

Four Main Types
 Counting spirals  In
a nutshell  Frequency of patterns
 Other patterns
Classification
The four main types of phyllotaxis usually recognized
are shown below. They can be detected and further classified by
the number of visible spirals (parastichies)
they display. All these patterns can be modeled by simple latticelike
mathematical structures. Spiral phyllotaxis is the most frequent.
Other patterns may exist that are not quite as regular, and seldom
mentioned by botanists.
Four Main Types of Phyllotaxis 




Distichous Phyllotaxis


Whorled Phyllotaxis












In distichous
phyllotaxis, leaves or other botanical elements grow one by one,
each at 180 degrees from the previous one.


In whorled
phyllotaxis, two or more (three in the example above) elements grow
at the same node on the stem. Elements in a node are evenly spread
around the stem, midway between those in the previous node. 


Spiral Phyllotaxis 

Multijugate
Phyllotaxis 











In spiral phyllotaxis, botanical elements grow one by one, each
at a constant divergence angle d from
the previous one. This is the most common pattern, and most often
the divergence angle d is close to the Golden Angle, which
is about 137.5 degrees. The latter case gives rise to Fibonacci
Phyllotaxis. 

In multijugate phyllotaxis, two or more botanical elements (two
in the example above) grow at the same node. Elements in a whorl
(group of elements at a node) are spread evenly around the stem
and each whorl is at a constant divergence angle d from
the previous one. Often, multijugate patterns look very similar
to spiral patterns and the only way to detect them is to count the
number of spirals visible in the pattern (see below).




Return to Top 


To further classify spiral
and multijugate patterns, one counts the number of visible spirals,
called parastichies, which join each element to
its nearest neighbors. These spirals normally come in two families,
yielding a pair of numbers, called parasitichy numbers.
If the parastichy numbers have no common divisor other than 1, the
pattern is a spiral phyllotaxis. If the parastichy numbers do have
a common divisor k, then the pattern is multijugate (more
precisely kjugate) and there are k elements at
each node. The notion of parastichy numbers can be extended to distichous
phyllotaxis with parastichy numbers (1,1) and whorled with parastichy
numbers (k,k).












This Aonium has parastichy numbers
(2, 3). Since 1 is the only common divisor of 2
and 3, this is a spiral pattern. Since spiral phyllotaxis can be
viewed as 1jugate, the notation 1(2,3) is also
used for this pattern.


This gymnocalycium has parastichy numbers
(10, 16), which have the common divisor k=2.
Hence this is a multijugate pattern (more precisely 2jugate). The
notation 2(5, 8) is also used to classify this
pattern. 


Classical Phyllotactic Patterns,
in a Nutshell 

The classification of a phyllotactic pattern (from
the 4 above kinds) is given by its pair of parastichy numbers (m,n).
If k is the greatest common divisor of m and n, and m=ki, n=kj,
the notation k(i, j) may be used. The number k is the number of
elements per whorl. When k =1, the configuration is called spiral,
when k>1 and i=j, it is whorled or kmerous
and when k>1, and i, j are distinct, it is called multijugate
(and more specifically kjugate).
As special cases, (1, 1) configurations are called distichous,
and 2whorled (or dimerous) patterns, 2(1,1) are called decussate.
The configuration is a Fibonacci
phyllotaxis when the parastichy numbers are successive
elements in the Fibonacci sequence.
To learn more about spiral phyllotaxis and parastichies,
visit the spiral lattice page.
Relative Frequency of Patterns 

Spiral phyllotaxis is the most common kind of phyllotactic
pattern. According to a survey by Hutchinson (see Meicenheimer
(1998)), in a sample of 650 dycotyledonous families, 50.6% showed
exclusively spiral or multijugate phyllotaxis, 14.7% showed exclusively
whorled phyllotaxis while 34.7% showed a mixture of these phyllotaxes.
According to a compilation by Jean
(1994) of many surveys of 650 species and 12500 specimens, among
plants showing spiral or multijugate phyllotaxis, about 92% showed
Fibonacci phyllotaxis (with
parastichy numbers successive elements in the Fibonacci sequence),
6% where bijugate 2(i,j) with i and j successive Fibonacci numbers,
1.5% were spiral phyllotaxis with Lukas parastichy numbers (Lukas
numbers form a Fibonaccilike sequence starting at 1, 3. Lukas gave
the Fibonacci numbers their name in the 19th Century).


Other Phyllotactic Patterns 
Return to Top 

One of the findings with the dynamical
model presented in this site is that there might be phyllotactic
patterns that show a regularity not of any of the above 4 kinds.
In these other patterns, that we have named periodic orbits,
the divergence angle is not constant but repeats periodically
a finite sequence of values. There is some evidence for the existence
of such patterns in plants. Note how the example shown below resembles
a regular lattice and seems to display families of parastichies
(mathematically, this pattern can be obtained by superimposing 2
lattices). However, in one family, these "parastichies"
are undulating. Nonetheless, this superficial similarity with lattice
patterns may be the reason why periodic orbits are usually not recognized
by botanists.



A periodic orbit of period 2 for the dynamical
model, with divergence angles 174.35 and 55.12 degrees that repeat
indefinitely. This is a (unrolled) cylindrical representation of
phyllotaxis. 






