In botany , phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phllon “leaf” and taxis “arrangement”). [1] Phyllotactic spirals form a distinctive class of patterns in nature .

**Leaf arrangement**

Two different examples of alternate (spiral) leaf patterns

The leaves on the basic stem are arranged opposite and alternate (also known as spiral ). If several leaves rise from the same level (at the same node ) on a stem , or appear to rise, the leaves may also shrink .

With an opposite leaf arrangement, two leaves emerge from the stem at the same level (at the same node ) on opposite sides of the stem. An opposite leaf pair can be thought of as a swirl of two leaves.

Each leaf arises at a different point (node) on the stem, with an alternate (spiral) pattern.

Distichus phyllotaxis, also known as “two-rank leaf arrangement”, is a special case of either opposite or alternate leaf arrangement, where leaves on one stem are arranged in two vertical columns on opposite sides of the stem. Examples include various bulbous plants such as bufon . It also occurs in the habits of other plants such as Gasteria or Aloe plants, and also in mature plants of related species such as Kumara plicatilis .

In an opposite pattern, if successive leaf pairs are 90 degrees apart, the habit is called a dicus . It is common in members of the family Crassulaceae [2] . Spinach also occurs in phyllotaxis in Aizoaceae . In the genera of the Azoaceae, such as Lithops and Conophytum , many species have only two fully developed leaves at a time, the old pair curling back and dying to make room for the new pair as the plant grows. Is. [3]

The curved arrangement is quite unusual on plants except especially in plants with short internodes . Examples of trees with spiny phyllotaxis are Brabejum stellatifolium [4] and the related genus Macadamia . [5]

A whorl can occur as a basal structure where all the leaves are attached at the base of the shoot and the internodes are small or none. A large number of leaves spread in a circle is called a basal whorl rosette .

**Repeating spiral**

The angle of rotation from leaf to leaf in a repeating spiral can be represented by the degree of complete rotation around the stem.

Alternate distichus leaves will have a 1/2 angle of full rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5, in sunflower, poplar, and pear, it is 3/8, in and willow and almond the angle is 5/13. ^{[6]}The numerator and denominator usually consist of a Fibonacci number and its successor. In the case of simple Fibonacci ratios, the number of leaves is sometimes called the rank, as the leaves are lined up in vertical rows. With large Fibonacci pairs, the pattern becomes complex and non-repetitive.

This happens with a basal configuration. Examples can be found in compound flowers and seed heads. The most famous example is the sunflower head. This phylotactic pattern creates an optical effect of a criss-crossing spiral. In the botanical literature, these designs are described by the number of spirals counterclockwise and by the number of spirals clockwise. These are also Fibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals have vortices.

**Determination**

The pattern of leaves on a plant is ultimately controlled by a local deficiency of the plant hormone auxin in certain regions of the meristem. ^{[7]} Leaves begin to form in localized areas where auxin is absent. ^{[ }^{disputed }^{– }^{discussion}^{ ]} When a leaf begins to form and growth begins, auxin begins to flow towards it, thus leading to the loss of auxin from another area on the meristem where a new leaf is to be started. This gives rise to a self-propagating system that is ultimately controlled by the ebb and flow of auxin in different regions of the meristem topography. ^{[8]}^{}^{}^{}

**History**

Some early scientists—notably Leonardo da Vinci—observed the spiral arrangement of plants. ^{[9] In} 1754, Charles Bonnet observed that the spiral phyllotaxis of plants was often expressed in both a clockwise and counter-clockwise Golden Ratio series. ^{[10]} Mathematical observations of phyllotaxis followed with the 1830 and 1830 work of Karl Friedrich Schimper and his friend Alexander Braun, respectively; Auguste Bravais and his brother Louis linked the phyllotaxis ratio with the Fibonacci sequence in 1837. ^{[10]}

Insight into the mechanism had to wait until Wilhelm Hofmeister proposed a model in 1868. A primordium, the nascent leaf, is formed in the least crowded part of the shoot meristem. The golden angle between successive leaves is the blinding result of this setback. Since three golden arcs add little more than enough to wrap a circle, this guarantees that no two leaves follow the same radial line from center to edge. The parent spiral is the result of the same process that produces clockwise and counterclockwise spirals that emerge in densely packed plant structures, such as *protea* flower discs or pinecone scales.

In modern times, researchers such as Mary Snow and George Snow ^{[11]} continued these lines of inquiry. Computer modeling and morphological studies have confirmed and refined Hoffmeister’s ideas. Questions remain about the details. Botanists are divided on whether the control of leaf migration depends on a chemical gradient between primordia or purely mechanical forces. Lucas rather than Fibonacci numbers have been observed in some plants ^{[ }^{citation needed}^{ ]} and sometimes leaf position appears to be random.^{}

**Mathematics**

Physical models of phyllotaxis derive from Airi’s use of packing hard regions. Gerrit van Ettersen depicted a grid visualized on a cylinder (rhombic lattice). ^{[12]} Daudi et al. showed that phylotactic patterns emerged as self-organizing processes in dynamic systems. ^{[13]} In 1991, Levitov proposed that the minimum energy configuration of repulsive particles in cylindrical geometry reproduces the spirals of vegetative phyllotaxis. ^{[14]} Recently, Nisoli et al. (2009) showed this to be true by building a “magnetic cactus” made of magnetic dipoles mounted on bearings vertically with a “stem”. ^{[15] }^{[16]}They demonstrated that these interacting particles can lead to novel dynamical phenomena beyond the yield of botany: a “dynamic phylotaxis” family of non-local topological soliton emerges in the non-linear regime of these systems, as well as Purely classical roton and maxon as well. in the spectrum of linear excitations.

The closed packing of the sphere produces a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the Fibonacci series and the golden section of classical geometry. ^{[17] }^{[18]}

**In art and architecture**

The phyllotaxis has been used as an inspiration for many sculptures and architectural designs. Akio Hizume has built and performed several bamboo towers based on the Fibonacci sequence that demonstrate phylotaxis. ^{[19]} Saleh Masumi has proposed a design for an apartment building where the balconies of the apartments project in a spiral arrangement around a central axis and each one does not directly shade the balcony of the apartment below.