Here's a whopper! "Of course DNA isn't random, any more than river drainage patterns or volcaic eruptions are random. But that doesn't mean any of those things are intelligently designed. The laws of physics and chemistry produce non-random patterns too."
After four lies have been debunked today, time to correct a misunderstanding. I will not say the commenter was lying this time, he is simply ignorant. So we can correct that!
No, DNA and the specified information inside is not a naturally occurring "non-random pattern" and we'll address this in two ways. First to demonstrate that the information in DNA is specific and intentional. Second to demonstrate that "random" patterns are actually not random, but predetermined by God to express numerical sequences.
The Long Answer:
When intelligent agents act they produce specified complexity. We know this because we understand that when intelligent agents act, they use choice. An essay by William Dembksi lays out in detail how we can understand the products of intelligent design by examining how designers work:
To see why CSI [complex-specified information] is a reliable indicator
of design, we need to examine the nature of intelligent causation. The
principal characteristic of intelligent causation is directed
contingency, or what we call choice. Whenever an intelligent cause acts,
it chooses from a range of competing possibilities. This is true not
just of humans, but of animals as well as extra-terrestrial
intelligences. A rat navigating a maze must choose whether to go right
or left at various points in the maze. When SETI (Search for
Extra-Terrestrial Intelligence) researchers attempt to discover
intelligence in the extra-terrestrial radio transmissions they are
monitoring, they assume an extra-terrestrial intelligence could have
chosen any number of possible radio transmissions, and then attempt to
match the transmissions they observe with certain patterns as opposed to
others (patterns that presumably are markers of intelligence). Whenever
a human being utters meaningful speech, a choice is made from a range
of possible sound-combinations that might have been uttered. Intelligent
causation always entails discrimination, choosing certain things,
ruling out others. Given this characterization of intelligent causes,
the crucial question is how to recognize their operation. Intelligent
causes act by making a choice. CSI is a reliable indicator of design because its recognition coincides
with how we recognize intelligent causation generally. In general, to
recognize intelligent causation we must establish that one from a range
of competing possibilities was actualized, determine which possibilities
were excluded, and then specify the possibility that was actualized.
What's more, the competing possibilities that were excluded must be live
possibilities, sufficiently numerous so that specifying the possibility
that was actualized cannot be attributed to chance. In terms of
probability, this means that the possibility that was specified is
highly improbable. In terms of complexity, this means that the
possibility that was specified is highly complex. All the elements in
the general scheme for recognizing intelligent causation (i.e.,
Actualization-Exclusion-Specification) find their counterpart in complex
specified information-CSI. CSI pinpoints what we need to be looking for
when we detect design. (Intelligent Design as a Theory of Information by William Dembski)
In summary, Dembski notes that intelligent agents can choose from one of
many competing possibilities. If the choice made is unlikely to occur
and sufficiently complex, then we can attribute that choice to design.
This comes from our understanding of how intelligent agents operate--not
from a negative argument against evolution. In The Design Inference, Dembski lays out a three-part "user-friendly" explanatory filter which we can use to detect intelligent design:
This explanatory filter recognizes that there are three causes for things: chance, law and design. The premise behind the filter is the positive prediction of design that designers tend to build complex things with low probability that correspond to a specified pattern. In biology, this could be an irreducibly complex structure which fulfills some biological function. This filter helps ensure that we detect design only when it is warranted. If something is high probability, we may ascribe it to a law. If something is intermediate probability, we may ascribe it to chance. But if it is specified and low probability, then this is the tell-tale sign that we are dealing with something that is designed. In these high information-situations, intelligent design theorist Stephen C. Meyer also emphasizes why intelligent design is the right explanation:
“
"Experience teaches that information-rich systems … invariable result
from intelligent causes, not naturalistic ones. Yet origin-of-life
biology has artificially limited its explanatory search to the
naturalistic nodes of causation … chance and necessity. Finding the best
explanation, however, requires invoking causes that have the power to
produce the effect in question. When it comes to information, we know of
only one such cause. For this reason, the biology of the information
age now requires a new science of design. (Stephen C. Meyer, Mere Creation, pg. 140).
“ "Indeed, in all cases where we know the causal origin of 'high information content,' experience has shown that intelligent design played a causal role." (Stephen C. Meyer, DNA and Other Designs)
"Intelligent design provides a sufficient causal explanation for the origin of large amounts of information, since we have considerable experience of intelligent agents generating informational configurations of matter."(Meyer S. C. et. al., "The Cambrian Explosion: Biology's Big Bang," in Darwinism, Design, and Public Education, edited by J. A. Campbell and S. C. Meyer (Michigan State University Press, 2003)
Intelligent design is thus a cause sufficient to produce the high levels
of information, i.e. irreducible complexity, found in biology.
Intelligent design is not merely a negative argument against evolution,
but is inferred because of its positive predictions of how we understand
designers to operate.
There are other examples of mutually exclusive predictions of design and descent, as is explained in the tables below. In each example, intelligent design is inferred because it makes positive predictions that match the evidence, despite the fact that descent makes the exact opposite prediction (which is not met by the evidence).
Inferring Intelligent Design using its Positive Predictions:
There are thus various examples where design makes positive predictions, but Darwinian evolution coincidentally makes the exact opposite prediction. Design proponents do not argue against evolution merely because that is what proves design, but because in these special cases, the falsification of evolution also entails a matched positive prediction of intelligent design theory, because intelligent design predicts the exact opposite of evolution. We thus detect intelligent design through findings its positive predictions based upon the way we understand intelligent agents to operate.
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Also, random patterns are NOT random. The Universe is designed to produce patterns that can be identified as non-random expressions of mathematical progressions. Random is NOT random.
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.
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.
The way to find if a number belongs to the Mandelbrot set is to put it through a particular formula (the details are shown in the In-Depth box in this article). In this way, we can check every possible number to see if it belongs to the Mandelbrot set, and then plot the results on a graph. We color the point black if it does belong to the Mandelbrot set; we give it a different color if it does not. For example, in Figure 1 we can see that the numbers 0 and -1 are part of the Mandelbrot set, whereas the number 1/2 is not.
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.
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.
No, DNA and the specified information inside is not a naturally occurring "non-random pattern" and we'll address this in two ways. First to demonstrate that the information in DNA is specific and intentional. Second to demonstrate that "random" patterns are actually not random, but predetermined by God to express numerical sequences.
There was a lot to chew on in my last post and I think that some people will not have read down to this so I am presenting it alone with a very short introduction.
"There are thus various examples where design makes positive predictions, but Darwinian evolution coincidentally makes the exact opposite prediction. Design proponents do not argue against evolution merely because that is what proves design, but because in these special cases, the falsification of evolution also entails a matched positive prediction of intelligent design theory, because intelligent design predicts the exact opposite of evolution. We thus detect intelligent design through findings its positive predictions based upon the way we understand intelligent agents to operate." - from article, below
Dear readers - Evolution was an hypothesis that Charles Darwin admittedly put together using the work of many other researchers, including Blyth and Wallace and Hutton. The problem with Evolution is that it is entirely backwards. It is like someone trying to explain gravity and suggesting that gravity causes a ball to roll uphill. But we know that, in normal circumstances, balls roll downhill and that is what we find in organisms. Organisms are losing genetic information and picking up harmful mutations along the way over time. Darwin had it backwards, it is devolution that is operating in organisms today and we will see as science investigates the cell further that the impossibility of the ball rolling uphill will become more and more obvious!
FAQ: How do we Detect Design?
| The Short Answer: We detect design by looking for the tell-tale signs that an intelligent agent acted. Intelligent agents tend to produce specified complexity when they act. We can then seek to detect design by looking for that specified complexity. Using an "explanatory filter" helps us to use normal logic to infer where design was a cause involved in creating an object. Design also could makes other predictions which can also help us to detect design. |
The Long Answer:
When intelligent agents act they produce specified complexity. We know this because we understand that when intelligent agents act, they use choice. An essay by William Dembksi lays out in detail how we can understand the products of intelligent design by examining how designers work:
This explanatory filter recognizes that there are three causes for things: chance, law and design. The premise behind the filter is the positive prediction of design that designers tend to build complex things with low probability that correspond to a specified pattern. In biology, this could be an irreducibly complex structure which fulfills some biological function. This filter helps ensure that we detect design only when it is warranted. If something is high probability, we may ascribe it to a law. If something is intermediate probability, we may ascribe it to chance. But if it is specified and low probability, then this is the tell-tale sign that we are dealing with something that is designed. In these high information-situations, intelligent design theorist Stephen C. Meyer also emphasizes why intelligent design is the right explanation:
“ "Indeed, in all cases where we know the causal origin of 'high information content,' experience has shown that intelligent design played a causal role." (Stephen C. Meyer, DNA and Other Designs)
"Intelligent design provides a sufficient causal explanation for the origin of large amounts of information, since we have considerable experience of intelligent agents generating informational configurations of matter."(Meyer S. C. et. al., "The Cambrian Explosion: Biology's Big Bang," in Darwinism, Design, and Public Education, edited by J. A. Campbell and S. C. Meyer (Michigan State University Press, 2003)
There are other examples of mutually exclusive predictions of design and descent, as is explained in the tables below. In each example, intelligent design is inferred because it makes positive predictions that match the evidence, despite the fact that descent makes the exact opposite prediction (which is not met by the evidence).
Inferring Intelligent Design using its Positive Predictions:
| Table 1. Ways Designers Act When Designing (Observations): |
| (1) Take many parts and arrange them in highly specified and complex patterns which perform a specific function. (2) Rapidly infuse any amounts of genetic information into the biosphere, including large amounts, such that at times rapid morphological or genetic changes could occur in populations. (3) 'Re-use parts' over-and-over in different types of organisms (design upon a common blueprint). (4) Be said to typically NOT create completely functionless objects or parts (although we may sometimes think something is functionless, but not realize its true function). |
| Table 2. Predictions of Design (Hypothesis): |
| (1) High information content machine-like irreducibly complex structures will be found. (2) Forms will be found in the fossil record that appear suddenly and without any precursors. (3) Genes and functional parts will be re-used in different unrelated organisms. (4) The genetic code will NOT contain much discarded genetic baggage code or functionless "junk DNA". |
| Table 3. Predictions of Darwinian Evolution (Hypothesis): |
| (1) High information content machine-like irreducibly complex structures will NOT be found. (2) Forms will appear in the fossil record as a gradual progression with transitional series. (3) Genes and functional parts will reflect those inherited through ancestry, and are only shared by related organisms. (4) The genetic code will contain much discarded genetic baggage code or functionless "junk DNA". |
| Table 4. Comparing the Evidence (Experiment and Conclusion): | ||||
| Line of Evidence | Prediction of Darwinian evolution | Prediction from intelligent design | Data | Best explaining hypothesis: |
| 1. Biochemical complexity | High information content machine-like irreducibly complex structures will NOT be found. | High information content machine-like irreducibly complex structures will be found. | High information content machine-like irreducibly complex structures are commonly found. | Design. |
| 2. Fossil Record | Forms will appear in the fossil record as a gradual progression with transitional series. | Forms will appear in the fossil record suddenly and without any precursors. | Forms tend to appear in the fossil record suddenly and without any precursors. | Design. |
| 3. Distribution of Molecular and Morphological Characteristics | Genes and functional parts will reflect those inherited through ancestry, and are only shared by related organisms. | Genes, DNA sequences, and functional parts will be re-used in different unrelated organisms. | Genes and functional parts often are not distributed in a manner predicted by ancestry, and are often found in clearly unrelated organisms. | Design. |
| 4. Biochemical Functionality | The genetic code will contain much discarded genetic baggage code or functionless "junk DNA." | The genetic code will NOT contain much discarded genetic baggage code or functionless "junk DNA." | Increased knowledge of genetics has created a strong trend towards functionality for "junk-DNA"; examples of DNA of unknown function persist, but function may be expected or explained under a design paradigm. | Design. |
There are thus various examples where design makes positive predictions, but Darwinian evolution coincidentally makes the exact opposite prediction. Design proponents do not argue against evolution merely because that is what proves design, but because in these special cases, the falsification of evolution also entails a matched positive prediction of intelligent design theory, because intelligent design predicts the exact opposite of evolution. We thus detect intelligent design through findings its positive predictions based upon the way we understand intelligent agents to operate.
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Also, random patterns are NOT random. The Universe is designed to produce patterns that can be identified as non-random expressions of mathematical progressions. Random is NOT random.
Fractals, Fibonacci Numbers, Patterns, Design...All of the Universe attests to a Designer God.
Words and music. Words first!
Fractals
Hidden Beauty Revealed in Mathematics
by Jason Lisle, Ph.D.
January 1, 2007
Featured In
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?
Figures 1 & 2 (click to enlarge)
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.
Figure 3 (click to enlarge)
Figure 4 (click to enlarge)
Figure 5 (click to enlarge)
Figure 6 (click to enlarge)
Figure 7 (click to enlarge)
In-Depth
The formula for the Mandelbrot set is zn+1 = zn2 + c. In this formula, c is the number being evaluated, and z is a sequence of numbers (z0, z1, z2, z3…) generated by the formula. The first number z0 is set to zero; the other numbers will depend on the value of c. If the sequence of zn stays small (zn ≤ 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 zn 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.
holds a Ph.D. in astrophysics from the University of Colorado at Boulder and is a popular author for Answers in Genesis-USA. Dr. Lisle uses his knowledge of the heavens and his biblical perspective to proclaim the handiwork of God in lectures, including 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 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 zn+1 = zn2 + c remains bounded.[1] That is, a complex number, c, is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn 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 i2 = −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..."