# Turtle Algebra and Algebraic Data Types

In this chapter we explore a new way of creating paths---turtle graphics---and learn some new ways of manipulating lists and functions.

<div class="callout callout-info"> If you run the examples from the SBT console within Doodle they will just work. If not, you will need to start your code with the following imports to make Doodle available.

``````import doodle.core._
import doodle.image._
import doodle.syntax.all._
import doodle.image.syntax.all._
import doodle.java2d._``````

</div>

## Turtle Graphics

So far our paths have used an absolute coordinate system. For example, if we wanted to draw a square we'd use code like

``````import doodle.core.PathElement._

val path =
Image.openPath(
List(moveTo(10,10), lineTo(-10,10), lineTo(-10,-10), lineTo(10, -10), lineTo(10, 10))
)``````

It's often easier to define paths in terms of relative coordinates, specifying how far we move forward or turn relative to our current location. This is how a turtle graphics system works. Here's an example.

``````import doodle.turtle._
import doodle.turtle.Instruction._

// Create a list of instructions for the turtle
val instructions: List[Instruction] =
List(forward(10), turn(90.degrees),
forward(10), turn(90.degrees),
forward(10), turn(90.degrees),
forward(10))

// Ask the turtle to draw these instructions, creating an Image
val image: Image = Turtle.draw(instructions)``````

So where's the turtle in all this? This model was developed in the 60s by Seymour Papert in the programming language Logo. The original Logo could control a robot that drew on paper with a pen. This robot was called a turtle, due to its rounded shape, and way of programming this robot became known as turtle graphics.

Using turtle graphics and another concept, known as an L-system, we can create images that mimic nature such as the plant in Figure turtles:plant. Controlling the Turtle→