a method for generating phyllotaxis

A Method for Generating Phyllotaxis

over Surfaces of Revolution

Jon McCormackCentre for Electronic Media Art

School of Computer Science and Software Engineering

Monash University, Clayton Australia

[email protected]

http://www.csse.monash.edu.au/~jonmc

15 November 2003

Abstract

This paper describes a method for generating phyllotaxis on surfaces of revo-

lution. It extends the work done by previous authors in using Lindenmayer

systems (L-systems) to model phyllotaxis in plants for the purposes of generat-

ing realistic visual models. The method described permits placement of small

elements in phyllotactic patterns over surfaces of revolution. An area-based

phyllotactic model is described which gives an analytic equation relating the

elements to be placed and the area they cover on the surface. From this rela-

tionship, it is possible to calculate where on the surface each element should be

placed, even if the individual elements vary in size. Geometric models can be

created using a turtle interpretation of parametric modules generated by an L-

system. Examples are shown for placement on spheres, oblate and prolate

spheroids. Example images of plants, generated using this model, are also il-

lustrated.

2 A Method for Generating Phyllotaxis: Jon McCormack

The power of the golden section to create harmony arises from itsunique capacity to unite different parts of the whole so that eachpreserves its own identity, and yet blends into the greater patternof a single whole.

— György Doczi, The Power of Limits (Doczi 1981)

1 Introduction

The range and diversity of natural structure and form is unparalleled in our

experience of the world. The plant kingdom has numerous examples of novel

architectural structure that have long fascinated artists, designers, biologists,

architects and engineers (Thompson 1942; Stevens 1974; Doczi 1981; Bloss-

feldt and Sachsse 1994). Many have been captured by the phenomenon of

phyllotaxis as an optimal packing scheme observed in many different species

of plant — a scheme that connects nature and mathematics.

This paper describes a method for generating phyllotaxis over surfaces of

revolution, allowing the generation of complex geometric models of organisms

that exhibit such patterns. The method described is suitable for integration

into modelling systems that use parametric Lindenmayer systems (L-systems)

to model plants (Prusinkiewicz and Hanan 1990; Hanan 1992). It has the ad-

vantage over previous methods in that it generates an analytic solution that

can be computed within a parametric L-system grammar itself, without the

need for external software.

1.1 Phyllotaxis

Phyllotaxis is the regular arrangement of plant lateral organs according to an

optimal packing scheme, based on the golden ratio. It is an interesting “self-

organizing” property of plant morphogenesis, demonstrated experimentally

1: Introduction 3

(Douady and Couder 1992) and computationally (Douady and Couder 1996).

Examples include leaves on a stem, cone scales on a cone axis, and florets or

achenes in a composite flower head. Phyllotaxis has been extensively studied

in the literature from a variety of perspectives that include art, architecture,

mathematics, biology and computer graphics (a recent overview can be found

in (Jean and Barabe 1998), and Ball gives an overview from a developmental

morphogenesis perspective (Ball 2001, Chapter 4).

Helmut Vogel was one of the first to develop a planar mathematical model

of phyllotaxis based on florets of equal area (Vogel 1979). The “numbers of

spirals that can be traced through a phyllotactic pattern are predominantly

the integers of the Fibonacci sequence” (Erickson 1983, page 54). Phyllotactic

patterns arrange the lateral organs in ranks or parastichies. In geometric

modelling of phyllotaxis using L-systems, lateral organs are typically distri-

buted over cylindrical or planar surfaces (Prusinkiewicz and Lindenmayer

1990, Chapter 4). Figure 1-1 shows examples of these placements and the L-

systems that generated them.

Planar Phyllotaxis:

†

#define a 137.5#define N 400

w : A(0)p1 : A(n) : n < 400 Æ +(137.5)

[ f( n ) sphere ] A(n + 1)

Cylindrical Phyllotaxis:

†

#define a 137.5#define h 0.1#define r 5.0#define N 200

w : A(0)p1 : n < N Æ [&(90) f(r) sphere]

f(h) /(a) A(n + 1)

Figure 1-1: Planar and Cylindrical phyllotaxis using L-systems. See section 2.1.2 for ex-planation of turtle commands.


An extension to the planer model is to change the element’s altitude, size and

orientation as a function of n. Changing the altitude of the elements as they

are placed gives an approximation of placement over a cone or section of a

cone.

While these two models have been successful in modelling a number of

plant species, there exist many examples of phyllotaxis that are not described

by this model (figure 1-2 for example).

Isopogon formosus Isopogon petiolaris

Figure 1-2: Plants from Isopogon exhibiting phyllotaxis over sections of approximate pro-late spheroids (from (Greig 1999))

1.2 Related Work

Fowler, Prusinkiewicz and Battjes proposed a collision-based model, distri-

buting primordia (undeveloped organs that eventually grow into florets) on a

surface of revolution whose generating curve is defined by one or more Bézier

curves (Fowler et al. 1992). Their model is “descriptive and explanatory”, cal-

culating the placement of elements using a collision-based method. In their

model the radial angle, f is incremented by

†

2pj -2 radians, where

†

j is the

golden ratio

†

1+ 5( ) 2. The placement along the generating curve at this

angle is determined by finding the minimum distance necessary to avoid colli-

sion with any previously placed element. In practice, this is computed using a

binary search technique. Due to the collision-based component of the model,

the method cannot directly be implemented with standard DOL-systems.

1: Introduction 5

Lintermann and Deussen describe a technique for phyllotaxis over a

sphere (Lintermann and Deussen 1999). They use an incremental approach

based on area approximation, similar to that presented here. However, their

results are limited to spheres or sub-sections of spheres only.

Prusinkiewicz et. al. present a method which operates on arbitrary sur-

faces of revolution (Prusinkiewicz et al. 2001), based on the model developed

by Ridley (Ridley 1986). This method uses an arc-length parameterisation of

planar curves, which then generate a surface of revolution. They use an in-

cremental method to approximate the integration of the surface area as ele-

ments are placed over the surface. The technique is implemented using a

derivation of Chomsky grammars and the incorporation of C-like1 code state-

ments into the modelling language itself.

Fleischer et. al. developed a cellular texture model that could “grow” cel-

lular elements over pre-defined surfaces (Fleischer et al. 1995). While not

specifically designed for creating phyllotactic patterns, the system gave each

element (cell) a simple program (time-varying first order differential equa-

tion) that could control its placement over the surface based on (for example)

simulation of chemical reaction-diffusion over the surface. The method could

be considered a generalization of the collision-based model of Fowler, Prus-

inkiewicz and Battjes, offering additional capabilities such as movement, ad-

hesion and changes in size due to cell-cell interaction.

The concept of simple cells running independent programs can be

achieved using a developmental cellular model, based on L-systems, devel-

oped by the author. Figure 1-3, Homage to Kurt Fleischer, replicates his

“thorny spheres” model, using this cellular developmental model (described in

(McCormack 2003)) The figure uses no predefined geometry or external sur-

faces; rather the surfaces are grown using a developmental L-system with

phyllotactic placement.

1 As in the C programming language (Kernighan and Ritchie 1988).


Figure 1-3: Varying Thorns in the style of Kurt Fleischer et. al. (c.f. (Fleischer et al. 1995,page 247)) This image was created using the developmental cellular model detailed in(McCormack 2003). The placement of the thorns is controlled using the area-based phyl-lotactic method described here. The thorns themselves are grown using the generalizedcylinder extensions. Simulation of reaction-diffusion-like equations determine the size,growth and direction of the bumps and thorns. Note the continuously varying thorn heightand curvature on the spheres.

1.3 Features of the Model

The main feature of the model presented here is that it gives an analytic solu-

tion to the placement of elements over a surface of revolution. Hence, it can

be incorporated into any parametric L-system description without the need

for additional software or standalone components, which must be then inte-

grated into the final model. Many of the plant organs and surfaces on which

they develop can be described within the L-system using the generalized cyl-

inder extensions introduced in (McCormack 2003). Hence the L-system speci-

fication is sufficient to define complete and complex three-dimensional geo-

metric models without the need for specialized external surface modelling

programs or collision-detection systems required in other approaches.

2: The Area-Based Phyllotactic Model 7

2 The Area-Based Phyllotactic Model

This section describes the area-based phyllotactic model in detail. Examples

over a number of different surfaces are described along with some practical

examples of the application of this technique.

2.1.1 Surfaces of Revolution

A surface of revolution is generated by rotating a two-dimensional curveabout an axis over the interval

†

0,2p[ ). For this discussion, it is assumed the

rotation is always about the z-axis, giving the generated surface a radial

symmetry in the xy plane (known as azimuthal symmetry). The generating

curve is assumed to be a continuous, single-valued function

†

fC : ¬ Æ ¬ , writ-

ten

†

fC z( ) defined over the domain

†

z Œ [zmin , zmax ] . For example, the generat-

ing curve

†

r2 - z2 over domain

†

z Œ [-r, r] generates a sphere of radius r.

Figure 2-1: Generating curve for a sphere.

We are interested in placing geometric elements over some section of the

generated surface, S, according to the golden section. The elements will be

placed using turtle commands generated by the production of L-system

strings.


Due to the azimuthal symmetry of the generated surface of revolution, the

area of placement can be defined in terms of the upper and lower limits

†

[zl , zu ] of the generating curve (see figure 2-1). The surface area of this gen-

erated surface,

†

AS is defined:

†

AS = 2p fC (z) 1+ fC¢(z)

Ê

Ë Á

ˆ

¯ ˜ 2

zl

zu

Ú dz (2.1)

Let the projected area2 of element i to be placed on the surface be

†

Aei, then

the function

†

AE (i) = Aek

k=1

i

Â (2.2)

gives the total area required for i elements. There is no requirement for each

element to have an equal projected area, however if this is the case, equation

(2.2) becomes

†

AE (i) = iAe (2.3)

where

†

Ae is the projected area of the individual element. Using equation (2.1)

we can calculate the maximum value of i needed to fill the area required.

Due to the azimuthal symmetry, we can paramatise the surface, S, over the

axis of rotation and azimuth distance

†

S = S(f, h) : 0 £ f < 2p, zl £ h £ zu (2.4)

Note that this is different from the conventional parametric form for a sur-

face. This is done for convenience in the calculations that follow. Figure 2-2

shows the parameters for a sphere.

2 The projected area can be calculated by orientating the element so that the xy plane corres-ponds to the tangential plane of the surface on which the element is to be placed. The projectedarea is then defined

†

dxdyR

ÚÚ , where R is the region over which the element is defined.


Figure 2-2: Sphere generated as a surface of revolution.

As each element, i, is placed on the surface, the parameters

†

fi and hi are re-

quired to determine the location of the element on the surface. The radial

angle,

†

fi is distributed

†

fi = i df

= i2p j - 1( )

j

= 3 - 5( )p i

(2.5)

where

†

j is the golden mean (

†

ª 1.61803 ). The numerical value of

†

fi equates to

approximately 137.508°. Using equation (2.1), the surface area of the surface

of revolution at

†

hi can be written:

†

AS (hi ) = 2p fC (z) 1+ fC¢(z)

Ê

Ë Á

ˆ

¯ ˜ 2

zu - hi

zu

Ú dz (2.6)

To ensure correct placement, it is required that

†

AE (i) ≡ AS (hi ) (2.7)

hence

†

hi = AS-1 AE (i)( ) (2.8)


Depending on the nature of the generating curve, equation (2.8) may not have

an algebraic solution, as finding an inverse function may be impossible. In

this case, numerical methods may be employed. Alternatively, another solu-

tion is to use an approximating polynomial of limited order (< 4), to which an

inverse can usually be found.

2.1.2 Turtle Interpretation

It is common to interpret the derivation strings generated by L-systems using

a turtle interpretation, named after the original turtle graphics concept of a

mechanical turtle, which moves over a two-dimensional drawing surface

(Abelson and DiSessa 1982). The turtle responds to commands such as “move

forward”, “turn left”, “turn right”, and so on. The turtle has a drawing pen

affixed to its base that can be raised and lowered (“pen up” and “pen down”

commands), giving it the ability to draw line images.

For three-dimensional drawing and construction of geometry, the turtle

concept is abstracted into three dimensions. The turtle maintains a current

state, consisting of a co-ordinate reference frame defined by a global position

vector and three orthogonal unit vectors: representing heading, up and left

directions of the turtle (figure 2-3). Other turtle instructions are used to in-

stantiate geometric primitives. The geometry generated can be output to an

image rendering system or real-time hardware display. Turtle commands also

accept real-valued parameters that control quantitative effects of the com-

mand.

Figure 2-3: Turtle co-ordinate reference frame has a position (

†

t ) and an orthogonal axissystem (

†

r H ,

†

r L and

†

r U vectors). Commands to manipulate this reference frame’s

orientation are also shown.


Some basic turtle commands and their interpretation are shown in table 2–1.

Turtle Position and Orientation modifying commands:

CommandParameterDefaults Description

f(l) l = l* Move forward by l units in the direction of

†

r H . The

new turtle position

†

¢ t = ¢ t x , ¢ t y , ¢ t z( ) is calculated

†

¢ t = t + lr H .

+(q)

†

q = d * Turn left by angle q degrees.

–(q)

†

q = d Turn right by q degrees (equivalent to +(-q))

&(q)

†

q = d Pitch down by q degrees.

^(q)

†

q = d Pitch up by q degrees.

\(q)

†

q = d Roll left by q degrees.

/(q)

†

q = d Roll right by q degrees.

|(n)

†

n = 1 Turn around n times. Equivalent to +(180n).

Table 2–1: Basic turtle commands affecting turtle position and orientation.

The position to place the element on the surface, S, can be expressed in terms

of the turtle commands:

†

/(fi)f(zu - hi) (̂90) f(fC (hi )) (2.9)

2.1.3 Orientation of the Element

The positional information described above provides a location to place ele-

ments on the surface of revolution. In addition, a specific orientation may also

be required. If this is the case, the element needs to be aligned to the surface

normal of the surface of revolution,

†

nS .

For surfaces with a parametric representation, the normal is the orthogo-

nal vector to the tangent plane. That is, for a surface

†

r(u, v) the normal vector

is

†

nr =∂r∂u

¥∂r∂v

= ru ¥ rv

(2.10)

* Globally defined values that can be changed by the user. Defaults to l = 1, and d = 90° (p/2).


In the case of a surface of revolution, the normal vector can be found by dif-

ferentiating the generating curve and then rotating this vector to the radial

location in

†

f . In practice, to align the element to the surface normal requires

orienting the turtle’s co-ordinate reference frame to the triplet

†

ru rv nr[ ] .

Following the turtle commands specified in (2.9), the turtle requires a rota-

tion,

†

r , based on the normal to the generating curve:

†

r = tan-1 fC¢(h)

Ê

Ë Á

ˆ

¯ ˜ (2.11)

Assuming the turtle is at the correct position on the surface with the heading

vector aligned with the z axis, a turtle command of ^(r) will align the turtle

to the surface normal.

3 Examples

This section presents examples of the method described in the previous sec-

tion on spheres, sub-spheres, and spheroids.

3.1 Example: Sphere

A sphere of radius r can be created by revolving the curve

†

fC (z) = fsphere (z) = r2 - z2 , z Œ -r, r[ ](3.1)

about the z axis over

†

[0, 2p) (figure 2-2). By equation (3.6) the area of the

surface defined from the boundaries indicated in figure 2-2 is

†

Asphere (hi ) = 2p r2 - z2 1+ -z

r2 - z2

Ê

Ë Á

ˆ

¯ ˜

2

zu - hi

zu

Ú dz

= 2p r dzzu - hi

zu

Ú= 2prhi

(3.2)

Using the equivalence relation of equation (3.7)

†

hi =AE (i)2pr (3.3)

If the elements all have the same projected area then

3: Examples 13

†

hi =iAe

2pr (3.4)

†

hi can be expressed in terms of

†

qi, the azimuth angle between the positive z

axis and the vector formed from the centre of the sphere to the position of

element i on the sphere

†

qi = cos-1 cosqu -iAe

2pr2

Ê

Ë Á

ˆ

¯ ˜ (3.5)

where

†

qu is the value of

†

q at

†

zu (i.e.

†

h0). Note that the angle

†

q is equivalent to

a parametric parameter when the sphere is represented parametrically, i.e.,

†

S(q,f ).

A B C

Figure 3-1: Phyllotaxis of various geometric elements over the surface of a sphere. Ele-ments placed are a sphere (A), Cone (B) and hexagon (C).

Figure 3-1 shows some examples. For the examples shown in this figure, each

element is of constant projected area (i.e. all elements are the same size).

Figure 3-1C is reminiscent of some of the skeletal forms of Radiolaria, studied

by Thompson (Thompson 1961, pages 151–159)3.

3 Thompson was motivated in his study by the work of Ernst Haeckel, who drew Auloniahexagona using hexagons. Thompson correctly stated that “no system of hexagons can enclosespace” (Thompson 1961, page 157). This was also acknowledged by Haeckel, who observedthat a few square or pentagonal facets are found among the hexagons. Twelve pentagons areneeded to correctly form a closed space, as was latter demonstrated in the architectural con-structions of R. Buckminster Fuller.


Figure 3-2: Elements arranged on a sphere from

†

p6

£ q £p2

.

Figure 3-2 shows examples where

†

zl < r and

†

zu < r , thus covering only a

section of the entire sphere.

Elements of constant area do not correctly simulate the growth of plant

elements such as florets (Jean 1982). Figure 3-3 shows an example where the

elements placed on the sphere do not have a constant area. In the case of this

figure, the element has an exponential growth function, whereby the radius of

each element (a sphere in the case of figure 3-3A) grows according to the

function

†

r = c1 1- e- c2k( ) (3.6)

where

†

c1 and c2 are constants. This function is designed to simulate the

growth of florets from the centre of a floral head, the youngest elements ap-

pearing at the apex of the sphere. For elements with a circular projected area

the value of

†

Aekfrom equation (3.2) is given by

†

Aek= pr2

= c1p 1- e- c2k( )2 (3.7)

The figure shows a graph of this function and the cumulative area function

†

AE (i). The graph also shows the value of this function for elements of con-

stant area for comparison.

3: Examples 15

A. Spheres of increasing size placedover a larger sphere. The growth of theindividual sphere’s radius is as speci-fied in equation (3.18).

B. Radius (r) and projected area (Aek) of individ-

ual elements increases as they are placed onthe surface.

C. Conical elements of increasing sizeplaced over the surface of a sphere.

D. Cumulative area of elements as their numberincreases, shown for elements of constantprojected area (dashed line) and increasingprojected area (solid line) as shown in C.

Figure 3-3: Elements of differing size placed over a sphere.


Figure 3-4: Illustrates the growth and placement of the elements. A ten step sequence ofshades of grey (darkest to lightest) shows the relative age of elements as they are placedon the surface.

3.2 Example: Oblate and Prolate Spheroids

Using the methodology described in section 2.1.1, as for the sphere we can

define a surface of revolution that describes a spheroid. Two types of sphe-

roids are described here, oblate or “squashed” and prolate or “stretched”

spheroids (figure 3-5). Both these shapes can be described by rotating an el-

lipse about its minor (for oblate) or major (for prolate) axis. The motivation

for using such shapes is the observed profiles of plant organs that were previ-

ously approximated by cylinders (c.f. figure 1-2). The cylindrical approxima-

tion leads to inconsistencies at the base and top.

Figure 3-5: Oblate and Prolate spheroids.

3: Examples 17

The implicit form of these spheroids is

†

x2 + y2

a2+

z2

c2- 1 = 0

(3.8)

where a is the equatorial radius and c is the polar radius. For an oblate

spheroid

†

a > c , for a prolate spheroid

†

a < c .

Proceeding as for the case of a sphere (section 3), the spheroid can be

formed by rotating the elliptical curve

†

fC (z) = fellipse (z) = a 1-zc

Ê

Ë Á

ˆ

¯ ˜

2

(3.9)

about the z axis over

†

[0, 2p) (figure 2-2). By equation (3.6) the area of the

surface defined from the boundaries indicated in figure 2-2 is

†

Aspheroid (hi ) = 2p fellipse (z) 1+ ¢ f ellipse (z)( )2

zu - hi

zu

Ú dz

= 2pa 1+a - c( ) a + c( )z2

c 4dz

zu - hi

zu

Ú(3.10)

Evaluating this definite integral gives:

†

Aspheroid (hi ) =

pa

zu 1+ zu2 a2 - c2

c 4

Ê

Ë Á

ˆ

¯ ˜ - zu - hi( ) 1+ zu - hi( )2 a2 - c2

c 4

Ê

Ë Á

ˆ

¯ ˜ +

c2 sinh-1 zu

a2 - c2

c2

Ê

Ë Á Á

ˆ

¯ ˜ ˜ - sinh-1 zu - hi( ) a2 - c2

c2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

Ê

Ë

Á Á

ˆ

¯

˜ ˜

a2 - c2

Ê

Ë

Á Á Á Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜ ˜ ˜ ˜

(3.11)

In the case of a prolate spheroid,

†

a < c and so

†

a2 - c2 will have a complex

root. However, due to the nature of the hyperbolic sine function, the equation

evaluates to a real solution that does not involve complex numbers. Figure

3-6 shows a graph of this equation for both oblate and prolate spheroids with

equatorial to polar radius ratios of 0.75:1 and 1.25:1 respectively. Finding the

inverse function needed to satisfy equation (3.8) is not possible analytically in

the general case. However, it is possible to approximate the function using a

polynomial, in which case an inverse can be found. Therefore:

†

hi = Pspheroid-1 AE i( )( ) (3.12)


where

†

Pspheroid ª Aspheroid is the polynomial approximation to equation (3.22). A

specialized function that returns the value of

†

hi when supplied the param-

eters

†

a, c, zu and AE i( ), can be made available to the generating L-system to

avoid the complex parameter manipulation necessary to incorporate this so-

lution into the L-system rules.

Figure 3-6: Surface area of oblate (solid line) and prolate (dashed line) spheroids as a func-tion of distance from the polar radius along the z axis.

Figure 3-7 shows simple conical elements placed over oblate and prolate

spheroids using the method described here.

Oblate Spheroid Prolate Spheroid

Figure 3-7: Oblate and Prolate spheroids with phyllotactic placement of elements overtheir surfaces.

3: Examples 19

3.3 Plant Examples

The use of the spheroids discussed in the previous section finds application in

the realistic visual modelling of a variety of different plants. For example,

figure 3-8 and figure 3-9 illustrate some possibilities. Figure 3-8 is typical of

Figure 3-8: Plant model created using the method described in this paper. No external ge-ometry was used to create this model. Textures were defined procedurally or sourcedfrom real specimens. Variation in the length, size and placement is achieved using sto-chastic functions. The image on the right shows a close-up revealing the model detail.

Figure 3-9: Model created by placing thorn-like elements over a prolate spheroid. The rateof curvature varies as the elements are placed using a Hermite interpolative function.


many members of the Myrtaceae family. The cluster of stamens is modelled

using generalized cylinders, with length, size and growth direction subject to

modulation using a normal (Gaussian) distribution.

4 Conclusions

A model for generative phyllotaxis over surfaces of revolution has been de-

scribed and application shown to the modelling of plants using L-systems. The

method differs from other techniques described in the literature in that it can

be directly incorporated into plant models described by parametric D0L-

systems, whereas other techniques, while more flexible from an interactive

design perspective, require additional software such as curve editors.

There are two different approaches to the problem of developing a general

model of phyllotaxis — the analytical mathematical approach such as that ad-

opted in this paper where the basic architecture is taken as an empirical

given and a model which creates that architecture specified. This certainly

gives good and efficient geometric models, but it does lack explanatory power

in terms of why such patterns form. Here the developmental, causal and

physical models provide better explanatory information (Douady and Couder

1992).

It is the authors opinion that both these methodologies have something to

offer, and that they are not opposed, but complementary. Further research

may better integrate a variety of approaches, to achieve models that are rich

in both descriptive and explanatory modes.

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a method for generating phyllotaxis

Documents

phyllotaxis phyllotaxis

examples of phyllotaxis

planar phyllotaxis

cylindrical phyllotaxis

phenomenon of phyllotaxis

model figure

modelling systems

areabased phyllotactic