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With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same
such as those of ''Gasteria'' or ''Aloe'' seedlings, and also in mature plants of related species such as ''Kumara plicatilis">Gasteria'' or ''Aloe">Gasteria">...
such as those of ''Gasteria'' or ''Aloe'' seedlings, and also in mature plants of related species such as ''Kumara plicatilis''.
In botany
Botany, also called plant science, is the branch of natural science and biology studying plants, especially Plant anatomy, their anatomy, Plant taxonomy, taxonomy, and Plant ecology, ecology. A botanist or plant scientist is a scientist who s ...
, phyllotaxis () or phyllotaxy is the arrangement of leaves
A leaf (: leaves) is a principal appendage of the stem of a vascular plant, usually borne laterally above ground and specialized for photosynthesis. Leaves are collectively called foliage, as in "autumn foliage", while the leaves, stem, ...
on a plant stem
A stem is one of two main structural axes of a vascular plant, the other being the root. It supports leaf, leaves, flowers and fruits, transports water and dissolved substances between the roots and the shoots in the xylem and phloem, engages ...
. Phyllotactic spirals form a distinctive class of patterns in nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, wave ...
.
Leaf arrangement
The basic arrangements of leaves on a stem are opposite and alternate (also known as spiral). Leaves may also be whorled if several leaves arise, or appear to arise, from the same level (at the samenode
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines ...
) on a stem.
With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines ...
), on opposite sides of the stem. An opposite leaf pair can be thought of as a whorl of two leaves.
With an alternate (spiral) pattern, each leaf arises at a different point (node) on the stem.
Distichous phyllotaxis, also called "two-ranked leaf arrangement" is a special case of either opposite or alternate leaf arrangement where the leaves on a stem are arranged in two vertical columns on opposite sides of the stem. Examples include various bulbous plants such as '' Boophone''. It also occurs in other plant habits
A habit (or wont, as a humorous and formal term) is a routine of behavior that is repeated regularly and tends to occur subconsciously.
A 1903 paper in the ''American Journal of Psychology'' defined a "habit, from the standpoint of psychology, ...
such as those of '' Gasteria">...
In an opposite pattern, if successive leaf pairs are 90 degrees apart, this habit is called decussate. It is common in members of the family Crassulaceae Decussate phyllotaxis also occurs in the Aizoaceae. In genera of the Aizoaceae, such as ''Lithops'' and '' Conophytum'', many species have just two fully developed leaves at a time, the older pair folding back and dying off to make room for the decussately oriented new pair as the plant grows.
If the arrangement is both distichous and decussate, it is called secondarily distichous.
The whorled arrangement is fairly unusual on plants except for those with particularly short internodes. Examples of trees with whorled phyllotaxis are '' Brabejum stellatifolium'' and the related genus ''Macadamia
''Macadamia'' is a genus of four species of trees in the flowering plant family Proteaceae. They are indigenous to Australia—specifically, northeastern New South Wales and central and southeastern Queensland. Two species of the genus are comm ...
''.
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 nonexistent. A basal whorl with a large number of leaves spread out in a circle is called a rosette.
Repeating spiral
The rotational angle from leaf to leaf in a repeating spiral can be represented by a fraction of afull rotation
The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to radians, 360 degrees or 400 gradians. As ...
around the stem.
Alternate distichous leaves will have an angle of 1/2 of a full rotation. In beech
Beech (genus ''Fagus'') is a genus of deciduous trees in the family Fagaceae, native to subtropical (accessory forest element) and temperate (as dominant element of Mesophyte, mesophytic forests) Eurasia and North America. There are 14 accepted ...
and hazel
Hazels are plants of the genus ''Corylus'' of deciduous trees and large shrubs native to the temperate Northern Hemisphere. The genus is usually placed in the birch family, Betulaceae,Germplasmgobills Information Network''Corylus''Rushforth, K ...
the angle is 1/3, in oak
An oak is a hardwood tree or shrub in the genus ''Quercus'' of the beech family. They have spirally arranged leaves, often with lobed edges, and a nut called an acorn, borne within a cup. The genus is widely distributed in the Northern Hemisp ...
and apricot
An apricot (, ) is a fruit, or the tree that bears the fruit, of several species in the genus ''Prunus''.
Usually an apricot is from the species '' P. armeniaca'', but the fruits of the other species in ''Prunus'' sect. ''Armeniaca'' are also ...
it is 2/5, in sunflowers, poplar, and pear
Pears are fruits produced and consumed around the world, growing on a tree and harvested in late summer into mid-autumn. The pear tree and shrub are a species of genus ''Pyrus'' , in the Family (biology), family Rosaceae, bearing the Pome, po ...
, it is 3/8, and in willow
Willows, also called sallows and osiers, of the genus ''Salix'', comprise around 350 species (plus numerous hybrids) of typically deciduous trees and shrubs, found primarily on moist soils in cold and temperate regions.
Most species are known ...
and almond
The almond (''Prunus amygdalus'', Synonym (taxonomy)#Botany, syn. ''Prunus dulcis'') is a species of tree from the genus ''Prunus''. Along with the peach, it is classified in the subgenus ''Amygdalus'', distinguished from the other subgenera ...
the angle is 5/13. The numerator and denominator normally consist of a Fibonacci number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
and its second successor. The number of leaves is sometimes called rank, in the case of simple Fibonacci ratios, because the leaves line up in vertical rows. With larger Fibonacci pairs, the pattern becomes complex and non-repeating. This tends to occur with a basal configuration. Examples can be found in composite flower
Flowers, also known as blooms and blossoms, are the reproductive structures of flowering plants ( angiosperms). Typically, they are structured in four circular levels, called whorls, around the end of a stalk. These whorls include: calyx, m ...
s and seed
In botany, a seed is a plant structure containing an embryo and stored nutrients in a protective coat called a ''testa''. More generally, the term "seed" means anything that can be Sowing, sown, which may include seed and husk or tuber. Seeds ...
heads. The most famous example is the sunflower
The common sunflower (''Helianthus annuus'') is a species of large annual forb of the daisy family Asteraceae. The common sunflower is harvested for its edible oily seeds, which are often eaten as a snack food. They are also used in the pr ...
head. This phyllotactic pattern creates an optical effect 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 mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...
. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.
Determination
The pattern of leaves on a plant is ultimately controlled by the accumulation of the plant hormoneauxin
Auxins (plural of auxin ) are a class of plant hormones (or plant-growth regulators) with some morphogen-like characteristics. Auxins play a cardinal role in coordination of many growth and behavioral processes in plant life cycles and are essent ...
in certain areas of the meristem
In cell biology, the meristem is a structure composed of specialized tissue found in plants, consisting of stem cells, known as meristematic cells, which are undifferentiated cells capable of continuous cellular division. These meristematic c ...
. Leaves become initiated in localized areas where auxin concentration is higher. When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from area on the meristem
In cell biology, the meristem is a structure composed of specialized tissue found in plants, consisting of stem cells, known as meristematic cells, which are undifferentiated cells capable of continuous cellular division. These meristematic c ...
close to where the leaf was initiated. 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
In cell biology, the meristem is a structure composed of specialized tissue found in plants, consisting of stem cells, known as meristematic cells, which are undifferentiated cells capable of continuous cellular division. These meristematic c ...
atic topography
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the landforms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary sci ...
.
History
Some early scientists—notablyLeonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
—made observations of the spiral arrangements of plants. In 1754, Charles Bonnet observed that the spiral phyllotaxis of plants were frequently expressed in both clockwise
Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...
and counter-clockwise golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
series. Mathematical observations of phyllotaxis followed with Karl Friedrich Schimper and his friend Alexander Braun
Alexander Carl Heinrich Braun (10 May 1805 – 29 March 1877) was a German botanist from Regensburg, Bavaria. His research centered on the morphology of plants and was a very influential teacher who worked as a professor of botany at the univers ...
's 1830 and 1830 work, respectively; Auguste Bravais and his brother Louis connected phyllotaxis ratios to the Fibonacci sequence
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
in 1837.
Insight into the mechanism had to wait until Wilhelm Hofmeister proposed a model in 1868. A primordium
A primordium (; : primordia; synonym: anlage), in embryology, is an Organ (anatomy), organ or tissue in its earliest recognizable stage of development. Cell (biology), Cells of the primordium are called primordial cells. A primordium is the simp ...
, the nascent leaf, forms at the least crowded part of the shoot meristem
In cell biology, the meristem is a structure composed of specialized tissue found in plants, consisting of stem cells, known as meristematic cells, which are undifferentiated cells capable of continuous cellular division. These meristematic c ...
. The golden angle
In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two Arc (geometry), arcs such that the ratio of the length of the smaller arc to the ...
between successive leaves is the blind result of this jostling. Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge. The generative spiral is a consequence of the same process that produces the clockwise and counter-clockwise spirals that emerge in densely packed plant structures, such as ''Protea
''Protea'' () is a genus of South African flowering plants, also called sugarbushes (Afrikaans: ''suikerbos''). It is the type genus of the Proteaceae family.
About 92% of the species occur only in the Cape Floristic Region, a narrow belt of mo ...
'' flower disks or pinecone scales.
In modern times, researchers such as Mary Snow and George Snow 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 chemical gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
s among the primordia or purely mechanical forces. Lucas number
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence ar ...
s rather than Fibonacci numbers have been observed in a few plants and occasionally, the leaf positioning appears to be random.
Mathematics
Physical models of phyllotaxis date back to Airy's experiment of packing hard spheres. Gerrit van Iterson diagrammed grids imagined on a cylinder (rhombic lattices). Douady et al. showed that phyllotactic patterns emerge as self-organizing processes in dynamic systems. In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis. More recently, Nisoli et al. (2009) showed that to be true by constructing a "magnetic cactus" made of magnetic dipoles mounted on bearings stacked along a "stem". They demonstrated that these interacting particles can access novel dynamical phenomena beyond what botany yields: a "dynamical phyllotaxis" family of non local topologicalsolitons
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such locali ...
emerge in the nonlinear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
regime of these systems, as well as purely classical rotons and maxons in the spectrum of linear excitations.
Close packing of spheres generates a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the Fibonacci series and the golden section
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
of classical geometry.
In art and architecture
Phyllotaxis has been used as an inspiration for a number of sculptures and architectural designs. Akio Hizume has built and exhibited several bamboo towers based on the Fibonacci sequence which exhibit phyllotaxis. Saleh Masoumi has proposed a design for an apartment building in which the apartmentbalconies
A balcony (from , "scaffold") is a platform projecting from the wall of a building, supported by columns or console brackets, and enclosed with a balustrade, usually above the ground floor. They are commonly found on multi-level houses, apartme ...
project in a spiral arrangement around a central axis and none shade the balcony of the apartment directly beneath.
See also
* Anisophylly, a type of leaf size difference on horizontal shoots * Available space theory *Decussation
Decussation is used in biological contexts to describe a crossing (due to the shape of the Roman numeral for ten, an uppercase 'X' (), ). In Latin anatomical terms, the form is used, e.g. .
Similarly, the anatomical term Chiasm (anatomy), chi ...
* Fermat's spiral
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance f ...
* L-system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some ...
* '' Orixa japonica''
* Parastichy
* Plastochron
* Repulsion theory
* Three-gap theorem
* Sphere packing in a cylinder
References
{{Patterns in nature Plant morphology Leaves