# 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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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 Benoît 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..."

## 10 comments:

Evolution fails as a theory of

biologybecause it can't account for a few accidents ofmathematics?[jokerlaugh.wav]

A few music videos, however well or badly made, don't make a scientific argument, Radar. Especially not when they get basic stuff wrong, as the video "The Fingerprint of God" does with the golden ratio. And none of them mention one rather important little fact: the Fibonacci sequence and phi (along with

e, pi, and several other important mathematical constants appear in biology because the sequences that generate those numbers happen to be the most efficient way to do certain things, and evolution is all about efficiency, doing the most with the fewest resources.I suppose if you have a hammer, everything looks like a nail.

If you believe in a designer, everything looks designed.

If you are foolish, you believe that created things made themselves. You believe that nothing created everything. Have you ever asked yourself where nothing came from? Have you then asked yourself how nothing could create everything? Do you see how foolish Darwinism is from the start?

Efficiency, Jon Woolf, is a hallmark of good design. Darwinists do not believe a Designer designed things, yet they ascribe powers of design to random chance. I am posting a series of articles that puts the lie to such nonsense. Mutations are destroying organisms, not building them and mutations will eventually kill all organisms off if we do not learn to intervene or God decides the end has come.

Accidents of mathematics? Are you serious? Have you heard of the greatest holder of the Lucasian Chair of Mathematics at Cambridge, arguably the author of the greatest scientific treatise ever in

Principia Mathematica, Sir Isaac Newton? Newton knew that mathematics was significant because the world was logically designed and as it happens not only is appearance often similar but design patterns as well.Newton first studied for the ministry and wrote 1.3 million words about God and faith. Yet he also found time to start the discipline of Physics and arguably invented calculus (Leibniz would also have a claim and it is likely that they both came up with a version independently)(Leibniz did change some dates on his notes, which hurt his case).

Polls of scientists usually rate Newton the greatest scientist ever and put Einstein second. Ironically, Newtonian physics was supplemented by Relativity as it is a very good estimate of the laws of motion but Einstein observed that it was not precise when great speed or gravity was involved and postulated his theories. Newtonian Law is precise enough to put a man on the Moon but eventually begins to deviate from truth in drastic circumstances.

Newton became a bit oddball at the end of his days, but later his body was exhumed it we discovered he had poisoned himself with mercury during the course of his many experiments, unknowingly bringing early death and confusion.

Stephen Hawking now sits in Newton's Chair, bringing dishonor to the position. His latest book is an illogical mess of bad science and philosophy even as he declares philosophy to be dead.

But it is Darwinism that will be dead. I will keep pointing out designs in the Universe and in organisms and keep reminding people that things do not design themselves nor do they create themselves. A man cannot pick himself up by his own bootstraps. Nor can he make himself and the boots before he tries. Darwinism = epic fail.

Newton? You mean the guy who believed in alchemy, published an entirely wrong theory of optics, was a Christian heretic of the most extreme sort (he rejected the concept of the Trinity), and made a habit of destroying anybody who dared to criticize his theories? That Newton?

Even the great Sir Isaac wasn't right about everything, Radar.

"Efficiency, Jon Woolf, is a hallmark of good design."

It's also a hallmark of processes that develop by trial and error.

"Darwinists do not believe a Designer designed things, yet they ascribe powers of design to random chance. "

No, they don't. Evolution isn't random.

Jon says "Evolution isn't random."

Fair enough. Then evolution, whatever it may be, has nothing to do with the development or variation of organisms.

You have two choices. Either there is random evolution or there is the opposite of random. In other words, design!

Yes, Jon, what happens in organisms is NOT random, which is where you are in trouble. Everything we now know about the inner workings of organisms is intricate design.

Also, Sir Isaac Newton did suffer from mercury poisoning, so if he tended towards a Unitarian view of God later in life God will cut him some slack. He did, after all, acknowledge Christ as Savior and God as Creator earlier in life and in fact studied first for the ministry.

Wow, evolution is not random...wow! Evolution is being assigned the creative powers and ingenuity of God now? Not surprising I guess. Hawking gives gravity the God throne. Why shouldn't Darwinists be consistent?

Darwinists pretend that Evolution is a random and undirected process and then give it credit for all designs and beauty found in nature. Evolution in the mind of a Darwinist is indistinguishable from God in terms of power but unlike God, a transcendent eternal being Who continues to uphold the Universe and is not bound by material temporal existence, Evolution is an uncaring force that despite having no explanation or reason or purpose is given credit for creation powers.

Somebody find me an honest Darwinist who will admit that he believes that random uncaring forces formed the Universe or let him unveil the pagan nature of deity he worships in place of God. If the Universe and the entire Solar System and the Earth and all of life did not *poof* accidently into existence then there had to be purpose and plan, therefore, a design. A design needs a Designer, a selection needs a Selector, no free lunch!

Umm... Radar... I'm pretty sure that actively demonstrating just how much you don't understand evolution isn't going to win you too many debates. You act like this is the first time anyone has told you that evolution isn't random. As you've been advised many times, only YEC's say that evolution is random. And it's pretty clear YEC's don't know the first thing about evolution.

-Canucklehead.

Pretty clear evolutionists don't know the first thing about evolution.

Your problem is this, the first time you agree that there is anything other than random going on in evolution you have agreed with design. Once you accept design you will have to designated a Designer. So please keep telling us evolution is not random so the world can figure out that evolution is not...period!

Good thing I reviewed these comments. Woolf, for instance, gives evolution credit for being an engineer? Seriously! Magic!

"Your problem is this, the first time you agree that there is anything other than random going on in evolution you have agreed with design. "

When sediment is sorted by running water, that's evidence of design?

When a storm kills birds with breastbone length x but not those with breastbone length 1.01x, that's evidence of design?

The existence of a timberline on any high mountainside is evidence of design?

Radar, you really have a genius for making your side sound even less intelligent than it actually is.

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