Category Archives: Math

Spurious Correlations With Time Series: What We Can Learn

There is a viral website being passed around. Harvard Law student Tyler Vigen made an online tool that allows one to choose a time series and to choose another time series that it is potentially highly correlated with. This has inspired many people to make graphs of time-varying variables that have little to do with each other but are still highly correlated.

For example, the amount of money spent on pets and the number of people who died by falling down the stairs may have little to do with each other, yet there is almost a perfect correlation of r=0.97.

money-spent-on-pets-us_people-who-died-by-falling-down-the-stairs

The media has covered the popularity of Vigen’s website through a variety of angles. Most of them suggest that we shouldn’t be too quick to trust graphs and that, once again, correlation is not causation.

I agree fully that we should be critical of charts and graphs, that we should be skeptical of their assumptions, and that we should look into their hidden meanings. There’s nothing wrong with that sentiment of skepticism.

I also agree that, on the most basic and technical level, correlation is not causation. What I think is less commonly understood is the fact that correlation is actually evidence for causation, since causation correlates with correlation.

But there are deeper and more more nuanced truths about these types of graphs that people tend to gloss over. It has to do with the fact that these data are time series data. Time series data have certain properties that are unique. Correlating time series is not the same as correlating non-time-varying data.

Why do we find spurious correlations?

The first and obvious thing to note is that times series can be misleading because they appear to be authoritative in their sample size. The graph above shows data from 2000 to 2009, which seems like a long time, but is actually only 10 data points. Correlations merely describe the association of two directional relationships from the respective means; they don’t even begin to tell us whether we should put much confidence on whether the association really exists.

A better way to do this would be to use a statistical test for your correlation value or to run a regression. If your regression coefficient is positive, you might be onto something.

But with time series data, that could still be wrong.

You see, times series is special; realizations of variables are not independent and can evolve according to a specific process across time. There’s a very peculiar process that is known to generate spurious correlations.

It’s called a random walk. It’s literally like walking around in random directions over time.

The most basic form of this data generating process can be described as the following:

\\x_{t} = x_{t-1} + \epsilon_t

In other words, we can make a random walk by choosing some initial value of x. The next period’s value will be the current period’s value added to a realization of a value from a normal distribution centered at zero. I repeat this to get my entire series. Since the values from the normal distribution are random, I can’t predict which way I’m going to go; hence, the random walk.

This is an example of a random walk, with initial value at zero and disturbances from a standard normal:

Graph

Long story short, it is hard (impossible) to predict which way the series will go. Here’s the million dollar question:

What happens when we look at the correlation of two completely unrelated random walks? For this exercise, I simulate two completely separate random walks of 100 time periods long. I calculate the correlation them and save that number. I do this 10,000 times and plot the distribution of the correlations. If it is true that two series are completely unrelated to each other, then we might expect that their correlations are zero, so the distribution should be tightly centered around zero, and not going anywhere close to 1 or -1. However, the distribution from the simulation looks like this:

distribution

It isn’t tight around zero at all! In fact, it is pretty common to get values over 0.7 or under -0.7, and extremely common to get values over 0.5 or under -0.5. A huge number of these things are spurious correlations, and we made it by merely relating two things that are most definitely not related.

Why does this trick work? Because correlations are covariances divided by variances. However, the variance of a random walk is infinite; if you walk around randomly, there’s no pressure to return to where you started from. Dividing by an infinite value is taboo by math standards. The correlation you calculate with completely separate random walks does not make sense.

It gets even worse. There are things called random walks with drift and random walks with trend. The former has been commonly used to describe prices of things (like stocks), which are unpredictable in the short run but predictable in the long run. The positive “drift” term basically means you can make money in the long-run. For example, the S&P 500 looks like a random walk with positive drift.

But the drift term, represented by a constant alpha, can be anything, positive or negative. It is just added to the series in every time period.

\\x_{t} = \alpha + x_{t-1} + \epsilon_t

So the next exercise is similar to the last one. Again, I make two completely separate random walks with the same parameters, but this time, I choose an an alpha value by taking a draw from a normal distribution of standard deviation 0.5 before I simulate each random walk. The distribution of the correlations for 10,000 correlations is described below.

distribution2

That’s right. It gets a lot worse. Not only are the values far from zero, but getting strong correlations near 1 and -1 is actually more common than not. This is because the drift term is chosen beforehand; if the two drift terms are of the same sign, then the two series tend to go the same direction. If the two drift terms are of opposite sign, the two series tend to go opposite directions. This phenomenon generates strong correlations, yet they are spurious correlations.

So what’s the solution to this?

First of all, we want to get rid of the infinite variance that is associated with the random walk. What happens when we make the series “mean revert” back to a constant? It turns out, if we have the following specification, the variance is no longer infinite.

\\x_{t} = 0.5x_{t-1} + \epsilon_t

This is because the 0.5 multiplicative factor in front of the lagged values of x “pulls” the series towards zero at each time period. Specifically, it halves the previous period’s value.

As it turns out, just doing this makes the distribution of the correlations much closer to what we want. In fact, a lot of the spurious correlations between two unconnected time series simply go away.

distribution3

So the bottom line is that it only makes sense to calculate correlations of time series that look a lot like the mean-reverting process described above, rather than the random walk and drifting processes described above. The problem with “money spent on pets” and “people who fall down the stairs” is that they may both be random walks with drift. (I’m not so sure about random walks with deaths, but there’s definitely drift.) The same reasoning goes for a lot of the series on Vigen’s websites.

If we suspect drift and/or random walk, the way to solve this problem is to first-difference the series. In other words, we generate a whole new series that is made of the difference between the current period’s value and last period’s value. In our random walk with drift process, it would essentially cancel out the x values AND the alpha values, leaving us with the difference in the realizations of epsilons, which is random. Correlating the two differenced series from two different random walks with drift will give us very close to zero correlation. On the other hand, if the two series are connected in some way; e.g., if the epsilons are truly correlated, then we should be able to detect it.

In short, there’s a lot going on with time series. Don’t believe every graph you see, but don’t dismiss the importance of correlation in statistics either.

Tolerating Imperfection

It wasn’t too long ago that I came upon an evangelism worksheet that posed the following question: if God is perfection, where in this 2-d space would you place yourself? With each subsequent question and scenario, the reader was supposed to draw stick figures. Pretty soon, the scenario became pretty predictable: there was an unbridgeable gap between you and God. The message? You deserved to burn in Hell forever, of course.

atheist_hell

Humanists approach the question of perfection differently. Contrary to popular accusation, we do not celebrate the fact that we are not perfect people. In fact, many of us will admit that the world is a dark place. Our minds are irrational, our societies are broken, and sometimes we do horrible things to each other. Indeed, we should fight for change.

Instead, humanists approach imperfection with a degree of proportion. You said a bad word when you were seven? You stole some money when you were eight? You shouldn’t have done it, but humanists don’t think you deserve to go to Hell.

We also see things in terms of the sentient creatures that are involved and what beings can be harmed, not in absolute theological terms. You think that contraception and/or gay sex is a sin? We beg to differ.

We recognize that rational pursuit of the goal to become better people means we should think critically about justice and punishment. We have to understand there are diminishing returns to enforcing moral values, and that telling people they will face much more “justice” than they deserve is downright cruel.

Vocabulary is tricky. I guess the idea I’m promoting isn’t that we should accept our imperfection and do nothing about it. It’s that we should tolerate them. It’s that we should approach them rationally in accordance with the principle of a compassionate understanding of human limitations.

Of course, you can read and think about this all you like, but there’s nothing like a Tim Minchin song to sum up the ethos of this post:

Reductionism: Still Awesome, Still Misunderstood

I realized that there’s a thing called “missing the point” about “missing the point”. After all, I kind of knew that my defense about the pure awesomeness of reductionism would be misconstrued  into something much much worse. (But who knew it would be my friend Andrew Tripp, of the Depaul Alliance for Free Thought?)

Such is the reality of internet discussion. So let’s clear up some of the charges.

For instance, Mr. Mei seems to hold the belief that at some point that I said reductionism, namely the understanding that everyone and everything is made up of atoms, is a horrible evil viewpoint that causes children to have nightmares. Or something.

Actually, not quite. As even the original author of my chosen definition of reductionism has said, such use of diction (i.e., “evil evil belief”) are for comic effect (to satirize a culture that seems to consistently misunderstand the concept). Also, I have to point out that that only a passing reference was made to Andrew’s post, and that the post itself was not an ad hominem, or even a specific reply, to what Andrew wrote. (If anything, I was more interested in the replies of a bunch of philosophically-minded Christian Facebook friends). But let’s move on.

I would love, LOVE for him or anyone else to show me where I said that.

By posing this question, it seems that Andrew doesn’t seem to understand what I thought the most distasteful part of his original post was, or what I’m criticizing and not criticizing. And the pride by which he innocently quotes Dan Fincke again in his reply (which is what I was objecting to) seems to confirm my point. Here’s the bulk of the relevant quotation, with my added bolding for emphasis:

There is a tendency to talk like the only level of explanation that is at all meaningful is on the physics level. Now, of course everything in our experience is ultimately physical and made up of atoms, which are further composed of subatomic particles. But that does not mean that atoms are the only level on which true things can be said. Those atoms combine in remarkably complex patterns that give rise to the objects of study in chemistry, biology, psychology, and sociology. Those emergent patterns are real. It’s not like in biology we say, “There’s no such thing as evolution because this organism and its descendants are really still just patterns of atoms”. The differences in the patterns of atoms that make up one organism and its offspring are significant. They are worth saying there is something new evolved in nature when an organism is distinct enough in the patterns of its properties from its ancestors. These are real subjects of study. Real differentiations in nature.It would be stupidity to judge those patterns as somehow artificial simply because there is a way to conceptualize the organisms in purely atomic terms that pay no attention to the features that are interesting on the biological level.

The biggest point here is that both philospher (Dan) and social activist (Andrew), by writing this piece or quoting it as if it were accurate, seem to really miss the point about the mere reality of reductionism. For one, they seem to think that reductionism means “judging patterns as somehow artificial”, and that somehow thinking about things atomically destroys features that are interesting on higher levels. However, that doesn’t seem to be what reductionism actually is, which is incidentally covered on my Number Three point about how “maps are really important”. I don’t just think this is just bad philosophy; it’s also quite misleading to a general audience.

But I think Andrew’s beef with the overall secular community is the following:

How does this work with reductionism, then? Well, take that viewpoint and mix it with the sorts of upper-crust white academics who sit at the top of the atheist movement…

…Thus, we have the movement’s near-total lack of engagement with issues of race, gender, and institutional violence.

This is what I refer to when I criticize over-the-top reductionism. Not reductionism itself, but the ability of it to intersect with old notions of color-blindness to allow otherwise rational atheists to ignore issues affecting marginalized communities.

Now this is just confusing. After proudly quoting an entire Dan Fincke paragraph about the mistaken idea that “reductionism” is about ignoring the importance of human-level maps, and after writing an entire post using the word “reductionism” straight-up, only by the end of Andrew’s reply do we get somewhat of a conditional statement: so we’re not talking about true reductionism but (fake?) over-the-top reductionism.

And hence my point on Number Two, lampooning how people think atheism causes mass murderers, or how Darwinism leads to racism. Somehow Andrew also missed the point here: the thing is, if people want to offer a critique of the misuse of a particular concept, they have to be careful.

Just as eugenicists don’t deserve to be described as “Darwinists” (and in my humble opinion “social Darwinist” is really really pushing it), people who think reductionism implies race-blindness (or any other normative position, implicit or explicit) don’t deserve to be described as “reductionists”. At the very least, we should use language that makes this distinction clear, not wobble around with confused definitions and popular misconceptions. After all, we don’t say of 1940’s America that too many (white) scientists “have an annoying tendency to be Darwinists“.

And to reiterate my points so that everything is really really clear.

1) Human-level maps are really really important. It means that our mental conceptions matter. It means that we have to work towards social justice. It means that race, gender, sexual orientation, political institutions, and the issues and correlations that tie everything together really matter.

2) You have to be careful about what you say is “real”. Especially when you’re talking about reductionism, saying that the patterns of “biology” are real and that you can’t describe it on a lower-level is really bad philosophy. Also don’t fall for the Mind Projection Fallacy, and wear lots of sunscreen.

3) Reductionism is still awesome, beautiful, and unbelievably cool. I think I covered this in my original post.

4) All we need is love. I dunno. I just felt like writing that.

Is There a God? Stephen Hawking

I can’t believe I missed this documentary. Very good scenes, lots of dramatic CGI work.

Can atheists make logical or moral arguments without the Christian God?

I normally would not bother with such questions, but I hope that the beauty of logic (and mathematics) could be appreciated more. Also, hopefully we can collectively learn to be more familiar with popular logical fallacies. One can start by learning about the more classical arguments for the existence of God. These include the Cosmological Argument and the Ontological Argument.

It was Bertrand Russell who noted of such arguments like the Ontological Argument that it “does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies.”

Russell’s observation could very well be true for many of the newer, less classical apologetics from very obscure theological traditions (that seem incidentally quite isolated from the larger evangelical community). One argument is called the Transcendental Argument for the Existence of God (TAG). It is known not for its popularity but for its ability to become very long and convoluted.

Unsurprisingly, there are many distinct formulations of this argument that one can find at places like CARM. One of the more popular ones, formulated by Matt Slick, has inspired many refutations which you can both read and watch. For the purposes of this post, I will focus on a less formal version covered on this blog. It’s a rehash of the ideas of a relatively little known theologian named Cornelius Van Til.

The argument is given below with cute diagrams.

The Transcendental Argument for the Existence of God

The atheist offers many criticisms of Christianity. There must be a basis for such criticisms.

Truth claims like “Christianity is not true” employ logic, science, and ethics.

Logic, science, and ethics need to be accounted for. At the base of logic, science, and ethics are transcendental truths about reality. These truths, like the idea that the universe is consistent across time (see Hume’s Problem of Induction), that the law of noncontradiction holds, etc. are transcendental because they are not contingent on experience or consciousness.

How do we account for what is true? EITHER we are autonomous beings capable of understanding truth OR God decides what is true. This is the position of “non-neutrality”. Christians and atheists cannot agree on anything because the very idea of truth they hold is different.

As it turns out, the diagram above is wrong. It is impossible for a secular worldview to account for the transcendent absolutes that are the foundations of logic, science, and ethics. Autonomous man is trapped in his own experience and cannot by itself rationally justify the transcendental absolutes that allow for logic, science, and ethics.

The only alternative left is a Christian God who accounts for the transcendent absolutes that give rise to logic, science, and ethics.

For the Christian, everything is what it is because God says – and is – so… God is thus the prior and authoritative interpreter of all facts, and the truth of a proposition is equivalent to how well it conforms to God’s interpretation of the facts.

Therefore, the atheist who uses logic, science, and ethics to argue against Christianity presupposes that Christianity is true. This cannot be.

This argument, if valid, is a very damning one. By the author’s own admittance, this would throw out not only all secular criticism of Christianity, but also the arguments of Christians who argue for the existence of God from a neutral point of view (one that says we agree with atheists on some things.)

It has a further implication.

The argument I provided doesn’t a priori prove the truth of Christianity, it just says it’s not rational to assert anything else to be true.  You could think that nothing is true, including the statement “nothing is true.”  You would be left with radical nihilism, which rejects that the meaninglessness of truth claims is a reason for rejecting them, and can’t assert anything to be true, even itself.

It builds a strong choice between Christianity and absolute atheistic nihilism. It means either Christianity is true, or everything we know is meaningless. It would mean that atheists cannot make objective truth claims, and therefore cannot make a logical or moral argument without the Christian God.

Imagine that the only restaurant in town is one that serves only nihilism and Christianity, and you can’t order both.

Where does the fallacy lie?

There are many troublesome and outright fallacious parts of this argument. Many of these objections are interconnected, but we will start with an obvious one.

The most noticeable thing is that the Christian God is not well-defined as a solution. That is, there is no good reason to think that God solves the problems presented by TAG to atheists.

The Christian God, under almost all interpretations, cannot do evil. It is therefore also bound by the ethical absolutes and is therefore neither independent of ethical absolutes nor a possible interpreter of such “facts” because he cannot decide otherwise. Also the principle that God cannot cease to be God (or cease to be perfect, moral, loving, etc.) because of his nature (consistent with the law of noncontradiction) is also a transcendental absolute that limits God’s independence. This Christian God, bound by transcendental absolutes, when used in an argument like TAG, is clearly an example of referring the problem upward. In short, one must still explain why the absolutes transcend God. It is clear that the proposition of God in TAG is merely a semantic slight-of-hand to try to avoid explaining what is still not explained. If the rest of TAG is valid (which it is not), the Christian has the same problem as the atheist because there is no real account of the absolutes that God is also subject to.

Non-neutrality is nonsensical.

The false dilemma given in TAG is either that truth is God’s truth as he interprets it, or truth is what humans seek to interpret it as. Both of these positions on “truth” are nonsensical because, by citing a proper “authority” on truth, they make truth contingent on conscious minds (either God’s mind or human minds). The only working idea of truth is that it is independent of all minds (not just of human minds). The objectivist formalization of this principle is called the Primacy of Existence. Truth is neither decided nor created by humans; rather, truth is what is according to reality, and reality is that which is primary over consciousness.

The dichotomy of Autonomous Man vs. Christian God is therefore not only a false dichotomy, but also nonsensical.

The argument uses a fallacy of equivocation on the words “interpretation” and “truth”. When TAG says that what is true is what God interprets it to be, the direct meaning is that God decides what is true. This is utterly and blatantly confused with the concept of human interpretation of truth, which is not a process of deciding truth, but is rather an exercise of trying to understand what is true in the objective world; it is a mental exercise. Therefore, denying atheists the right to understand the objective world while allowing Christians the ability to understand God’s interpretation is a dishonest form of special pleading.

The assumption of truth-making “authority” is an incorrect premise.

TAG uses the connotation of the word “authority” and presupposes that there is something that grants authority to make truth claims (Man or God). This premise is not only unproven but is also clearly incorrect. Because the purpose of TAG is to show that only by assuming the Christian God can we make valid truth claims, we can merely show how valid truth claims can be made without the need to even mention authority.

As noted above, I believe that the assumption that Existence is Primary is necessary to make any truth claims. In fact, Christianity is nonsensical if the Primacy of Existence is not assumed. Otherwise, God’s existence would be contingent on God (or something else) deciding that he himself exists, which contradicts the fact that God must be eternal.

On the contrary, if we assume that the existence of all things is what it independently is; that is, all interpretations of X do not affect the actual existence of X. In this context, valid truth claims are simply made based on how well they conform to existence; they do not require ill-defined concepts like “authority” and “autonomy”.

TAG fundamentally misunderstands logical systems.

TAG at its core asks that logical systems must be fully accounted for, that is, they must be consistent and complete. Its solution of the Christian God, as I have shown, is not a solution but an incoherent deux ex machina. But a more fundamental problem is that it is searching for a solution to a problem that probably has no solution (see Kurt Godel).

Neutrality is correct.

There are two main reasons why neutrality is correct (and why non-neutrality is wrong). The first is that TAG fundamentally does not allow for probabilistic arguments. The need for a completely satisfactory answer to the Problem of Induction, for example, is a search for logical certainty about the validity of the inductive method (and therefore logical certainty that the sun will rise tomorrow given enough experience, for example). This is not only inconsistent with the spirit of scientific inquiry, but it also mistakenly excludes positions of neutrality.

Secondly, the Christian cannot be consistent without agreeing with many of the assumptions that atheists commonly hold. As in all logical systems, it is impossible to derive logical truth without some unprovable assumption. Atheists choose to believe assumptions like the Primacy of Existence and the reality that we are not Brains in Vats. Christians also believe this too, but also in addition to other theological claims that I have shown are neither necessary nor coherent.

TAG assumes neutrality. 

TAG is made to convince the atheist that there are only two justifiable choices (nihilism and Christianity) based on commonly agreed knowledge. If it did not assume neutrality, then all attempts to refute TAG would automatically be wrong because the atheist cannot assert it to be wrong. TAG would then be a logical fallacy called begging the question.

Further comments and fallacies

Christianity does not solve the problems given by inductive skepticism; instead it exacerbates the problem. Hume’s critique of induction is that given that we see that the sun rises in the morning everyday, there is no reason to think that the sun actually rising tomorrow will happen rather than its negation, that is, the sun not rising tomorrow. Christianity, incidentally, says that it is indeed possible for the conclusions of induction to be invalid. That is, at any time, and at any moment, God may choose to suspend the laws of nature. This has happened not only at events like Joshua’s halting of the sun and , but also at events fundamental to the truth of Christianity, like the Resurrection of Jesus. Christianity itself is not consistent with the uniformity of the Universe.

Introducing a new form of pseudoscientific model that says the universe is uniform as long as God says otherwise is not only another form of special pleading, but it renders science as we know and use it today completely irrelevant (hence pseudoscientific). A scientific and inductive method that is accepting of supernatural phenomenon renders science absurd and opens the floodgates to unfalsifiable explanations like intelligent design and faith-healing. Christianity cannot account for science as we know it.

Let us ignore all the above problems and grant that TAG is 100% valid.

TAG does not hint at the existence of God and is therefore poorly named. TAG merely questions the logical absolutes that we hold and asks for an account of them. The proper conclusion of TAG (assuming it is 100% valid, which it is not) is that either that the Universe is completely nihilistic or that there is some mind accounting for logic, science, and ethics. This mind does not have to be a “god” in the perfect, all-knowing, all-powerful sense. This entity merely needs to be capable enough to interpret facts and create a rational, objective universe where logic, science, and ethics are valid.

Let us go even further and assume that TAG is 100% valid and does actually hint at the existence of God.

Then this God is not necessarily Christian. To show uniqueness would require not only serious theological interpretation of all current ideas of God, but also all existing and future conceptions of God. Specifically, TAG does not show how this supposed “God” sent a son down to Earth. It does not demonstrate the validity of the Adam and Eve “metaphor” or the revelation that Jesus is returning one day. It does not show much of what is essential in the Apostle’s Creed and Nicene Creed to be true. TAG demonstrates almost nothing that is important in Christianity.

Therefore, even if we make the absurd assumption that TAG is valid, its final conclusion of reducing the only possibilities to nihilism and Christianity is a false dilemma, and a very bad one.

Final comments

It was Bertrand Russell, of course, who wrote that many of the Ontological arguments reduce to “bad grammar” and “bad syntax”. I think this is a good paradigm for many of these “arguments for God’s existence”.

I personally find it hard to believe that the Christian God spread his message so well as to send his own Son to die in an illiterate and superstitious part of Palestine, only to not give any reasonable argument for his existence to be passed on. Rather, we have questionable 20th century “experts” on theology to tell us exactly how to think on their fallacious grounds.

I agree with many Christians who say that God “transcends all reason”. He is illogical, incoherent, and so blatantly nonsensicalthat human comprehension is impossible (and perhaps human knowledge of his existence is unattainable). It is clear that belief in God requires faith, and, as Kierkegaard might note, a mega-gigantic leap of faith. After all, it is faith that gives “evidence for things not seen, and the substance of things hoped for.” In short, it makes people think there is evidence when there is not, and it gives things for people to believe just to fulfill their wishes.

I hope you enjoyed it. If you didn’t, the good news is that I’m never posting one of these apologetic refutations ever again.

Math, Infinity, and Beautiful Design

In his book The Blind Watchmaker, Dawkins discusses the appearance of design.

Biology is the study of complicated things that give the appearance of having been designed for a purpose.

In the context of evolution, the appearance of design makes sense. But I want to explore an area of geometry and mathematics that relates simple algorithms to beautiful appearances of design.

Welcome to the world of fractals. This is a world where I can draw a bounded snowflake that has a boundary of infinite length. It’s a world of paradox, a world where it doesn’t matter how big you are, because you won’t notice the difference when you zoom in. It’s a world of infinity, a world of endless iteration. More importantly, it’s a world of simplicity in terms of beginnings, and stunning beauty and complexity in terms of ends.

The Koch snowflake shows how simple the rules are for fractals to be constructed. One starts with an equilateral triangle. The next iteration involves the union with another triangle, which forms triangles on the periphery. Now we union the smaller triangles with rotated versions of themselves, which forms even smaller triangles on the periphery, and so on…

The resulting snowflake is not only aestheticallly pleasing, but it can also be shown to have infinite length.

—-

The most iconic figure of fractal geometry, however, is the Mandelbrot Set.

Few people know or understand how simple the rules are that define this set. Formally, the M-set consists of all the elements C in the complex plane such that the iteration Z_n+1 = (Z_n)^2 + C is bounded, starting with Z_o = (0,0).

Computers can calculate by repeating the iteration over and over for different values of C to see if the norm of Z_n goes toward infinity.

The resulting graph of the M-set is mesmerizing. It’s a set that features very complicated arrangements of symmetry and self-similarity.

The more beautiful renditions of the M-set are created by assigning colors to values that allow Z_n to diverge, depending on the speed of divergence. As we can see below, there are all kinds of intricate “designs” in these renditions. There are seahorses and islands, shells and spirals, peninsulas and antennas.

So what can computer generated graphics show about nature, evolution, and biology life? Do fractals have any useful function besides being “beautiful”?

1) Fractals allow things that work on a large scale to be reproduced on a smaller scale (or maybe vice versa). The heart pumps blood to larger blood vessels, which branch out to smaller vessels, to smaller capillaries, and so on, and each branching pattern mimics the level above it. No designer had to use his intelligence to think about each step along the way. No God had to draw a blueprint of the location of every tiny blood vessel. Fractals allow simple patterns that work to be expanded and repeated into smaller, more intricate settings without any need to think about them, without any design.

2) Fractals are nearly ubiquitous in nature; they are found in places ranging from snowflakes and seashells to leaves and mountains.

3) Fractals are optimal for many purposes. For example, it was proven mathematically that the self-similarity of fractals allowed them to serve as the one and only optimal solution for antenna design. The fact that so many living things have fractal qualities also suggests that natural selection favored the advantages that fractals gave to certain species.

Of course, the creationists are really looking for trouble here. They look at the word “infinity” and not only do they think “intelligent design”, they think Yahweh, Jesus, and one version of one book in particular.

And they trip all over their watchmaker-design argument. After all, everything that is complex and specified must have a clear designer, right? And what do they say when we can’t find a designer for such things as fractals?

Their conjectured unknowable and untouchable god was, in fact, the very knowable Judeo-Christian God of Creation. And His methods actually express themselves as geometry. But it is a geometry that the Greeks could scarcely have imagined.

Just assert and assert and assert the superiority of your God and your One True Religion over and over again.

Let’s give them a round of applause.

This. Is. Awesome.

Euler’s Identity

Due to a math problem set, I have absolutely no time to write a lengthy post today. But I’ll share what really rattled my mind in high school:

e^{i \pi} + 1 = 0\,\!

My first thought was that this identity doesn’t make sense. How can you take e to an imaginary power?

Anyways, the fun part is staring at it and appreciating all the elements involved here in their elegant simplicity:

i – an imaginary number

1 – the identity in multiplication

0 – the identity in addition

pi – the commonly used irrational number

e – the commonly used transcendental number

Now the not-so-fun part is proving it (from the power series definition of e^x). Good luck with that.

 

P.S. Actually both pi and e are transcendental.