# Words and music. Words first!

# Fractals

## Hidden Beauty Revealed in Mathematics

### by Jason Lisle, Ph.D.

### January 1, 2007

Did you know that amazing, beautiful shapes have been built into numbers? Believe it or not, numbers like 1, 2, 3, etc., contain a “secret code”—a hidden beauty embedded within them. Numbers have existed from the beginning of creation, yet researchers have only recently discovered the hidden shapes that the Lord placed within them.

^{1}Such beauty defies a secular explanation but confirms biblical creation.

The strange shape in Figure 1 is a sort of “map.” Most maps that we think of are representations of something physical, like a roadmap or a map of a country. But the map in Figure 1 does not represent a physical object; instead it represents a set of numbers. In mathematics, the term “set” refers to a group of numbers that have a common property. For example, there is the set of positive numbers (4 and 7 belong to this set; -3 and 0 do not).

A few decades ago, researchers discovered a very strange and interesting set called “the Mandelbrot set.”

^{2}Figure 1 is a map (a plot) that shows which numbers belong to the Mandelbrot set.

### What do these images mean?

A “set” is a group of numbers that all have a common property. For example, the numbers 4 and 6 are part of the set of even numbers, whereas 3 and 7 do not belong to that set. The Mandelbrot set is a group of numbers defined by a simple formula which is explained in the In-Depth box in this article. Some numbers belong to the Mandelbrot set, and others don’t.Figure 1 is a plot—a graph that shows which numbers are part of the Mandelbrot set. Points that are black represent numbers that are part of the set. So, the numbers, -1, -1/2, and 0 are part of the Mandelbrot set. Points that are colored (red and yellow) are numbers that do not belong to the Mandelbrot set, such as the number 1/2. Although the formula that defines the Mandelbrot set is extremely simple, the plotted shape is extremely complex and interesting. When we zoom in on this shape, we see that it contains beautiful spirals and streamers of infinite complexity. Such complexity has been built into numbers by the Lord.

The Mandelbrot set (Figure 1) is infinitely detailed. In Figure 2, we have zoomed in on the “tail” of the Mandelbrot set. And what should we find but another (smaller) version of the original. This new, smaller Mandelbrot set also has a tail containing a miniature version of itself, which has a miniature version of itself, etc.—all the way to infinity.

Evolution cannot account for fractals. These shapes have existed since creation and cannot have evolved since numbers cannot be changed.The Mandelbrot set is a very complex and detailed shape; in fact it is infinitely detailed. If we zoom in on a graphed piece of the Mandelbrot set, we see that it appears even more complicated than the original. In Figure 2, we have zoomed in on the “tail” of the Mandelbrot set. And what should we find but another (smaller) version of the original; a “baby” Mandelbrot set is built into the tail of the “parent.” This new, smaller Mandelbrot set also has a tail containing a miniature version of itself, which has a miniature version of itself, etc.—all the way to infinity. The Mandelbrot set is called a “fractal”

^{3}since it has an infinite number of its own shape built into itself.

In Figure 3, we have zoomed into a region called the “Valley of Seahorses.” By zooming in on one of these “seahorses” we can see that it is a very complex spiral (

*see Figure 4*). If we continue to zoom in, the order and beauty continue to increase as shown in Figures 5 and 6. As we zoom in yet again, we see in Figure 7 another “baby” version of the original Mandelbrot set at the center of the intersecting spirals; it appears virtually the same as the original shape, but it is 5 million times smaller.

Where did this incredible organization and beauty come from? Some might say that a computer produced this organization and beauty. After all, a computer was used to produce the graphs in the figures. But the computer did not

*create*the fractal. It only produced the map—the representation of the fractal. A graph of something is not the thing itself, just as a map of the United States is not the same thing as the United States. The computer was merely a tool that was used to discover a shape that is an artifact of the mathematics itself.

^{4}

God alone can take credit for mathematical truths, such as fractals. Such transcendent truths are a reflection of God’s thoughts. Therefore when we discover mathematical truths we are, in the words of the astronomer Johannes Kepler, “thinking God’s thoughts after Him.” The shapes shown in the figures have been built into mathematics by the Creator of mathematics. We could have chosen different color schemes for the graphs, but we cannot alter the shape—it is set by God and His nature.

Evolution cannot account for fractals. These shapes have existed since creation and cannot have evolved, since numbers cannot change—the number 7 will never be anything but 7. But fractals are perfectly consistent with biblical creation. The Christian understands that there are transcendent truths because the Bible states many of them.

^{5}A biblical creationist expects to find beauty and order in the universe, not only in the physical universe,

^{6}but in the abstract realm of mathematics as well. This order and beauty is possible because there is a logical God who has imparted order and beauty into His universe.

### Infinite Complexity?

This sequence of images (Figures 3–7) shows what happens as we continually zoom in on a very small region of the Mandelbrot set. We start by zooming in on the highlighted region of the Mandelbrot set called the “Valley of Seahorses” (Figure 3). By zooming in on one of these “seahorses” we can see that it is a very complex spiral (Figure 4). We continue to zoom in (the region is indicated by the grayscale inset) in Figures 5, 6 and 7. Figure 7 shows a “baby” Mandelbrot set; it is virtually identical to the original shape, but it is 5 million times smaller.

### In-Depth

The formula for the Mandelbrot set is z_{n+1}= z

_{n}

^{2}+ c. In this formula, c is the number being evaluated, and z is a sequence of numbers (z

_{0}, z

_{1}, z

_{2}, z

_{3}…) generated by the formula. The first number z

_{0}is set to zero; the other numbers will depend on the value of c. If the sequence of z

_{n}stays small (z

_{n}≤ 2 for all n), c is then classified as being part of the Mandelbrot set. For example, let’s evaluate the point c = 1. Then the sequence of z

_{n}is 0, 1, 2, 5, 26, 677… . Clearly this sequence is not staying small, so the number 1 is not part of the Mandelbrot set. The different shades/colors in the figures indicate how quickly the z sequence grows when c is not a part of the Mandelbrot set.

The complex numbers are also evaluated. Complex numbers contain a “real” part and an “imaginary” part. The real part is either positive or negative (or zero), and the imaginary part is the square-root of a negative number. By convention, the real part of the complex number (RE[c]) is the x-coordinate of the point, and the imaginary part (IM[c]) is the y-coordinate. So, every complex number is represented as a point on a plane. Many other formulae could be substituted and would reveal similar shapes.

*Distant Starlight*and

*Creation Astronomy*.

# Proof of God Using Math Without Words

Posted on Nov 28th, 2011

Thanks to Scott Keltner of Eudora, Kansas, says the writer of Marv's Blog. Marv is the author of

Having watched the above, I then noticed the ones below:

You think I am going to quit because of resistance? Are you kidding? Dare to live challenging the norm, dare to think critically and to think for yourself! Eternity beckons and I intend to

**push the envelope**

watch it bend!watch it bend!

The Ruling Paradigm holds tight to Darwinism, a concept that is not only ludicrous but also incredibly boring. God as a concept is Wonders, and wonders are what He has created! When this world is ended I will join God and

**we'll ride the spiral to the end and may just go where no one's been!****Laterilis from Lateralus by Tool**

NCSE and talkorigins and Darwinists in general? Thought Police.

Important sites like Creation.com and brilliant individuals like Ian Juby and even small fry like me?

We are the Resistance!

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For those unsure of Mandelbrot a youtube for the Valley of the Seahorses and also the infinite continuation of the Mandelbrots below and the description of the formula here...from wikipedia.

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For those unsure of Mandelbrot a youtube for the Valley of the Seahorses and also the infinite continuation of the Mandelbrots below and the description of the formula here...from wikipedia.

"The

**Mandelbrot set**is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape. The set is closely related to the Julia set (which generates similarly complex shapes), and is named after the mathematician BenoĆ®t Mandelbrot, who studied and popularized it.More technically, the Mandelbrot set is the set of values of

*c*in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial*z*_{n+1}=*z*_{n}^{2}+*c*remains bounded.^{[1]}That is, a complex number,*c*, is part of the Mandelbrot set if, when starting with*z*_{0}= 0 and applying the iteration repeatedly, the absolute value of*z*_{n}remains bounded however large*n*gets.For example, letting

*c*= 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand,*c*=*i*(where*i*is defined as*i*^{2}= −1) gives the sequence 0,*i*, (−1 +*i*), −*i*, (−1 +*i*), −*i*, ..., which is bounded and so*i*belongs to the Mandelbrot set.Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization..."