Today we are going to cover fractals. Fractals are pictures that, when you look at a small area of the picture, it looks similar to the overall picture (and other small areas, too). This property is called "selfsimilar". Many things in nature can be drawn better with fractals than with simple polygons. For example, think about a pine tree. The main branch (or trunk) has smaller branches coming off it. Those branches have smaller branches coming off of it. And so on, until you get to the pine needles. That is sort of selfsimilarity is exactly what fractals are.
Fractals are surprisingly easy to draw in Logo.
Before we can draw fractals, we must master a new programming technique called "recursion". Recursion is when a command calls itself. Recursive commands all follow the same pattern: they do a little work, then call themselves with simpler inputs. This, in turn, will do a little more work and call itself with even simpler inputs. When the input is so simple that there's essentially nothing to be done, the command just stops without doing any work (and without calling itself). Don't worry if you don't see why recursion is useful, that will become clear when we go through a few activities.
Anything you can do with a REPEAT loop can also be done with recursion. For example, let's take our routine for drawing a square from the first lesson.
TO SQUARE REPEAT 4 [ FORWARD 100 RIGHT 90 ] END SQUARE
Now let's write a recursive version of it.
TO SQUARE.RECURSIVE :SIDES.TO.GO ; base case: do nothing IF :SIDES.TO.GO = 0 [ STOP ] ; recursive case: draw a side and call recursively FORWARD 100 RIGHT 90 SQUARE.RECURSIVE :SIDES.TO.GO  1 END SQUARE.RECURSIVE 4
SQUARE.RECURSIVE inputs the number of sides of the square to draw. This seems a bit silly, since all squares have four sides, but it's part of how the recursion works.
Observe that there are two parts to the recursive function: the "base case" and the "recursive case". As SQUARE.RECURSIVE works, it uses the recursive case. The recursive case draws one side of the square, then calls SQUARE.RECURSIVE with an input that is one smaller than what it received. It's one smaller because the input represents the number of sides to draw and the command just drew one of them.
The "base case" is used when SQUARE.RECURSIVE should stop calling itself. This happens when there are no more sides left to be drawn, that is, when :SIDES.TO.GO is 0. Without the base case, Logo would draw the sides of a square forever.
Activity: Type in the SQUARE.RECURSIVE and call it.
Now let's use recursion to draw a fractal. We'll start with a square. We'll draw a smaller square in every corner of this square. In each of these smaller squares, we'll draw even smaller squares. And so on.
SQUARE.FRACTAL follows the pattern of RECURSIVE.SQUARE, but it recurses on a :DEPTH input, instead of :SIDES.TO.GO. :DEPTH represents the number of times the fractal is repeated. The base case is when :DEPTH = 0. The recursive case draws a square with a familiar REPEAT instruction. After drawing each side, it calls RECURSIVE.SQUARE to draw a smaller sqaure in each corner.
TO SQUARE.FRACTAL :LENGTH :DEPTH ; base case: no more squares IF :DEPTH = 0 [ STOP ] ; recursive case: draw a square such that each corner ; of the square has SQUARE.FRACTAL in it. REPEAT 4 [ FORWARD :LENGTH RIGHT 90 SQUARE.FRACTAL :LENGTH * 0.4 :DEPTH  1 ] END SQUARE.FRACTAL 200 4 
Activity: Run SQUARE.FRACTAL. You may want to run SLOWDRAW 10 to see how the picture is drawn in slow motion.
TO TRIANGLE.FRACTAL :LENGTH :DEPTH ; base case: ; just move forward, no more squares IF :DEPTH = 0 [ FORWARD :LENGTH STOP ] ; recursive case: ; draw a triangle such that each side of ; the triangle has TRIANGLE.FRACTAL in it. REPEAT 3 [ FORWARD :LENGTH / 3 TRIANGLE.FRACTAL :LENGTH / 3 :DEPTH  1 FORWARD :LENGTH / 3 RIGHT 120 ] END TRIANGLE.FRACTAL 200 4 

TO SNOWFLAKE.SIDE :LENGTH :DEPTH IF :DEPTH = 0 [ FORWARD :LENGTH STOP ] SNOWFLAKE.SIDE :LENGTH / 3 :DEPTH  1 LEFT 60 SNOWFLAKE.SIDE :LENGTH / 3 :DEPTH  1 RIGHT 120 SNOWFLAKE.SIDE :LENGTH / 3 :DEPTH  1 LEFT 60 SNOWFLAKE.SIDE :LENGTH / 3 :DEPTH  1 END TO SNOWFLAKE :LENGTH :DEPTH REPEAT 3 [ SNOWFLAKE.SIDE :LENGTH :DEPTH RIGHT 120 ] END SNOWFLAKE 200 4 

TO PLANT :SIZE :ANGLE IF :SIZE < 1 [ STOP ] RIGHT :ANGLE FORWARD :SIZE REPEAT 4 [ PLANT :SIZE / 2 DIFFERENCE RANDOM 160 80 ] BACK :SIZE LEFT :ANGLE END PLANT 100 0 

TO CURLY.FRACTAL :SIZE IF :SIZE < 0.5 [ STOP ] REPEAT 360 [ IF REPCOUNT = 5 [ LEFT 90 CURLY.FRACTAL :SIZE / 2 RIGHT 90 ] IF REPCOUNT = 10 [ LEFT 90 CURLY.FRACTAL :SIZE / 5 RIGHT 90 ] IF REPCOUNT = 15 [ LEFT 90 CURLY.FRACTAL :SIZE / 5 RIGHT 90 ] IF REPCOUNT = 20 [ LEFT 90 CURLY.FRACTAL :SIZE / 4 RIGHT 90 ] IF REPCOUNT = 25 [ LEFT 90 CURLY.FRACTAL :SIZE / 5 RIGHT 90 ] IF REPCOUNT = 30 [ LEFT 90 CURLY.FRACTAL :SIZE / 8 RIGHT 90 ] FORWARD :SIZE RIGHT REPCOUNT ] RIGHT 180 END CURLY.FRACTAL 10 

TO CRISSCROSS :SIZE :DEPTH IF :DEPTH = 0 [ STOP ] SETPENCOLOR :DEPTH REPEAT 4 [ FORWARD :SIZE / 2 CRISSCROSS :SIZE / 3 :DEPTH  1 FORWARD :SIZE / 2 BACK :SIZE RIGHT 45 FORWARD :SIZE BACK :SIZE RIGHT 45 ] SETPENCOLOR :DEPTH + 1 END TO CRISSCROSSPICTURE SETSCREENCOLOR 0 CRISSCROSS 100 4 END CRISSCROSSPICTURE 
For an indepth look at several different fractals in Logo, check out Vladimir Batagelj's essay entitled Drawing spacefilling curves in logo.