Turtle Geometry is extremely useful in drawing simple shapes quickly, and also in creating fractals of almost any type. Look here for an example on how to use turtle geometry and grammar re-writing to generate the Koch Curve. (Note: This example is at a fairly advanced level and assumes knowledge of the turtle geometry basics found on this page.)
Turtles in Geometry Explorer can understand these basic commands:
The turtle starts out with a specified heading and length. The heading is the direction in which the turtle will move. The length is how far the turtle should move when told to go forward or backward. The heading and length are given by a vector, that is a pair of points. The vector's length is just the distance between the points and the vector's heading is given by an arrow from the first point towards the second.
A turtle must also know what angle to turn by when it turns. This is specified by a set of three points: the initial, vertex, and terminal points of an angle.
As an example, let's use a turtle to construct a
regular
pentagon. First, we need to define a
turtle.
As described above we need to define a vector
and
angle. The angle that will be needed for our pentagon is one of
72 degrees. To construct a 72 degree angle first construct a segment
AB.
Then select A and set it as a center of rotation by choosing "Center''
under the Mark menu in the Transform Panel.
Then,
click on "Rotation'' under the Custom menu in
the Transform Panel and type in 72 for the angle and hit
the Okay button. Then, select point B and hit the rotation button in
the
Transform Panel. We now have a 72 degree angle BAC.
We need to store this angle
for future
use. To store the angle select B, A, and C (in that order)
and choose "Turtle Turn Angle'' under the Turtle menu
in
the main window. We next construct a
segment
DE that will serve as a vector for our turtle. Create a segment
DE
near angle BAC and select D and E (in that order) and then choose
"Turtle Heading Vector'' under the Turtle menu.
We have now stored an angle and vector for a turtle.
To
create the turtle, however, we need to specify
a position,
i.e. a point, at which the turtle will start at. Create a point F
anywhere
on the canvas and select it. At this point the "Create Turtle At
Point'' menu under
the Turtle menu will be active. Once we choose
this menu item a small green turtle will
be created at point F.
Also, a Turtle Controller window will pop up. We
will
use just the section of this window labeled
"Simple
Turtle''. (For more information on turtles and
the
use of the Controller window see the chapter on Turtle Geometry in the
pdf User Guide.) The "Simple Turtle" section has a set of seven
buttons
that we can use to control the movement of our turtle.
We are now ready to construct our pentagon. We do
this
by carrying out a sequence of five Draw Forward
--
Turn Left ("<--'') actions. Carry out these
actions
by pressing the Draw Forward button followed by the Turn Left button
five
times, getting the pentagon shown below.
Suppose that we move point D. By lengthening the
distance
from D to E we lengthen all of the turtle movements.
Also,
by changing the heading of DE we change the
orientation
of the pentagon drawn by the turtle. However,
moving point A does not change angle BAC and thus has no effect
on the pentagon.
The regular pentagon has many fascinating
properties.
One of these has to do with the ratio of a
diagonal
of the pentagon to a side. Select points I,G, and H
as shown and choose "Ratio'' from the Measure menu. This ratio is
called
the golden ratio. This ratio pops up in
many
surprising places in nature.