Benoit Mandelbrot once said “Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.” Really what he was driving at here is the extreme limitations of traditional Euclidian geometry to model objects that we see throughout nature. This is precisely why fractals are so important. Fractals are our only bridge between the world of orderly mathematics and the chaos of natural phenomenon. For Mandelbrot, fractals were a way of the describing the “roughness” of natural objects and chaotic systems. These types of objects are nearly impossible to model using traditional Euclidian geometry and this is why Mandelbrot’s work is so important. From the most basic two dimensional fractals to the breathtaking Mandelbrot set, Mandelbrot’s fractals have left a lasting impact on the field of mathematics and have opened up a whole new window for mathematical investigation.

Abstract

Although many fractals are very mathematically advanced, others can be produced using simple geometry. In our investigation of fractals, we focus on the development of one such fractal and use algebra and calculus to find volume and surface area.

The menger sponge (pictured to the right) is formed by starting with a cube and progressively removing the center cubes at each iteration, as pictured below.

Methods and Results

As we move past the simple, geometric fractals we enter the realm of complex number fractals. Here complexity and beauty mix with very interesting theoretical applications. Many of these fractals appear random but are also punctuated with intricate patterns.

The volume of the cube decreases because, although the number of cubes increases exponentially, the size of these cubes decreases faster.

From the graph we can see that surface area increases exponentially tending toward infinity. Furthermore the volume decreases exponentially tending toward zero.Pairing these two ideas we find a very interesting result. At infinite iterations the menger sponge has infinite surface area but zero volume!

Volume of menger sponge

This “randomness” or “chaos” gives us unique insight into chaos theory and chaotic systems. Chaotic systems are those that are mathematically deterministic but nearly impossible to predict because of extremely sensitivity to initial conditions. In chaotic systems small changes in the initial values can have drastic effects on the outcome.

Chaotic systems occur all around us, from disease to weather patterns, even global climate could be considered a chaotic system. What’s important to note is that chaotic system are mathematically deterministic. In other words, just like the Mandelbrot set has a function that its developed from, chaotic systems often have algorithms that determine their behavior. This differentiates chaos from random chance. However, this extreme sensitivity to initial conditions means that even an understanding of the system doesn’t translate to being able to predict it.Even if we can’t predict chaos, fractals give us a unique way to view chaos from a mathematical point of view and one of our only chances to model it.

One of Benoit Mandelbrot’s most important contributions to math was the Mandelbrot set. Unlike geometric fractals the Mandelbrot set is actually a set of numbers. When plotted in the complex plane these numbers form the famous fractal image shown on the left.

The Mandelbrot set is created using the function f(z) = z2+cA number c is included in M if when iterated at zero the iterates never exceed two. This process is detailed below.

The figure to the left shows c = -1.9 under iteration. As we can see it never exceeds 2 so -1.9 is in the Mandelbrot set. Also notice the rather chaotic behavior under iteration. This is common with numbers in the Mandelbrot set.

Taking an even closer look we can see something even more intriguing. Notice how the colors blend together and are intermingled. As it turns out if we change c very slightly it can have very drastic and often unforeseen effects on the number of iterations needed to escape. We could also say that the Mandelbrot set has a “chaotic” nature to it.

References

Surface Area of Menger Sponge

𝑆𝑛 =2 × 20𝑛 + 4 × 8𝑛

9𝑛

𝑉𝑛 =20

27

𝑛

𝑆𝑛+1 =8

9𝑆𝑛 +

24

9

20

9

𝑛−1

Faculty Mentor: Bruce Teague

Let c = 1

𝑧1 = (0)2+1 = 1

𝑧2 = (1)2+1 = 2

𝑧3 = (2)2+1 = 5

𝑧4 = (5)2+1 = 26

𝑧5 = (26)2+1 = 677

Let c = -1

𝑧1 = (0)2−1 = −1

𝑧2 = (−1)2−1 = 0

𝑧3 = (0)2−1 = −1

𝑧4 = (−1)2−1 = 0

𝑧5 = (0)2−1 = −1

Values tend to infinity so 1 is not in the Mandelbrot set

Values never exceed 2 so -1 is in the Mandelbrot set

Under iteration there are two different behaviors that can emerge. Either the numbers will get progressively larger as we iterate the function or they will never exceed the number 2, as shown on the right. The Mandelbrot set is composed of all the numbers for which the later applies.

Even though there are only two kinds of behavior we can observe, there is incredible variation. A number may quickly diverge or it may do so slowly.

The photo above shows a close up of one piece of the Mandelbrot set. Notice the black sections. These sections represent numbers in the Mandelbrot set. All the other colors are numbers outside the Mandelbrot set. The picture is generated by counting the number of iterations that it takes for these numbers to escape (get larger than 2). The numbers are then color coded based on the number of iterations they take to escape.

IntroductionFractals are mathematical curves or geometric figures that exhibit self-similar patterns. In most cases these patterns reoccur on progressively smaller scales. Fractals can be seen all throughout nature from a delicate snowflake to the amazon river basin. French mathematician Benoit Mandelbrot is largely credited with the discovery of fractal geometry because of his tremendous contributions to this field of mathematics.

Many people are now speculating that the whole universe might be a fractal.

Regardless of whether this is true or not, fractals are all around us and if we look carefully perhaps we can see a whole new type of beauty and elegance in nature.

At each iteration we remove the middle cubes on each face and the one in the very center. This decreases the number of cubes from 27 to 20. If the larger cube has a dimension of L, each of the smaller cubes then will have a dimension of L/3. Then Ln = (1/3)n. This formula gives the length of a cube at iteration n. To get volume of one cube we at n iterations we cube the length of a side. So volume of one cube is Vn = [(1/3)n ]3 or equivalently (1/27)n multiplying by the number of cubes we get that the formula for volume is (20/27)n.

Number of iterations

Expanded form Surface Area

0 S0 = 6 6

1𝑆1 =

8

9(6) +

24

9

20

9

1−1 8

2𝑆2 =

8

9(8) +

24

9

20

9

2−1 352

27≈ 13.04

3𝑆3 =

8

9

352

27+24

9

20

9

3−1 6016

243≈ 24.76

4𝑆3 =

8

9

6016

243+24

9

20

9

4−1 37376

729≈ 51.27

# of iterations Equation Expanded Form Volume

0 S0 = 1 1 1

1𝑆1 =

1

27

1

× 201 =1

27× 20

20

27≈ 0.74

2𝑆2 =

1

27

2

× 202 =1

729× 400

400

729≈ 0.55

3𝑆3 =

1

27

3

× 203 =1

19683× 8000

8000

19683≈ 0.41

Surface Area of Menger sponge

If we let each face have an area of 1, the initial surface area is 6. At the first iteration each face is split into 9 new faces and 8 are kept, so surface area is 8/9 of original. Additionally we now have four new faces on the inside which we multiply by 6 (number of sides). Finally after the first iteration we now have 20 more cubes than before so we add 20n-1 all this is scaled down by a factor of 9 because the size of the cubes is decreasing. This process is illustrated in the table and graph on the right.

Volume

Surface Area

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