Seoul. You can read more about the picture here: http://gallery.bridgesmathart.org/exhibitions/2014-bridges-conference/bobb. The same page discusses another version of the picture, also by Robert Bosch, but this time illustrating the Travelling Salesman problem. +Patrick Honner has posted about the other version of the picture here: https://plus.google.com/+PatrickHonner/posts/ALvhM8JK5kJ.

Robert Bosch’s website: http://www.dominoartwork.com

Wikipedia on Leonardo Fibonacci: http://en.wikipedia.org/wiki/Fibonacci

The On-Line Encyclopedia of Integer Sequences on the Fibonacci numbers: http://oeis.org/A000045

Fibonacci numbers in nature: http://en.wikipedia.org/wiki/Fibonacci_number#In_nature

As well as featuring in this picture, the Fibonacci number 6765 is the name of an asteroid: http://en.wikipedia.org/wiki/6765_Fibonacci

“We’re also a band.” (http://en.wikipedia.org/wiki/The_Fibonaccis)

(Found via +Patrick Honner.)

#art #artist #mathematics #scienceeveryday

2

Logarithmic Spirals

You see logarithmic spirals every day. They are the natural growth curves of plants and seashells, the celebrated golden curve of ancient Greek mathematics and architecture, the optimal curve for highway turns. Peer into a flower or look down at a cactus and you will see a pattern of logarithmic spirals criss-crossing each other like so:

This elegant spiral pattern is called phyllotaxis and it has a mathematics that is equally lovely. One reason why the logarithmic spiral appears in nature is that it is the result of very simple growth programs such as
grow 1 unit, bend 1 unit
grow 2 units, bend 1 unit
grow 3 units, bend 1 unit
and so on…

Any process which turns or twists at a constant rate but grows or moves with constant acceleration will generate a single logarithmic spiral. An equally similar cellular automata program will generate phyllotaxis.

http://alumni.media.mit.edu/~brand/logspiral.html

Various naturally occurring flowers, their polar graphs (upper row) and the constraining superpolygons (middle row). From left to right: Tradescantia, Nicotiana sp., strawberry, Geranium

Glass Peacock

Peacock imagery is something I seem to revisit every so often. This is probably the most literal version I’ve done so far, using a couple of stacked-up orbit trap shapes to make the eyes.

"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”

phyllotaxy started following you

WHEEEEEEE 8DDD

Thanks for following!  I have 50 followers omg

"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).  (c) For smaller G values (G ≈ 0.1) higher order Fibonacci modes are obtained.  Here φ = 139º and parastichy numbers are (5, 8)."

Phýllondelic -

Inspired by Phyllotaxis, the mathematic arrangement of leaves and seeds within plants (and nature in general) which utilises the golden ratio and fibonacci to produce the most energy and space efficient designs.

I really hope I wasn’t assigned this advisor solely because I’m queer and she’s married to a woman

this makes no academic sense at all

Fibonacci pattern of dots, creating a phyllotactic portrait of Fibonacci. This was created by Robert Bosch, an artist who also is a professor of mathematics, from Oberlin College in Ohio, USA. From the artist notes:

Using a simple model of phyllotaxis (the process by which plant leaves or seeds are arranged on their stem), I positioned dots on a square canvas. By varying the radii of the dots, I made them resemble Fibonacci. Incidentally, the number of dots, 6765, is a Fibonacci number. So are the number of clockwise spirals (144) and counterclockwise spirals (233) formed by the dots.