Saturday, May 28, 2016

220: Cognitive BSes

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Hi everyone— before we start, just wanted to remind you, the Math Mutation book is out!   To order, you can follow the link at or just search for it on Amazon.    And if you ever pass through the Portland, Oregon metro area, I’ll be happy to autograph your copy.   If you like it, posting a positive review on Amazon would be really helpful.  Now on to today’s topic.

As you may recall, one of the topics we covered in some previous podcasts, and in the Math Mutation book, is the idea of “Cognitive Biases”.   These are well-known ways in which the human brain has a natural instinct to think in ways that violate basic laws of logic and mathematics.   One classic example is the Anchoring bias:  if asked a question that has a quantitative answer, you will tend to give an estimate close to numbers you recently heard.   For example, suppose I arrange separate discussions with two people to estimate how many listeners Math Mutation has.   With the first one, I start by asking “Does Math Mutation have more than 100 listeners, or fewer than 100?”.   But with the second one, I open with “Does Math Mutation have more than 1 million listeners, or fewer than 1 million?”   If I then ask both of them to estimate the total number of listeners, the first will probably come up with a much smaller estimate than the second, even though neither has any objective information to justify a particular number.   

After reading the chapter in my book, my old Princeton classmate Tim Chow pointed out that calling this a “Cognitive Bias” might not be justified.    Sure, the listener technically has no information to support the larger number in the second case— but in cases where we are talking to another human being, we trust them to provide relevant information.   This includes both direct statements of facts, and implications that might not be directly stated.   If I ask you whether Math Mutation has more or fewer than 100 listeners, I am implicitly communicating the information that the 100 number is pretty close, even though I have not rigorously declared this to be a relevant fact.   So if this number isn’t close, I have essentially misled you with false information— the fact that you trusted me and used the wrong number is my fault, not some flaw in your mental logic.    Thus, this “Cognitive Bias” is really a social manipulation.

Now, if you’re familiar with the literature on this topic, you might point out an interesting experiment that seems to refute this.   In this experiment, subjects saw a roulette wheel spin, then were asked the percentage of the United Nations countries that were in Africa.   Even though there is no logical reason for them to suspect the roulette wheel had advanced knowledge of geopolitics, their answers were still biased towards the results they saw on the wheel.   Many similar experiments have been carried out.   Doesn’t this provide irrefutable proof that this really is a cognitive bias?

Not so fast.   This is a very artificial situation.   Maybe when asked to guess a number about which they have absolutely no idea, they just grab any arbitrary number they can think of, which will tend to be one they saw recently.   They aren’t following some flawed cognitive process, they just don’t have any reason to pick any particular number.   Again, this doesn’t really indicate a mathematical flaw in their reasoning— they don’t think the number they picked has a particular logical justification.   Not knowing an answer, they just defaulted to what was at the top of their head.   

Most of the other well-established Cognitive Biases are open to similar criticisms.  Another example is the Conjunction Fallacy:   suppose I tell you that Joe is a Princeton mathematics graduate and chess champion, and then ask you to choose the more likely of two statements.  1.  “Joe is now a physics professor.”  2. “Joe is now a physics professor and head of the local Math Mutation fan club.”    You will likely choose option #2, since it seems like this kind of guy should be a Math Mutation fan.   But on reflection, option 2 must be strictly less likely than option 1, as it takes the same basic fact and adds an additional, more restrictive, condition.  But again, there is information being communicated between the lines:  if I give you those two choices, you probably interpret #1 as implicitly stating that Joe is NOT president of the Math Mutation fan club.   I didn’t say that, but the additional choice in the second option made this a very reasonable inference.   Once again, it can be seen as more of a social manipulation, where I leveraged typical communication conventions to imply something without actually stating it, and the implication is not strictly justified by mathematical logic.

We should point out, though, that even if the so-called “Cognitive Biases” are not truly flaws in the logic of the human brain, they are still important psychological effects to be aware of for many reasons.   For example, let’s take a look at some practical applications of the Anchoring bias.  It’s well known that when negotiating prices in business, your opening offer can set an anchor that affects the entire discussion.   Negotiating business contacts usually have some level of trust in each other, and taking advantage of this to establish a good anchor is a smart, though slightly manipulative, technique.    On another note, suppose you’re a surveyor trying to get accurate estimates in a survey or questioning experts on a difficult topic.   You need to be careful not to include some kind of number in the question that might unintentionally influence the result.   So knowing about Anchoring is still very useful, whether you call it a true cognitive bias or a simple persuasion technique.    In general, I still believe the Cognitive Biases are worth studying and raising awareness of, though maybe more as social or linguistic phenomena than as true flaws in the human mind.

And this has been your math mutation for today.


Sunday, May 1, 2016

219: A Portal to the Past

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I was sad to hear of the recent passing of Umberto Eco, one of my favorite contemporary novelists.   Due to his long career as a professor of “semiotics”, his novels drew on a wide variety of historical, mathematical, and scientific ideas from throughout the past millennium.    One that I read relatively recently was “The Island of the Day Before”, his 1994 story of a soldier marooned during the 1600s, the historical period during which the governments of Europe were desperately searching for a reliable way to measure longitude.   You may recall our discussion of this multi-century quest back in episode 108.    The key plot twist is that the main character, an Italian soldier named Roberto Della Griva, gets marooned on a ship that is trapped within sight of the International Dateline, and an island which sits beyond it.   

Della Griva attaches a mystical significance to this line, as is implied by the book’s title.   He somehow convinces himself that if he could just cross it, it would mean he was traveling back in time.   If he will manage to swim across the line tomorrow, would it mean that today he should see himself swimming in the distance?  Here is one amusing passage from the book:   

“Indeed, as he sees it distant not only in space but also (backwards) in time, from this moment on, whenever he mentions that distance, Roberto seems to confuse space and time, and he writes, "The bay, alas, is too yesterday," and ,"How much sea separates me from the day barely ended," and even, "Threatening rainclouds are coming from the Island, whereas today it is already clear . . . .”

While these speculations are rather absurd, this discussion got me curious about the actual history of the International Dateline.   It has been recognized since ancient times that the time of day is slightly different as you travel to the east or west, but the idea that you might travel all the way around the world and gain or lose a day was only really conceivable in relatively recent eras of human history.    The real history of the Dateline can probably be properly said to have started with Magellan’s circumnavigation of the globe.   When the handful of survivors of that three-year voyage arrived home in 1522, they were surprised to discover that despite careful logging of their travels, they had lost a day.     Here is a description from one of them:

On Wednesday, the ninth of July [1522], we arrived at one these islands named Santiago, where we immediately sent the boat ashore to obtain provisions. [...] And we charged our men in the boat that, when they were ashore, they should ask what day it was. They were answered that to the Portuguese it was Thursday, at which they were much amazed, for to us it was Wednesday, and we knew not how we had fallen into error. For every day I, being always in health, had written down each day without any intermission. But, as we were told since, there had been no mistake, for we had always made our voyage westward and had returned to the same place of departure as the sun, wherefore the long voyage had brought the gain of twenty-four hours, as is clearly seen.

As you can see, even though they were caught by surprise at first, the sailors were able to quickly realize their fallacy.   Because they were traveling in the same direction as the sun, they had experienced one less day, but each day they had experienced was slightly longer.   So there was no actual time travel, just an accounting error.    A similar phenomenon was observed later by other circumnavigators, such as English explorer Francis Drake.    This incident also inspired the famous surprise ending of Jules Verne’s “Around the World in 80 Days”.   Even though the reasons for the gain or loss of a day upon circumnavigation were well known, I suppose it is vaguely possible than an uneducated sailor like Eco’s character could have attached a more mystical significance to the effect.

But even these odd experiences of sailors were not really common enough to motivate standardization of time zones and an international dateline, until the era of trains came along in the 19th century.   Suddenly it was possible to move quickly and continuously between areas with different local times.    Finally in October 1884, representatives from 25 nations met at an international conference in Washington, DC, and came up with the system of time zones we know today, based on longitudinal lines starting from the Greenwich meridian, and times derived by adding or subtracting from the Greenwich Mean Time, or GMT.   To increase the chances of universal adoption, it was agreed that local islands and nations can move the timelines for convenience, which is why we see those squiggly time zone boundaries today instead of simple longitudinal lines.     Amusingly, the French seemed to be insulted by the idea of a location in England defining the time zones: until 1911, instead of referring to Greenwich Mean Time, they referred to Paris Mean Time minus nine minutes and 21 seconds, which was equivalent.

Unfortunately, since these meridians and zones are all artificial labels, I’m afraid Della Griva’s dream of somehow using them for time travel would never quite pan out.   Some modern writers ponder this idea as well, but I think they are mostly tongue-in-cheek.   For example, travel author Bill Bryson has written:  

“I left Los Angeles on January 3 and arrived in Sydney fourteen hours later on January 5. For me there was no January 4. None at all. Where it went exactly I couldn’t tell you. All I know is that for one twenty-four-hour period in the history of earth, it appears I had no being.   I find it a little uncanny, to say the least. I mean to say, if you were browsing through your ticket folder and you saw a notice that said, ‘Passengers are advised that on some crossings twenty-four-hour loss of existence may occur’…, you would probably get up and make inquiries, grab a sleeve, and say, ‘Excuse me.’”   

But somehow I don’t think Bryson really believes he lost a day of his life due to the shifting of a few time labels.    I would be more concerned about the portion of my existence that is wasted while crammed into a tiny airline seat for half a day.

In a more serious vein, there was a solar eclipse last month that started on March 9 and ended on March 8, but it was actually traveling forward in time the whole way, although it achieved its peculiar timeline by crossing the International Dateline as it travelled.   Also, you shouldn’t completely lose hope— one form of time travel across the dateline is possible.   Remember that under the theory of relativity, if you travel by airplane at a high speed, you really do lose a tiny fraction of a second, as time slows down for you in relation to your friends on the ground.   However, that works across any line, not just one artificial one.

And this has been your math mutation for today.


Saturday, April 30, 2016

The Book Is Out!

Hi everyone-- the Math Mutation book, "Math Mutation Classics:  Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available.  You can order it from Amazon at this link.    A perfect Mother's or Father's day gift for a geeky parent!

By the way, if you like it, don't forget to post a review on Amazon.   And of course, if you're in the Portland, Oregon metro area, I'll be happy to autograph your copy sometime.


Sunday, March 27, 2016

218: Itching for the I Ching

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Recently I was reading a biography of John Cage, the quirky avant-garde 20th-century classical composer who I have mentioned in a few previous podcasts.    One of the most fascinating aspects of Cage’s composing was his attempt to introduce random elements into his music, starting in the 1950s, in order to free himself from preconceived patterns.   He experimented with numerous sources of randomness, including die rolls, ambient noise from the environment, and even imperfections in the paper he was writing on.   But one method that absorbed his interest for a long time was the ancient Chinese book of divination known as the I Ching.    Once Cage discovered the I Ching, it became his main guide in the selection of random numbers.   Some of his compositions required thousands of random numbers to be completed.   As a result, many of his visitors noted that anyone who stepped through Cage’s door was soon drafted into tossing coins for a few hours to generate I Ching trigrams for use in Cage’s music.

The I Ching,  or Book of Changes, is said to be one of the world’s oldest books, written around 3000 years ago.   It is based around interpreting the significance of various patterns of whole and broken lines, traditionally determined by tossing yarrow sticks, or by an equivalent method based on tossing coins.   The most fundamental set of patterns generated by the I Ching are the eight trigrams, patterns of three lines, which may each be solid or broken.   Each of the eight trigrams has several possible meanings, such as the mind, the spirit, emotions, or bodily sensations.   Pairs of these trigrams can be combined into one of 64 possible hexagrams, for an even richer set of possible meanings to interpret.    Being a listener of this podcast, you have probably realized by now that the combinations of three or six lines, each of which can be solid or broken, is precisely equivalent to a three- or six- digit binary number, if you interpret the solid lines as 1s and the broken ones as 0s.   So essentially, the I Ching is a divination system based on random numbers, expressed in binary, or base-2, notation, between 0 and 63.     Now I’m sure Chinese scholars will say I’m shortchanging the deep philosophy of the system, since these random divinations are accompanied by thousands of pages of interpretive text.   But it’s undeniable that these numbers are the basis.

Because of this numerical aspect, it’s actually not uncommon among historians of science to credit the ancient Chinese for first coming up with the idea of the binary number system, which is critical to modern computers.   Personally I’m a bit skeptical of this aspect of I Ching studies:  while the ancient book discussed many ways to combine and interpret the trigrams and hexagrams, they weren’t using these as a basis for a numerical system or for calculations of mathematical significance.   On the other hand, the legendary Gottfried Leibniz, co-inventor of calculus and early designer of ideas for calculating machines, did credit the I Ching for inspiring the idea of binary arithmetic in some 17th-century writings.    I think this may have been largely due to the fact that there were no other precedents for this idea in Leibniz’s time, though.   Most likely, he was astonished to see some basic ideas of his base-2 new arithmetic system in this ancient text, though he probably would still have developed the binary system if unaware of these writings.

As I read more about the I Ching online though, I was surprised to see that its description as a system of binary numbers is actually a bit of an oversimplification.   The reason is that the I Ching describes a complex procedure for generating the lines,  not the simple 0/1 coin toss you would have guessed.   When using the coin method to generate a solid or broken line, you are to toss three coins, with one side of each coin considered the “yin” side and the other the “yang” side.   Each yin toss has a value of 2, which each yang toss has a value of 3.   You then add the values together, to get a total between 6 and 9.   A 6 or 8 is a broken line, while a 7 or 9 is a solid line.    But there is more to it:  the less probable 6 or 9 values indicate that their line is “moving”, while the 7 or 8 lines are “stable”.   While the symbolic trigrams or hexagrams are still drawn with mainly solidness or brokenness visible, you need to note which are moving and which are stable, as this can make a major difference in the results of your divination.    Thus, one might say that the I Ching is really a base-4 divination system rather than binary.   In some of his writings, John Cage actually claimed to be using these stable and moving aspects to guide his randomly generated music.

But on top of the base-4 complication, there is yet one more mathematical wrinkle.   While the totally random methods such as tossing sticks and coins are the most commonly used, one online scholar notes that the I Ching describes another, more complex, method for generating the next 6/7/8/9 line number based on the current one, using a series of mathematical calculations.    These calculations are actually pseudo-random, similar to the Linear Congruential Generation algorithms used by modern computers.  This means that the results are deterministic, though hard enough to predict that they appear random.   Furthermore, according to this online analysis, the official I Ching algorithm is somewhat biased:  while solid and broken lines are equally likely, the 9 is much more probable than the 6, meaning that solid lines are significantly more likely to be “moving” than broken ones.   I’m sure New Age mystics would say there is some deep meaning in this, and that Yang is more mobile than Yin, or something like that.   Being a bit more of a cynic, I would lean towards the interpretation that the ancient Chinese were just not mathematically advanced enough to notice the problem.

Anyway, I’m not sure how all this was supposed to lead to John Cage generating better music:  while I really enjoy reading about his bizarre random methods, trying to listen to the resulting music for more than a few seconds at a time is not a very pleasant experience.   It’s also amusing that Cage put so much energy into generating numbers using I Ching methods, when he could have bought books of pre-generated random numbers, which were available for engineering and cryptographic applications for decades before the advent of modern computers, and saved a lot of time.   But I wonder if Cage’s avant-garde admirers would have claimed to like his music as much, if he told them the source was the Rand Corporation rather than ancient Chinese mysticism.

And this has been your math mutation for today.


Friday, March 4, 2016

Where Are The Missing Episodes?

Hi everyone--  you may have noticed that a bunch of episodes recently disappeared from this blog & the corresponding podcast feed.    Each of the disappeared episodes is included in the new Math Mutation book, which will be released in Spring 2016, and is already available for pre-order from Amazon.

Apologies to those of you who  are searching for those episodes.   I really wanted to keep them up on the web, but the publisher's contract requires their removal, out of fear that their free availability would hurt book sales.   I tried to push back, but it's a standardized contract & there's not much wiggle room.

Note that the other episodes will remain on this site, and I will continue to release new ones (at least) monthly and post them as usual.

Thanks again for your interest, and please check out the book page on Amazon!

Wednesday, February 24, 2016

217: The Oxford Calculators

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Before we start, just a quick reminder that I’ll be taking down a bunch of old episodes from the web any day now, due to their inclusion in the upcoming Math Mutation book.   So be sure to download any old ones you’re interested in ASAP.    Now, on to today’s topic.

Often we think of the centuries before Copernicus and Galileo as a huge black hole of science, with little of note happening other than medieval knights spending all day jousting at dragons.   But actually, the pioneers of the Scientific Revolution who became household names, like Copernicus and Galileo, did not spontaneously arise from a vacuum like some quantum particle.   They owed a major debt to a number of earlier scholars who re-examined the knowledge passed down by the ancients, and explored the possibility of more carefully applying mathematical methods to understand the natural world.   One of the most important such groups were the cluster of philosophers from the 1300s, led by John Dumbleton, Richard Swineshead, Thomas Bradwardine, and William Heytesbury, who became known as the Oxford Calculators.   

The most critical contribution of the Oxford Calculators was probably the concept that nearly any physical attribute worth studying could be measured and quantified.   Nowadays we take this for granted from the beginning of our science education in school, but this was by no means obvious for most of human history.   Among the ancient Greeks, Aristotle had discussed measurements related to size and motion, but had not addressed the issue of whether other aspects of the physical world could be measured numerically.   The ancients seemed to have an implicit assumption that many aspects of reality, like heat and light, could only be discussed qualitatively.    The Oxford Calculators challenged this approach, and tried to quantify their discussions whenever possible, believing that it really should be possible to quantitatively specify nearly everything you could discuss.   They were primarily thinking of themselves as theologians and philosophers, so did not conduct the actual experiments that would be critical to the real advances of the later scientific revolution, but even thinking in these terms was a major step.

Another important contribution was their willingness to reopen discussion of matters that had been supposedly solved by the ancients, and question teachings that had been passed down since Aristotle.   One example is Artistotle’s belief that the velocity of an object would be proportional to the force exerted on it, and inversely proportional to its resistance.   In modern terms, we would write this as V = kF/R, with V = velocity, K = some constant, F = force, and R = resistance.    One of the Oxford Calculators, Thomas Bradwardine, pointed out that there was something fundamentally wrong with this formula:  if the force and resistance precisely balanced, we should expect an object not to move at all.   However, the Aristotelian formula would impart a constant velocity, since F/R would equal 1 in such a case.   Bradwardine proposed an alternate formula, which we would now write as V = k log (F/R).   This was also horribly wrong, but at least fixed a key flaw in Aristotle’s teachings, so we need to give him some credit.   Since the log of 1 is 0, Bradwardine’s approach would at least predict objects not to move when force and resistance are balanced, a pretty important characteristic for a basic theory of motion.   This wrong formula also helped advance the definition of logarithms, a major mathematical building block for further developments centuries later.

The Oxford Calculators’ most significant concrete contribution was probably the Mean Speed Theorem, an idea which is often incorrectly attributed to Galileo, but was actually first published by William Heytesbury in 1335.  This came out of attempts to understand accelerated motion, quite a challenge before the development of calculus, our key tool for understanding changing quantities.   You may remember the basic formula for the distance traveled under constant acceleration:  S = 1/2 A T squared, where S is the distance traveled, A is acceleration, and T is time.   Nowadays, we can trivially derive this using calculus.   But Heytesbury was able to essentially figure out the same formula centuries before calculus was available, reasoning that the total distance traveled by an object under constant acceleration would be equal to the average speed multiplied by the time.   In modern terms, the average speed is just 1/2 A T, and the time is T, so Haytesbury’s reasoning leads us to the same 1/2 A T squared formula, without the need for calculus.  Heytesbury didn’t write it precisely in the modern form, but managed to achieve the correct result, centuries before Galileo.    

Now, we should point out that while they provided critical stepping stones, the achievements of the Oxford Calculators do not diminish the significance of Galileo’s work or others involved in the later Scientific Revolution.   If their successors sometimes failed to recognize or mention them, it wasn’t out of malice or deceit, but mostly because the modern system of scientific citation and of publication credits had not yet been established.    With modern scholarship, we can give the Oxford Calculators the credit they deserve, without needing to take away credit from anyone else.   They played a small but critical role in laying the foundations for the rapid mathematical and scientific advances that continue to benefit and astonish us in our own time.

And this has been your math mutation for today.


Wednesday, February 3, 2016

216: Bowie Meets Escher

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Apologies that this podcast is a little late.   A bunch of my time has been taken up by a fun new project:  putting together a Math Mutation book.   Yes, I’ve actually found a publisher crazy enough to want one!  I’ll announce more details here as we get closer to the book’s release.    One slight downside though:  due to the realities of publishing contracts, any episodes that end up in the book will have to be taken down from the web.   Sorry about that; I pushed back a little, but they were firm on this aspect of the contract.   So, if you were planning to catch up on old episodes, be sure to download them ASAP, before they start disappearing.   Now, on to today’s topic.

With the sad passing of the musical pioneer David Bowie, it seems appropriate to try to create one more Math Mutation episode that focuses on him somehow.   You may recall that he has come up before, in our discussion on random song selection back in podcast 193.   But this time I thought it would be fun to talk about something a little different:  the climactic scene in the movie Labyrinth.   As you may recall, Labyrinth was a 1980s movie that starred Bowie, Jennifer Connelly, and lots of muppets.   Bowie played the Goblin King in this move, where the only other speaking human was Connelly, as a scared teenage girl trying to rescue her baby brother, who Bowie had kidnapped.   To catch up with him, she had to traverse a bizarre maze filled with strange traps and spooky muppet monsters.   When the girl finally catches up with the Goblin King, he is in a huge maze with staircases in every direction, clearly inspired by M.C.Escher’s classic 1953 lithograph Relativity.

Before we get into the movie, let’s talk about the original lithograph.   Relativity is one of Escher’s less absurd works, in that the 3-D structure he depicts is actually self-consistent, and can theoretically be built in three dimensions.   It centers around a triangular group of staircases, with various doorways, windows, and secondary staircases nearby, and faceless figures walking up and down in various locations.   Where the Escher mind-bending comes in is that there are multiple distinct sources of gravity in the picture, with each of the walking figures independently subscribing to one or the other, even if on the same staircase.   For example, in the staircase at the top, two figures seem to have their feet near the same stair, but the “tops” of the stairs to one of them are the “fronts” of the stairs to the other, so they are standing perpendicular to each other.   Similarly, the doors and windows each seem perfectly reasonable on their own, but all together don’t make much sense, creating multiple different impressions of which way is “up”.

As with many Escher prints, generations of college math majors have put this poster up on their walls, and enjoyed the absurd questioning of basic artistic and mathematical rules.   But is there a deeper meaning to the lithograph?   One blogger suggests that it is questioning the nature of who actually controls reality:  “Who controls the world, and reality, in this painting? It seems that the human-like figures do. By going about their everyday business they show no desire to change it. Perhaps Escher is trying to say something about human nature.  It seems as though as long as these beings can eat, walk, read, and go about their normal lives they are content to go along with the distorted world they live in, however ridiculous it is…  If we care enough to wake up and see what's going on, we will have the power to change it.”    This seems to be the most interesting analysis I can easily find on the web, and ties in nicely with some of the fan interpretations of Labyrinth.

Getting back to the movie:   as I mentioned, the climactic scene involves a chase through a Relativity-like maze, complete with inconsistent gravity from various angles.   This was before the days of cheap CGI effects, so the filmmakers actually built a large Relativity-like set, and used camera tricks to make it look like Bowie, Connelly, and the baby were subject to varying gravity in multiple directions.   Like most of the movie, this scene seemed to come out of nowhere, with nothing earlier specifically alluding to it.  Many critics panned the movie for basically that reason, just being an accelerating series of oddities with no underlying rules— initially it wasn’t much of a box office success, though it is now considered a cult classic.  In the years since it came out, legions of fans have tried to discover a deeper underlying meaning.   

The easiest interpretation is that this is just another in a long line of absurd children’s tales, with crazy magic and monsters that don’t really have much deeper meaning.   A slightly more convincing interpretation is that it’s a coming-of-age tale, where the girl learns to take on the maturity and responsibility to make her own decisions.   This would put it squarely in the typical space of many popular fantasy stories.   However, there are darker possibilities.   One website, “Vigilant Citizen”, claims that the entire movie is an allegory for mind control, with each of the obstacles in the labyrinth being somehow related to the internal world of a brainwashing victim.    This also ties in well with the resolution of the scene, where Connelly tells Bowie “You have no power over me”, and as a result finds herself safely teleported home with her baby brother.   But is an interpretation this dark really appropriate for what is largely regarded as a children’s movie?

Ultimately, I’m not sure which view is correct.  Was the final Relativity stairway chase in Labyrinth a metaphor for pulling free of mind control, or a gentler coming-of-age ritual?   Or was it just another case of the legendary Bowie choosing to be weird for weirdness’s sake?    We’ll never be able to answer those questions completely, but I have no doubt that future generations will continue to enjoy both Escher’s Relativity and David Bowie’s Labyrinth.

And this has been your math mutation for today.