The squareflake is a 'similarity' fractal related to the Koch snowflake fractal. Set the Level slider to zero, click on the green flag and the script draws the initiator (line segment) on all fours sides of the square. Set the Level slider to 1, click on the green flag, and the script draws the squareflake generator on all four sides of the square. As you set the Level slider to Levels, 2, 3, and 4, the perimeter increases. What about the area? Can you observe and prove to yourself (without using mathematics) that even though the perimeter increases as the Level increases, the AREA always remains equal to the area of the original square? The squareflake has an infinite perimeter bounding an area that never changes! Now wonder such curves were called pathological (sick) when first discovered. The squareflake has a dimension of 1.5, midway between a line and an area.
Click on 'See inside' to see the squareflake generator script (not a working script in the project) and the [recursive curve] script to the generator's left, that sets up recursion. The scripts describe a recipe for constructing your own similarity fractal. Remix to create your own generator (use paper and pencil) and model it with move and turn blocks. In the recursion script to the left of the generator, mirrors the generator by replacing every [move] block in the generator with a [recursive call] block, adjust the turns to match the turns in the generator, and you're done!
