phyllotaxy

3

Geometry at work: Sea animals, Fruits and Vegetables and Plants

The Nautilus

The nautilus (from the Latin form of the original Ancient Greek ναυτίλος, ‘sailor’) is a pelagic marine mollusc. Although not a golden spiral, the nautilus shell presents one of the finest natural examples of a logarithmic spiral.

Geometry of fruits and vegetables

When sliced in half, the majority of the depicted fruits and vegetables will display a geometric shape or pattern, based on symmetry or platonic solids.

The arrangement of leaves

Phyllotactic spirals form a distinctive class of patterns in nature, depicting the  arrangement of leaves on a plant stem.
The basic patterns are alternate, opposite, whorled or spiral, many of them arranged based on consecutive fibonacci numbers.

Fibonacci Spiral Wooden Pendant

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Approximate logarithmic spirals frequently occur in nature, for example the arms of spiral galaxies, nautilus shells or phyllotaxis of leaves.

Like standard jigsaw puzzles, this puzzle has only one solution, but instead of every piece being a different shape and approximately the same size, every piece is the same shape and a different size. The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature, for example on pine cones or sunflowers. The puzzle has 8 spirals in one direction, and 13 in the other.

Fibonacci Spiral Wooden Pendant

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Approximate logarithmic spirals frequently occur in nature, for example the arms of spiral galaxies, nautilus shells or phyllotaxis of leaves.

9

It is Tu Bish’vat apparently! Which means Jewish tree appreciation!Here are the pomegranates and a carambola (star fruit) now compared to December around where they first germinated!

I’ve been also noticing something interesting about the oldest pomegranate’s growth, in that it’s been alternating between 2 and 3 leaves per node, while the others have just been adding 2 leaves in opposite phyllotaxis, so it’ll be cool to see individual differences in all of them as they grow up. The second one is much taller with thinner and longer leaves, though I’m wondering if I just accidentally gave it more nitrogen than the others.

The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,[6] although the sequence had been described earlier as Virahanka numbers in Indian mathematics.[7][8][9] By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The sequence described in Liber Abaci began with F1 = 1.

Fibonacci numbers are closely related to Lucas numbers {\displaystyle L_{n}} in that they form a complementary pair of Lucas sequences {\displaystyle U_{n}(1,-1)=F_{n}} and {\displaystyle V_{n}(1,-1)=L_{n}}. They are intimately connected with the golden ratio; for example, theclosest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, … .

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings,[10] such as branching in trees,phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple, ReportoftheWeek videos, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone’s bracts.