Phyllotaxy
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In
botany Botany, also called plant science (or plant sciences), plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "bot ...
, phyllotaxis () or phyllotaxy is the arrangement of leaves on a
plant stem A stem is one of two main structural axes of a vascular plant, the other being the root. It supports leaves, flowers and fruits, transports water and dissolved substances between the roots and the shoots in the xylem and phloem, stores nutrie ...
. 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 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, line ...
) 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, line ...
), 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 Bulb or Bulbs may refer to: Common uses * Bulb, a food-storage structure within some plants ** Ornamental bulb, a kind of perennial plant * Light bulb, an electric lamp Maritime * Bulb keel, a type of keel * Bulbous bow, a part that sticks ou ...
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.
such as those of '' Gasteria'' or '' Aloe'' seedlings, and also in mature plants of related species such as '' Kumara plicatilis''. 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 The Crassulaceae (from Latin ''crassus'', thick), also known as the stonecrop family or the orpine family, are a diverse family of dicotyledon flowering plants characterized by succulent leaves and a unique form of photosynthesis, known as Cr ...
Decussate phyllotaxis also occurs in the
Aizoaceae The Aizoaceae, or fig-marigold family, is a large family of dicotyledonous flowering plants containing 135 genera and about 1800 species. They are commonly known as ice plants or carpet weeds. They are often called vygies in South Africa and New ...
. In genera of the Aizoaceae, such as ''
Lithops ''Lithops'' is a genus of succulent plants in the ice plant family, Aizoaceae. Members of the genus are native to southern Africa. The name is derived from the Ancient Greek words () 'stone' and () 'face', referring to the stone-like appearanc ...
'' 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. 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, native to northeastern New South Wales and central and southeastern Queensland specifically. Two species of the gen ...
''. A whorl can occur as a
basal Basal or basilar is a term meaning ''base'', ''bottom'', or ''minimum''. Science * Basal (anatomy), an anatomical term of location for features associated with the base of an organism or structure * Basal (medicine), a minimal level that is nec ...
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 a full rotation around the stem. Alternate distichous leaves will have an angle of 1/2 of a full rotation. In
beech Beech (''Fagus'') is a genus of deciduous trees in the family Fagaceae, native to temperate Europe, Asia, and North America. Recent classifications recognize 10 to 13 species in two distinct subgenera, ''Engleriana'' and ''Fagus''. The ''Engl ...
and
hazel The hazel (''Corylus'') is a genus 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. (1999 ...
the angle is 1/3, in oak 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 al ...
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 the Northern Hemisphere in late summer into October. The pear tree and shrub are a species of genus ''Pyrus'' , in the family Rosaceae, bearing the p ...
, it is 3/8, and in
willow Willows, also called sallows and osiers, from the genus ''Salix'', comprise around 400 speciesMabberley, D.J. 1997. The Plant Book, Cambridge University Press #2: Cambridge. of typically deciduous trees and shrubs, found primarily on moist ...
and almond the angle is 5/13. The numerator and denominator normally consist of a
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
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 A flower, sometimes known as a bloom or blossom, is the reproductive structure found in flowering plants (plants of the division Angiospermae). The biological function of a flower is to facilitate reproduction, usually by providing a mechanism ...
s and
seed A seed is an embryonic plant enclosed in a protective outer covering, along with a food reserve. The formation of the seed is a part of the process of reproduction in seed plants, the spermatophytes, including the gymnosperm and angiosper ...
heads. The most famous example is the sunflower 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 numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. 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 local depletion of the plant hormone auxin in certain areas of the
meristem The meristem is a type of tissue found in plants. It consists of undifferentiated cells (meristematic cells) capable of cell division. Cells in the meristem can develop into all the other tissues and organs that occur in plants. These cells conti ...
. Leaves become initiated in localized areas where auxin is absent. When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from another area on the
meristem The meristem is a type of tissue found in plants. It consists of undifferentiated cells (meristematic cells) capable of cell division. Cells in the meristem can develop into all the other tissues and organs that occur in plants. These cells conti ...
where a new leaf is to be 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 The meristem is a type of tissue found in plants. It consists of undifferentiated cells (meristematic cells) capable of cell division. Cells in the meristem can develop into all the other tissues and organs that occur in plants. These cells conti ...
atic
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary s ...
.


History

Some early scientists—notably
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially re ...
—made observations of the spiral arrangements of plants. In 1754,
Charles Bonnet Charles Bonnet (; 13 March 1720 – 20 May 1793) was a Genevan naturalist and philosophical writer. He is responsible for coining the term ''phyllotaxis'' to describe the arrangement of leaves on a plant. He was among the first to notice parth ...
observed that the spiral phyllotaxis of plants were frequently expressed in both
clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
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. Biography He studied botany in Heidelberg, Paris and Munich. In 1833 he began teac ...
's 1830 and 1830 work, respectively;
Auguste Bravais Auguste Bravais (; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied ...
and his brother Louis connected phyllotaxis ratios to the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
in 1837. Insight into the mechanism had to wait until Wilhelm Hofmeister proposed a model in 1868. A
primordium A primordium (; plural: primordia; synonym: anlage) in embryology, is an organ or tissue in its earliest recognizable stage of development. Cells of the primordium are called primordial cells. A primordium is the simplest set of cells capable ...
, the nascent leaf, forms at the least crowded part of the shoot
meristem The meristem is a type of tissue found in plants. It consists of undifferentiated cells (meristematic cells) capable of cell division. Cells in the meristem can develop into all the other tissues and organs that occur in plants. These cells conti ...
. 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 arcs such that the ratio of the length of the smaller arc to the length of the l ...
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''). Etymology The genus ''Protea'' was named in 1735 by Carl Linnaeus, possibly after the Greek god Proteus, who could change his form a ...
'' 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 of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
s among the primordia or purely mechanical forces. Lucas 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 Airy may refer to: * Sir George Biddell Airy (1801–1892), British Astronomer Royal from 1835 to 1881, for whom the following features, phenomena, and theories are named: ** Airy (lunar crater) ** Airy (Martian crater) ** Airy-0, a smaller crater ...
'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 topological
solitons In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
emerge in the
nonlinear In mathematics and science, a nonlinear 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, mathematicians, and many other ...
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 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 apartment
balconies A balcony (from it, balcone, "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. Types The traditional Maltese balcony is ...
project in a spiral arrangement around a central axis and none shade the balcony of the apartment directly beneath.


See also

*
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 chiasma is named a ...
* Fermat's spiral *
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 som ...
* '' Orixa japonica'' * Parastichy * Plastochron * Three-gap theorem *
Sphere packing in a cylinder Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude ...


References

{{Patterns in nature Plant morphology Leaves