🤔 Turtles and Strings and L-Systems¶
This section describes a much more interested example of string iteration and the accumulator pattern. Even though it seems like we are doing something that is much more complex, the basic processing is the same as was shown in the previous sections.
In 1968 Aristid Lindenmayer, a biologist, invented a formal system that provides a mathematical description of plant growth known as an L-system. L-systems were designed to model the growth of biological systems. You can think of L-systems as containing the instructions for how a single cell can grow into a complex organism. L-systems can be used to specify the rules for all kinds of interesting patterns. In our case, we are going to use them to specify the rules for drawing pictures.
The rules of an L-system are really a set of instructions for transforming one string into a new string. After a number of these string transformations are complete, the string contains a set of instructions. Our plan is to let these instructions direct a turtle as it draws a picture.
To begin, we will look at an example set of rules:
A -> B
Rule 1 Change A to B
B -> AB
Rule 2 Change B to AB
Each rule set contains an axiom which represents the starting point in the transformations that will follow. The rules are of the form:
left hand side -> right hand side
where the left hand side is a single symbol and the right hand side is a sequence of symbols. You can think of both sides as being simple strings. The way the rules are used is to replace occurrences of the left hand side with the corresponding right hand side.
Now let’s look at these simple rules in action, starting with the string A:
A B Apply Rule 1 (A is replaced by B) AB Apply Rule 2 (B is replaced by AB) BAB Apply Rule 1 to A then Rule 2 to B ABBAB Apply Rule 2 to B, Rule 1 to A, and Rule 2 to B
Notice that each line represents a new transformation for entire string. Each character that matches a left-hand side of a rule in the original has been replaced by the corresponding right-hand side of that same rule. After doing the replacement for each character in the original, we have one transformation.
So how would we encode these rules in a Python program? There are a couple of very important things to note here:
Rules are very much like if statements.
We are going to start with a string and iterate over each of its characters.
As we apply the rules to one string we leave that string alone and create a brand new string using the accumulator pattern. When we are all done with the original we replace it with the new string.
Let’s look at a simple Python program that implements the example set of rules described above.
Try running the example above with different values for the
parameter. You should see that for values 1, 2, 3, and 4, the strings generated follow the
example above exactly.
One of the nice things about the program above is that if you want to implement a different set of rules, you don’t need to re-write the entire program. All you need to do is re-write the applyRules function.
Suppose you had the following rules:
A -> BAB
Rule 1 Change A to BAB
What kind of a string would these rules create? Modify the program above to implement the rule.
Now let’s look at a real L-system that implements a famous drawing. This L-system has just two rules:
F -> F-F++F-F
This L-system uses symbols that will have special meaning when we use them later with the turtle to draw a picture.
Go forward by some number of units
Go backward by some number of units
Turn left by some degrees
Turn right by some degrees
Here is the
applyRules function for this L-system.
def applyRules(ch): newstr = "" if ch == 'F': newstr = 'F-F++F-F' # Rule 1 else: newstr = ch # no rules apply so keep the character return newstr
Pretty simple so far. As you can imagine this string will get pretty long with a few applications of the rules. You might try to expand the string a couple of times on your own just to see.
The last step is to take the final string and turn it into a picture. Let’s
assume that we are always going to go forward or backward by 5 units. In
addition we will also assume that when the turtle turns left or right we’ll
turn by 60 degrees. Now look at the string
F-F++F-F. You might try to
use the explanation above to show the resulting picture that this simple string represents. At this point its not a very exciting
drawing, but once we expand it a few times it will get a lot more interesting.
To create a Python function to draw a string we will write a function called
drawLsystem The function will take four parameters:
A turtle to do the drawing
An expanded string that contains the results of expanding the rules above.
An angle to turn
A distance to move forward or backward
def drawLsystem(aTurtle,instructions,angle,distance): for cmd in instructions: if cmd == 'F': aTurtle.forward(distance) elif cmd == 'B': aTurtle.backward(distance) elif cmd == '+': aTurtle.right(angle) elif cmd == '-': aTurtle.left(angle)
Here is the complete program in activecode. The
main function first creates the
L-system string and then it creates a turtle and passes it and the string to the drawing function.
Feel free to try some different angles and segment lengths to see how the drawing changes.
Here is a dragon curve. Use 90 degrees.:
FX X -> X+YF+ Y -> -FX-Y
Here is something called an arrowhead curve. Use 60 degrees.:
YF X -> YF+XF+Y Y -> XF-YF-X
The Sierpinski Triangle. Use 60 degrees.:
FXF--FF--FF F -> FF X -> --FXF++FXF++FXF--
L-Systems and Lists¶
Let’s return to the L-systems and introduce a very interesting new feature that requires the use of lists.
Suppose we have the following grammar:
X X --> F[-X]+X F --> FF
This L-system looks very similar to the old L-system except that we’ve added
one change. We’ve added the characters ‘[’ and ‘]’. The meaning of these
characters adds a very interesting new dimension to our L-Systems. The ‘[’
character indicates that we want to save the state of our turtle,
namely its position and its heading so that we can come back to this position
later. The ‘]’ tells the turtle to warp to the most recently saved position.
The way that we will accomplish this is to use lists. We can save the
heading and position of the turtle as a list of 3 elements.
y] The first index position in the list holds the heading,
the second index position in the list holds the x coordinate,
and the third index position holds the y coordinate.
Now, if we create an empty list and every time we see a ‘[’ we append the
list that contains
[heading, x, y] we create a history of saved places
the turtle has been where the most recently saved location will always be at
the end of the list. When we find a ‘]’ in the string we use the pop
function to remove the the most recently appended information.
Let’s modify our
drawLsystem function to begin to implement this new
When we run this example we can see that the picture is not very interesting, but notice what gets printed out, and how the saved information about the turtle gets added and removed from the end of the list. In the next example we’ll make use of the information from the list to save and restore the turtle’s position and heading when needed. We’ll use a longer example here so you get an idea of what the kind of drawing the L-System can really make.
Rather than use the
inst string supplied here, use the code from the string examples above (eg. ActiveCode: 3 (act_lsys_1)), and write your own applyRules function to implement this L-system.
This example only uses 6 expansions. Try it out with a larger number of
expansions. You may also want to try this example with different values for
the angle and distance parameters.
Here are the rules for an L-system that creates something that resembles a common garden herb. Implement the following rules and try it. Use an angle of 25.7 degrees.
H H --> HFX[+H][-H] X --> X[-FFF][+FFF]FX
Here is another L-System. Use an Angle of 25 and see what you get.
F F --> F[-F]F[+F]F
Post Project Questions
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Based on my own interests and needs, the things taught in this project...
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