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About Lsystems
Lsystems (Lindenmayersystems, named after Aristid Lindenmayer, 19251989), allow definition of complex shapes through the use of iteration. They use a mathematical language in which an initial string of characters is matched against rules which are evaluated repeatedly, and the results are used to generate geometry. The result of each evaluation becomes the basis for the next iteration of geometry, giving the illusion of growth.
The Lsystem SOP lets you simulate complex organic structures such as trees, lightning, snowflakes, flowers, and other branching phenomena.
Placing an LSystem in the viewer
To...  Do this 

Place the LSystem anywhere in the scene 

Place the LSystem at the origin 
Note
Lsystems can be moved once they are placed by either dragging them in the scene view or changing the values in the parameter editor.
Default LSystems can be chosen from the Gear dropdown menu at the geometry level.
LSystem Handles
The LSystem does not have its own rotation parameters; however, you can rotate it at the object level using the handles in the scene view or by adding a Transform node at the geometry level.
Note
The origin of the LSystem is at the base of the tree, not the middle of the object.
Fractal properties
There are several factors which combine to organize plant structures and contribute to their beauty. These include:

symmetry

selfsimilarity

developmental algorithms
With Lsystems, we are mostly concerned with the latter two. Selfsimilarity implies an underlying fractal structure which is provided through strings of Lsystems. Benoit Mandelbrot describes selfsimilarity as follows:
"When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called selfsimilar."
Lsystems provide a grammar for describing the growth of selfsimilar structures in time. Lsystem rules determine the underlying structures of growth in a way that is analogous to the way that DNA is thought to determine biological growth. This growth relies on the principle of selfsimilarity to provide extremely compact descriptions of complex surfaces.
Rewriting
The central concept of Lsystems is rewriting. This works by recursively replacing an initial state (the initiator) with rewritten geometry (the generator), reduced and displaced to have the same end points as those of the interval being replaced.
In 1968, Astrid Lindenmayer introduced a string rewriting mechanism termed "Lsystems". The grammar of Lsystems is unique in that the method of applying productions is applied in parallel and simultaneously replaces all letters in a given "word".
The simplest example of a rewriting grammar is where two "words" or
strings are used, built from the two letters: a
and b
, which may
occur many times in a string. Each letter is associated with a
rewriting rule. The rule a = ab
means that the letter a
is to be
replaced by the string ab
, and the rule b = a
means that the
letter b
is to be replaced by a
.
If we start the process with the letter b
(the premise), and
follow it through in time, we see a certain pattern emerges by
following the rewriting rules:
Production rule syntax
The general form of an Lsystem rule is:
[left_context<] symbol [>right_context] [:condition]=replacement [:probability]
Where…
left_context
An optional string that must precede the symbol for this rule to match.
symbol
The symbol to replace. For example, if the symbol is A
,
occurrences of A
in the initial string will be replaced with
replacement (if this rule matches).
right_context
An optional string that must follow the symbol for this rule to match.
condition
An optional expression that must be true for this rule to match.
replacement
The string that will replace the symbol (if this rule matches).
probability
The optional chance (between 0 and 1) that this rule will be executed.
For example, using 0.8
means this rule will execute 80% of the time.
Tip
You can use >
in production rules instead of =
. The meaning is
identical.
Turtle commands
We can combine this stringmanipulation system with a graphics routine that interprets the strings as commands for a drawing "turtle" with a position (XYZ) and heading (angle). By following the commands, the turtle traces out a shape as it moves.
Examples of simple turtle commands:
F
Move forward a step, drawing a line connecting the previous position to the new position.
f
Move forward without drawing.
+
Rotate right 90 degrees.

Rotate left 90 degrees.
(In the actual Lsystem node, the angle of the +
and 
commands
is configurable.)
With these simple rules, we can easily come up with a string that causes the turtle to draw a shape such as the letter "L". For example, assuming the turtle is initially facing upwards, we would use the following string to create the letter "L":
Rewriting turtle command strings
By iteratively running a turtle command string through rewrite rules, you can generate surprisingly complex geometry. The power of selfreference in rewrite rules can create extremely intricate figures.
As a very simple example of selfreference, consider an Lsystem
with the initial string A
and the rule A=F+A
. The rule means
"Wherever you see 'A', replace it with 'F+A'". Because the
replacement will contain within it the trigger for the rule, each
generation will cause the string to grow in a cascade effect:
This generates a growing list of repeated "move forward, then turn" commands. With a turn angle less than 90 and a sufficient number of generations, this Lsystem will approximate an arc or circle. You could use this behavior as the basis for curling a sheet of paper or curling a scorpion’s tail. Or, you could randomize the turn angle and create a squiggly line, which you could use as the basis for a bolt of lightning.
(Be sure not to confuse the turtle command string F+A
with the
mathematical statement F plus A. In the context of Lsystems, the
+
symbol means "turn", not "add".)
Another example: the following figure is called a quadratic Koch island. Beginning with these values:
Initial string (premise) 

Rewrite rule 

Angle 

…the turtle generates the following for three generations:
Note
The work required for Houdini to calculate successive
generations increases exponentially. If you try the island
example in Houdini, make sure the Generations parameter is
not greater than 3
.
Tip
You can press on an Lsystem node to see the node’s current string. This can be very useful in debugging rule substitution.
Branches
The systems described so far generate a single continuous line. To describe things like trees, we need a way to create branches.
In Lsystems, you create branches with the square brackets ([
and
]
). Any turtle commands you put inside square brackets are
executed separately from the main string by a new turtle.
For example, the turtle commands F [+F] F [+F] [F]
is interpreted
as:

Go forward.

Branch off a new turtle and have it turn right and then go forward.

Go forward.

Branch off a new turtle and have it turn right and then go forward.

Branch off a new turtle and have it turn left and then go forward.
This creates the following figure:
Another example: the command string F [+F] [F] F [+F] FF
creates
the following figure:
3D
The systems described so far generate flat geometry.
To move the turtle in 3D, you use the &
(pitch up), ^
(pitch
down), \\
(roll clockwise), and /
(roll counterclockwise)
commands.
For example, the initial premise FFFA
and the rule A= " [&FFFA]
//// [&FFFA] //// [&FFFA]
.
This creates the following 3D figure:
`.
The rule creates three branches at every generation. The pitch up
commands (&
) split the branches off from the vertical. The roll
commands (/
) make the branches go out in different directions.
(Note the A
at the end of each branch that ensures new copies of
the rule will grow from the ends of the branches.)
The "
command makes the F
commands half length in each
generation, which makes the branches shrink further out.
Use multiple Lsystem rules
In the previous section we used the rule
A= " [&FFFA] //// [&FFFA] //// [&FFFA]
.
Obviously this rule has redundancy. Since Lsystems are about replacing symbols with strings, we can simply replace the repeated strings with a new symbol, and then create a new rule for that symbol:
Rule 1 

Rule 2 

Because the branches are now defined in one place, if you want to change the branch instructions you only need to edit one string.
Note that the tworule system will take twice as many generations to produce the same result. This is because each generation performs one rule substitution.
So, whereas the single rule A= " [&FFFA] //// [&FFFA] //// [&FFFA]
grows by expanding A
at each generation, the dual rules of
A= " [B] //// [B] //// [B]
and B= &FFFA
work by alternating
between replacing A
with " [B] //// [B] //// [B]
and replacing
B
with &FFFA
.
Turtle command reference
Normally turtle symbols use the current length/angle/thickness etc. to determine their effect. You can provide explicit arguments in brackets to override the normal values used by the turtle command.
The following list shows the bracketed arguments. Remember that you can simply use the singlecharacter command without the arguments and Houdini will simply use the normal values.
F(l,w,s,d)
Move forward (creating geometry) distance l of width w using s cross sections of d divisions each.
H(l,w,s,d)
Move forward half the length (creating geometry) distance l of width w using s cross sections of d divisions each.
G(l,w,s,d)
Move forward but don’t record a vertex distance l of width w using s cross sections of d divisions each.
f(l,w,s,d)
Move forward (no geometry created) distance l of width w using s cross sections of d divisions each.
h(l,w,s,d)
Move forward a half length (no geometry created) distance l of width w using s cross sections of d divisions each.
J(s,x,a,b,c) K(s,x,a,b,c) M(s,x,a,b,c)
Copy geometry from leaf input J, K, or M at the turtle’s position after scaling and reorienting the geometry. The geometry is scaled by the s parameter (default Step Size) and stamped with the values a through c (default no stamping). Stamping occurs if the given parameter is present and the relevant Leaf parameter is set. The x parameter is not used and should be set to 0. Note that point vector attributes in the leaf inputs will be affected by the turtle movements.
T(g)
Apply tropism vector (gravity). This angles the turtle towards the negative Y axis. The amount of change is governed by g. The default change is to use the Gravity parameter.
+(a)
Turn right a degrees. Default Angle.
(a)
Turn left a degrees (minus sign). Default Angle.
&(a)
Pitch up a degrees. Default Angle.
^(a)
Pitch down a degrees. Default Angle.
\\(a)
Roll clockwise a degrees. Default Angle.
/(a)
Roll counterclockwise a degrees. Default Angle.

Turn 180 degrees
*
Roll 180 degrees
~(a)
Pitch / Roll / Turn random amount up to a degrees. Default 180.
"(s)
Multiply current length by s. Default Step Size Scale.
!(s)
Multiply current thickness by s. Default Thickness Scale.
;(s)
Multiply current angle by s. Default Angle Scale.
_(s)
Divide current length (underscore) by s. Default Step Size Scale.
?(s)
Divides current width by s. Default Thickness Scale.
@(s)
Divide current angle by s. Default Angle Scale.
'(u)
Increment color index U by u. Default UV Increment's first parameter.
#(v)
Increment color index V by v. Default UV Increment's second parameter.
%
Cut off remainder of branch
$(x,y,z)
Rotates the turtle so the up vector is (0,1,0). Points the
turtle in the direction of the point (x,y,z)
. Default behavior
is only to orient and not to change the direction.
[
Push turtle state (start a branch)
]
Pop turtle state (end a branch)
{
Start a polygon
.
Make a polygon vertex
}
End a polygon
g(i)
Create a new primitive group to which subsequent geometry is added. The group name is the Group Prefix followed by the number i. The default if no parameter is given is to create a group with the current group number and then increment the current group number.
a(attrib, v1, v2, v3)
This creates a point attribute of the name attrib. It is then
set to the value v1, v2, v3
for the remainder of the points on
this branch, or until another a command resets it. v2 and
v3 are optional. If they are not present, an attribute of
fewer floats will be created. The created attribute is always of
float type and with zero defaults. For example, the rule
a("Cd", 1, 0, 1)
added to the start of the premise will make
the Lsystem a nice pugnacious purple.
Use modeled geometry in an Lsystem
Houdini lets you create a copy of some geometry at the turtle’s location using certain commands. You can use this to create leaves and flowers on an Lsystem shrub, for example.

Connect the output of the geometry you want to stamp to one of the Lsystem node’s inputs.

Use the corresponding command (
J
,K
, orM
) in a turtle command string to insert the geometry.
Input  Turtle command 

1  J 
2  K 
3  M 
If you connect a leaf surface to the Lsystems input 1 and a flower to input 2, you can use the following to create a bush with leaves and flowers:
Premise 

Rule 1 

Rule 2 

Rule 3 

Rule 1 prefaces the K
and J
commands with f
(move forward
without drawing) to offset the geometry a little bit. Otherwise, the
leaf would be attached at its center, rather than the edge.
Symbol variables
Each symbol can have up to five userdefined variables associated with it. You can reference or assign these variables in expressions. Variables in the matched symbol are instanced while variables in the replacement are assigned.
For example, the rule A(i, j)=A(i+1, j1)
will replace each A
with a new A
in which the first parameter (i
) has been
incremented and the second parameter (j
) decremented.
Parameters assigned to geometric symbols (for example, F
, +
, and
!
) are interpreted geometrically. For example, the rule: F(i, j)
= F(0.5*i, 2*j)
will again replace each F with a new F containing
modified parameters. In addition to this, the new F will now be
drawn at half the length and twice the width.
Tip
The variables in the predecessor can also be referenced by the
condition or probability portions of the rule. For example, the
rule A(i):i<5 = A(i+1) A(i+1)
will double each A a maximum of
five times (assuming a premise of A(0)
).
Control length over time
To create an Lsystem which goes forward x percent less on each iteration, you need to start your Premise with a value, and then in a rule multiply that value by the percentage you want to remain.
Premise 
A(1)

Rule 
A(i)= F(i)A(i*0.5)

This way i is scaled before A is reevaluated. The important part is the premise: you need to start with a value to be able to scale it.
Stamp variables onto input geometry
The third argument to the J
/K
/M
commands is passed to the
connected geometry.
Tip
You can use this trick to get around the limitation of only three geometry inputs on a Lsystem.
Create all the different models you want (say, 20 different
types of leaves) and connect them to a Switch
node. Set the switch node’s Select input
parameter to stamp("/path/to/lsystem", "lsys",0)
.
Connect the switch node to the J input of an Lsystem node. Now
you can insert any of the 20 leaf types using
J(,,leaf_number)
.

Create a Circle node and set the number of divisions to
stamp("/path/to/lsystem", "lsys", 3)
.Because the default number of divisions is 3 (the second argument in the expression), this creates a triangle.

Connect the output of the circle node to the J input of an Lsystem node.

In the Lsystem rules, you can use
J(,,number)
to pass number to the J geometry. For example,J(,,4)
produces a square,J(,,5)
produces a pentagram.
Create groups within Lsystems
The g
command puts all geometry currently being built into a
group.
The group name is composed of a prefix set on the Funcs tab and
a number. Default prefix is lsys
, producing group names like
"lsys1". You can specify the number as an argument to the g
command.
For example, g[F]
puts geometry from the F
into a group (named
using ). Otherwise, the default index is incremented appropriately.
The current group is associated with the branch, so you can do
things like gF [ gFF ] F
to put the first and last F
into group
0, and the middle (branched) FF
into group 1.
To exclude a branch from its parent’s group, use g(1)
.
Edge rewriting
In The Algorithmic Beauty of Plants, many examples use a technique called edge rewriting which involve left and right subscripts. A typical example is:
Generations 
10

Angle 
90

Premise 
F(l)

Rule 1 
F(l) = F(l)+F(r)+

Rule 2 
F(r)=F(l)F(r)

However, Houdini doesn’t support the F(l)
and F(r)
syntax. You
can modify the rules to use symbol variables instead.
For the F turtle symbol, the first four parameters are length, width, tubesides, and tubesegs. The last parameter is userdefinable. We can define this last parameter so 0 is left, and 1 is right:
Generations 
10

Angle 
90

Premise 
F(1,1,3,3,0)

Rule 1 
F(i,j,k,l,m) :m=0 = F(i,j,k,l,0)+F(i,j,k,l,1)+

Rule 2 
F(i,j,k,l,m) :m=1 =F(i,j,k,l,0)F(i,j,k,l,1)

After two generations this produces: Fl+Fr+FlFr
There should not
be any difference between this final string and: F+F+FF
Another approach is to use two new variables, and use a conditional statement on the final step to convert them to F:
b 
ch("generations")

Premise 
l

Rule 1 
l:t<b=l+r+

Rule 2 
r:t<b=lr

Rule 3 
l=F

Rule 4 
r=F

The produces the following output:
Generation  String 

0 
l

1 
F

2 
F+F+

3 
F+F++FF+

Limit Lsystem growth inside a shape
The Lsystem node’s metatest input lets you generate rules that will cause the system to stop when it reaches the edges of a defined shape, like a topiary hedge.

Create a metaball or merged metaballs that define the volume in which the Lsystem can grow.

Connect the metaball node’s output to the Metatest input of the Lsystem node.

Use a conditional statement (
:
) with an "in" test. For example:
Example
Premise 

Rule 1 

Rule 2 


This Lsystem checks to see if the next iteration of growth will be within the Metatest bounds, and if not it prunes the current branch.

Rule 1 executes 80% of the time when the branch is within the metatest boundary.

Rule 2 executes when the branch is not within the metatest boundary (the
!
negates thein(x,y,z)
condition). The%
command ends the branch.
Note
If the Lsystem start point is not inside the metaball envelope, it will stay dormant. Once you have set up your Lsystem and metaballs, make sure you transform them together so the Lsystem is not accidentally moved outside the metaball.
Arrange geometry instances with Lsystems
Lsystems can be a powerful tool for arranging modeled geometry. By using an Lsystem as the template input to a Copy SOP, you can place a copy of a model at every point of the Lsystem.
For example, you could use the "arc approximation" Lsystem from the
Lsystem basics (premise=A
, rule=A=F+A
) to arrange
a series of spheres in an arc or circle. This gives you parametric
control of the bending and spacing of the arc of spheres.
Further reading
If you have any serious interests in creating Lsystems, you should obtain the book:
The Algorithmic Beauty of Plants by Przemyslaw Prusinkiewicz and Aristid Lindenmayer (1996, SpringerVerlag, New York. Phone 212.460.1500. ISBN: 0387946764)
It is the definitive work on the subject. It contains many Lsystems examples along with ideas and theories about modeling realistic plant growth.
Speed up calculations
Modeling something like a whole tree as single large LSystem may cause collision resolution calculations to be single threaded. The wire solver will look for pieces of wire objects that can be solved independently and divide the work among the available cores. One large connected LSystem means the work cannot be divided into smaller work units.
Try using a Wire Glue Constraint DOP to constrain a point on the LSystem where the branches join together near the root of the tree (constrain the point to its world space position). This will cause the wire solver to see the separate branches as distinct pieces that can be solved independently. Since the constrained point will not move, any motion on one of the branches will not affect the other branches. If possible, reducing the number of points in the wire object should also speed up the calculations.
Parameters
Geometry
Type
The type of geometry to create as the turtle moves.
Tip
You can create a tube path from a skeleton Lsystem using a Polywire SOP.
Skeleton
Draw polylines.
Tube
Draw tubes.
Generations
The number of times to repeat the rulesubstitution. If you specify a fractional number and Continuous angles and/or Continuous length are on (below), Houdini scales the geometry generated by the last substitution to give smooth growth between generations.
Start Position
This is the starting point position for the turtle.
Random Scale
If nonzero, randomly scales all the lengths specified by F and other similar turtle functions.
Random Seed
The seed to use for the random number generator. By varying this on a Lsystem using random rules (Ie: random scale, ~, or probabilistic rules) one can generate different instances of the Lsystem.
Continuous Angles
If set, the angles rotated by the last generation’s turtle operations will be scaled by the amount into the generation.
Continuous Length
If set, the lengths taken by the last generation’s turtle operations will be scaled by the amount into the generation.
Continuous Width
If set, the widths generated by the last generation’s turtle operations will be scaled by the amount into the generation.
Apply Color
If set, the Lsystem will output a color attribute on each point. The color value will be found by looking up into the Image File at the current U & V positions. The current U & V is altered with the ' and # turtle operations.
Image File
This is the image file which is used for the Apply Color operation. The image files used by the pic() expression is under Funcs tab.
UV Increment
These determine the default U and V increments of the ' and

turtle operations.
Point Attributes
If the type is Skeleton, this is available. Turning this on will cause the creation of many point attributes to be created to track how each point was generated:
width
The width of the tube that would have been generated.
segs
The number of segments that would be made on a tube.
div
The number of divisions the tube would be divided into.
lage
The vertical increment from the root of the tree. This is affected by the Tube::Vertical Increment parameter. It is similar to arc, but not dependent on edge length.
arc
The arc length from the root of the turtles path to this point.
up
The up vector of the turtle at this point.
gen
The generation that created this point.
Tube
Rows
The number of rows to divide tubes into. A value of 3 will cause the tubes to be swept triangles.
Cols
The number of columns to divide the tubes in. A value of 4 means one F will create 4 cross sections.
Tension
How straight the tubes should sweep to their destination point.
Branch Blend
How much a new branch should inherit off an old branches direction.
Thickness
Default width of the tubes.
Thickness Scale
How much the ! operation will affect the thickness.
Apply Tube Texture Coordinates
If checked, the tubes will generate uv texture coordinates.
Vertical Increment
The amount each tube will increment the V texture coordinate.
Values
Step Size
The default size of a movement, such as F, command.
Step Size Scale
The number used by the " command.
Angle
The default angle for an angle, such as /, command. This also becomes the variable a in the expression.
Angle Scale
The number used by the ; command.
Variable b
The value of the expression variable b.
Variable c
The value of the expression variable c.
Variable d
The value of the expression variable d.
Gravity
The amount of tropism from the T command. Also becomes the value of the expression variable T.
Number Of Variables
This multiparm allows the assignment of an arbitrary number of new expression variables.
Variable Name
The name of the expression variable. This is a single character. Check the Local Variables section to see what variables are already reserved.
Variable Value
The value of the expression variable.
Funcs
Pic Image File
The image file to use with the pic() expression function.
Group Prefix
The prefix used by g command.
Channel Prefix
The prefix used by the chan() expression function.
Leaf Param A
This is the name of the stamp parameter to stamp the leaf with. The value of the stamp comes from the J, K, or M operation. It can be read upstream using the stamp() function.
Leaf Param B
This is the name of the stamp parameter to stamp the leaf with. The value of the stamp comes from the J, K, or M operation. It can be read upstream using the stamp() function.
Leaf Param C
This is the name of the stamp parameter to stamp the leaf with. The value of the stamp comes from the J, K, or M operation. It can be read upstream using the stamp() function.
Rules
Read Rules From File
If this is set, the rule fields are ignored. Instead, the Rule File is read and used as the rules.
Write Rule Parameters to File
This will write all the current rules to the Rule File.
Rule File
The name of the file to use as a source of rules. This file should have one line per rule. Blank lines and lines that start with '#' will be ignored, so comments may be added to the rule file with '#'.
Context Ignore
This is a list of symbols. They will be ignored when trying to determine contexts.
Context Includes Siblings
By default, the context of each branch only includes the
symbols in that branch. Any subbranches or parent
branches will be skipped over. Given the rule A>B=F
, A[B]
will not resolve as B is in a subbranch. A[Q]B
will
resolve because the [Q]
is ignored. If you change the
Context Ignore to have []
, this effect is removed and
A[Q]B
will not resolve but A[B]
will. The Context
Includes Siblings flag restores the preHoudini 10
behavior of context of sibling branches being included.
For example, [A]Q[B]
will resolve if this flag is set, but
not resolve if it is not set.
Premise
The initial state of the Lsystem. This is the state of the Lsystem at generation 0.
Rule #
A rule to apply to the Lsystem. Applying the toggle will disable the rule, removing it from the generation procedure.
Inputs
Leaf J,K & M
This geometry is used by the J, K & M rules
Meta Test Input
This geometry is used by the in() function to determine bounding regions (for topiaries)
Locals
a
The value of the Angle parameter.
b
The value of the b parameter.
c
The value of the c parameter.
d
The value of the d parameter.
g
The age of the current rule, initially 0.
i
The offset into the current Lsystem string where the rule is being applied.
t
The iteration count, initially 0.
x, y, z
Current turtle position in space.
A
Arclength from the root of the tree to the current point.
L
Current length increment at the point.
T
The value of the Gravity parameter.
U
Color map U value.
V
Color map V value.
W
Width at the current point.
See also 