All pictures are from Acheron 2.0, a free explorer of geometrical fractals. You can download Acheron 2.0 here    
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  Introduction Back to Top

  The euclidean geometry uses objects that have integer topological dimensions. A line or a curve is an object that have a topological dimension of one while a surface is described as an object with two topological dimensions and a cuve as an object with three dimensions. This geometry adequately describes the regular objects but failed to be applicable when it comes to consider natural irregular shapes.

  Benoit B. Mandelbrot introduced a new concepts, that he called fractals, that are useful to describe natural shapes as islands, clouds, landscapes or other fragmented structures. According to Mandelbrot, the term fractals is derived from the latin adjective fractus meaning fragmented. According to Mandelbrot, a fractal can be defined as 'a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension'. This clever mathematical definition, albeit quite obscure for non-initiated people, means that a fractal curve is a mathematical function that produce an image having a topological dimension between one and two. Intuitively, fractals can be seen as curves partially filling a two-dimentional area. These curves are often described as space-filling curves.

  Fractals curves exhibit a very interesting property known as self-similarity. If you observe precisely the details of a fractal curve, it appears that a portion of the curve replicates exactly the whole curve but on a different scale. Mathematicians have in fact created geometrical fractal curves long before the introduction of the fractal geometry by Benoit Mandelbrot. Some of these curves are well-known as the Von Koch's snowflake or the triangle of Sierpinsky.

Von Koch Curve

Von Koch Curve
Mandelbrot Curve

Mandelbrot Curve
Minkowski Curve

Minkowski Curve
Hilbert Curve

Hilbert Curve
Cesaro Curve

Cesaro Curve
Sierpinski Curve

Sierpinski Curve
Sierpinski Objects

Sierpinski Objects
Peano Curve

Peano Curve
Squares Curve

Squares Curve
Heighway Curve

Heighway Curve

  Sample from Acheron of a Von Koch Curve   The Squares Curve is a nice fractal curve, build using a recursive procedure. I saw a sample of this curve in the well-known book 'Algorithms in C' written by Robert Sedgewick (Addison-Wesley Publ. ISBN 0-201-51425-7). It was not named from its inventor so I call it the Squares Curve. As almost all the geometric fractal curves, this curve shares the fascinating property of having an infinite curve length in a finite area..  

Construction Back to Top

The starting point of the recursive method for drawing the Squares curve is a simple square.

First Iteration in Squares Curve

Use the four corners of the square as the center of 4 smallers squares, each having half the size of the main square. The first iteration gives:

Second Iteration in Squares Curve

The same procedure gives already a nice picture at the second iteration:

Third Iteration in Squares Curve

Properties Back to Top

  • Curve Length

    The following reasoning concerns the curve for which only the outline is drawn. This gives a close curve with an univocal perimeter.

    Take the initial square and name N the length of its side. The perimeter of the 'curve' is N * 4.

    On the first iteration, the four corners are replaced by four smaller squares. So, the length of the curve is now equal to the sum of the segments common between recursion 0 (initial square) and recursion 1 plus the length of the newly added segments. The total length of the two segments removed at each corner is N/2, so the total removed is (N/2) * 4. The total length of the segments making the smaller squares is N/2 * 3 and 4 are added, one on each corner. Looking only at the added segments, the length increase is:

    Linc = (N/2)*3*4 - (N/2)*4 = (N/2)*8

    On the second iteration, the four small squares added at the first iteration will be replaced by four smaller squares. Here, the length of the segments removed on each square corner is equal to N/4 and the length of the smaller squares added is equal to (N/4) * 3. Looking only at the added segments, the length increase is:

    Linc = (N/4)*3*3*4 - (N/4)*3*4 = (N/4)*24

    The formula for the length increase can be generalized as:

    Linc = (N/2Rec) * 8 * 3(Rec - 1)

        where Rec is the iteration number (starting at 0)

    Here is a summary of the length increase and total length of the curve.
    Total Length
    0......N * 4
    1(N/2) * 8N * 4N * 8
    2(N/4) * 24N * 6N * 14
    3(N/8) * 72N * 9N * 23
    4(N/16) * 216N * 13.5N * 36.5
    5(N/32) * 648N * 20.25N * 56.75

    The Ratio of the length increase between two successive iterations is:

    Ratio = ((N/2(Rec+1)) * 8 * 3Rec) / ((N/2Rec) * 8 * 3(Rec-1))
    Solving the equation gives Ratioinc = 1.5, demonstrating what is quite obvious from the figures in the above table.

    The formula of the length increase can then be generalized to:

    Linc = N * 4 * rRec-1   where r = 1.5

    The total length of the curve is equal to the original length plus the sum of all the length increases.

    Using the following identitiy,

    1 + x + x2 + x3 + ... + xn = (xn+1 - 1) / (x - 1)

    the total length can be generalized:

    LTot = N * (( rRec * 8) - 4)

    Graphically, it gives a nice view of the ever increasing length:
    Total Length of Squares Curve

  • Area

    Take the initial square and name N the length of its side. The area of the 'curve' is noted N2. Using a reasoning analogous to the one followed for the determination of the curve length, the formula for the curve area is obtained.

    The area increase at each iteration can be generalized as:

    Areainc = (4 * 3Rec) / 4Rec+1

    Solving for the Ratio of the area increase between two successive iterations gives: Ratio = rRec   where r = 0.75

    The total area of the curve can then be expressed as:

    AreaTot = N2 + ( 1 + rRec + r(Rec+1) + .. r(Rec+n))

    Using the following identitiy,

    1 + x + x2 + x3 + ... + xn = (xn+1 - 1) / (x - 1)

    the total area can be generalized:

    AreaTot = N2 * 4 * ( 1 - rRec+1)

    As rRec+1 tends to Zero when iteration increases, the area tends to 4 times its original value.

    Graphically, it gives a nice view of the finite area:
    Total Area of Squares Curve

  • Fractal Dimension

    The fractal dimension is computed using the Box-Couting Method equation:

      D = log (N) / log ( r)

    The following picture helps finding the figures required by the formula:
    Squares Curve Dimension using Box-Counting Method

    Replacing r by 14 ( as the grid is 14 * 14) and N by 148 ( the number of small squares covered by the fractal curve) in the the Box-Counting equation gives:

      D = log(148) / log(14) = 1.89356

  • Self-Similarity

    Looking at two successive iterations of the drawing process provides graphical evidence that this property is also shared by this curve.
Variations Back to Top

All Variations described are available using Acheron 2.0

  • Iteration Level

    Eight recursion levels are available. Above this iteration number, the overall aspect of the curve remains essentially unaffected.

  • Curve Style

    Three ways for rendering the curve are available:

    • Normal
      Normal Style for Squares Curve
    • Filled
      Filled Style for Squares Curve
    • Outline
      Outline Style for Squares Cuvre

Author Biography Back to Top

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