CIS 110Recursive Graphics |
Programming Assignment |

Write a program that plots a Sierpinski triangle, as illustrated below. Then develop a program that plots a recursive pattern of your own design.

**Part 1. **
The *Sierpinski triangle* is another example of a fractal
pattern like the H-tree pattern from Section 2.3 of the textbook. The
Polish mathematician Wacław Sierpiński described the pattern
in 1915, but it has appeared in Italian art since the 13th century.
Though the Sierpinski triangle looks complex, it can be generated with
a short recursive program. Your task is to write a
program `Sierpinski.java` with a recursive
function `sierpinski()` and a `main()` function that
calls the recursive function once, and plots the result using standard
drawing. Think recursively: `sierpinski()` should draw one
colored triangle (pointed downwards) and then call itself recursively
3 times (with an appropriate stopping condition). When writing your
program, exercise modular design: include a (non-recursive)
function `filledTriangle()` that draws a filled equilateral
triangle of a specified size at a specified location.
** Use any color, or many colors!** But leave the background
white, to save toner when we print it out.

Your program shall take one integer command-line argument `N`,
to control the depth of the recursion. All of the drawing should fit
inside the unit box (*x*- and *y* coordinates between 0
and 1). First, make sure that your program draws a single colored
triangle when `N` is set to 1. Then, check that it draws four
colored triangles when `N` is set to 2. Your program will be
nearly (or completely) debugged when you get to this point. Below are
the target Sierpinski triangles for different values of `N`.
(Drawing the outline triangle is optional.)

%java Sierpinski 1%java Sierpinski 2%java Sierpinski 3%java Sierpinski 4%java Sierpinski 5%java Sierpinski 6

**A diversion: fractal dimension.**
In grade school, we learn that the dimension of a line segment is one,
the dimension of a square is two, and the dimension of a cube is
three. But you probably didn't learn what is really meant
by *dimension*. How can we express what it means mathematically
or computationally? Formally, we can define the *Hausdorff
dimension* or *similarity dimension* of a self-similar
figure by partitioning the figure into a number of self-similar pieces
of smaller size. We define the dimension to be the log (# self
similar pieces) / log (scaling factor in each spatial direction). For
example, we can decompose the unit square into 4 smaller squares, each
of side length 1/2; or we can decompose it into 25 squares, each of
side length 1/5. Here, the number of self-similar pieces is 4 (or 25)
and the scaling factor is 2 (or 5). Thus, the dimension of a square
is 2 since log (4) / log(2) = log (25) / log (5) = 2. We can
decompose the unit cube into 8 cubes, each of side length 1/2; or we
can decompose it into 125 cubes, each of side length 1/5. Therefore,
the dimension of a cube is log(8) / log (2) = log(125) / log(5) = 3.

We can also apply this definition directly to the (set of white points in) Sierpinski triangle. We can decompose the unit Sierpinski triangle into 3 Sierpinski triangles, each of side length 1/2. Thus, the dimension of a Sierpinski triangle is log (3) / log (2) ≈ 1.585. Its dimension is fractional—more than a line segment, but less than a square! With Euclidean geometry, the dimension is always an integer; with fractal geometry, it can be any fraction. Fractals are widely applicable for describing physical objects like the coastline of Britain.

**Part 2. ** In this part you will design and create your
own recursive fractal. Your job is to write a
program `Art.java` that takes one integer command-line
argument `N` to control the depth of recursion
and produces a recursive
pattern of your own choosing. It should work and stay within the
unit box for values 1 through 7; you must not call `setXscale`
or `setYscale`. You are free to choose a
geometric pattern (like, but not too similar to, `Htree`
or `Sierpinski`) or a random construction (like, but not too
similar to, `Brownian`). Originality and creativity in the
design will be a factor in your grade. Please use a white background
(unless your design includes a background image) to save toner. If
you do have a background image, it is allowed to go outside the
unit box, provided your actual drawing stays within the box.

**Checklist. **
Don't forget to read the checklist.

**Submission. **
Submit
`Sierpinski.java`, `Art.java`, and, if desired, a zip
file containing any supplementary image files needed
by `Art.java`. Finally, submit a
readme_sierpinski.txt file and answer the questions.