Pareidolia poser

By Phil Plait | April 21, 2009 7:26 am

Question for you: which of these two images shows dots that are placed at random, and which does not?

Random dots

The problem with questions like this is that you already know it’s the one on the left that’s random, and the one on the right isn’t, since you know I’m trying to trick you. But what’s going on?

Our brains love to find patterns in random noise. Look at the clumping of the dots on the left; surely that’s not random? But it is. The distance between dots will average out to some number, but statistically you expect there to be some deviation from that average, so that some dots will be closer together (making clumps) and some farther apart (making voids). That’s what’s happening on the left.

On the right, the random pattern that was generated was modified so that the dots would not be too close together. If a dot’s position was found to be too close to another, its position was redone until it was a minimum distance from all other dots. What’s left is a pattern that we think looks more random, but is in fact highly non-random.

A more detailed explanation of these images is at the blog In The Dark, and he uses it to talk about galaxy distributions. However, it also tells us a lot about our brains. We are instinctively lousy at statistics.

Another great example is this one: imagine you flip a coin ten times, and you keep track. Which of these sequences is more likely?

HHHHHTTTTT

or

TTHHTHHTTH

The answer is they are both exactly as likely. You have a 50/50 shot at a heads or tails on each throw, so any 10-throw sequence is just as likely as any other! But we look at the second sequence and see no information in it. We assume it’s just random, and therefore more likely than a sequence where we perceive there is information, like five heads in a row followed by five tails. But each is just as likely.

We view the entire Universe through our senses, and the data are processed by our brains. This gloppy computer is highly sophisticated, but also highly unreliable to give us unbiased information. We see patterns where they don’t exist, we see cause where they may be none, and we see intent where there may be randomness.

That’s why pareidolia — seeing faces or other familiar objects in random patterns like oil stains, wood grain, and the odd piece of bark of pastry item — cracks me up. The brain of a human will interpret that pattern into something familiar, and if that person is religious, they see a religious icon. But they don’t seem to hang the same connotation on seeing Abe Vigoda in a nebula and Lenin in a shower curtain, or Ben Grimm in a supernova, or any of a hundred examples I can find easily.

You can be fooled. Remember that, always. It pertains to a lot in life, and a lot in the life of an active skeptic. Fooling people is easy. Getting them to see? That’s what’s hard.

CATEGORIZED UNDER: Cool stuff, Pareidolia

Comments (82)

Links to this Post

  1. Random Placements « Risebakery’s Weblog | April 24, 2009
  1. Gblade

    Used to seeing random patterns, I though you were trying to trick me into thinking the right one isn’t random (opposite of what you say in the second paragraph). Though, more than likely both are not random unless they used a random physical process for their number generator. Pseudo-random.

  2. OMG! I see Jesus!

    Or maybe Darth Vader. It’s hard to be sure.

  3. mapnut

    Excellent post. I got the coin-flipping question right but picked the wrong dot field. I thought the one on the left looked clumped, but should have recognised the ones on the right were avoiding each other. The one on the left reminded me of a study I did on the distribution of clams, a long time ago. My fellow students and I had to rise at 4:30 am to catch low tide, and dig a transect a cubic foot at a time, recording the position of every clam. It was not fun! Then we applied a primitive computer program to determine whether or not the distribution was clumped. It was, slightly. Hardly seemed like the conclusion justified the backache.

  4. MPG

    If I squint I can see a rabbit on the right-hand side of the random dots picture. And the torso and thighs of a woman in a one-piece swimsuit in the centre. Oh, and a skull, middle-top. Hours of fun (for the easily pleased)!

  5. Todd W.

    Reminds me of the post I made to an ID proponent. I blindly typed several lines of text and a single line of purposefully chosen (or designed) text (all lines were pretty much jumbles of keyboard characters with no actual words). I then asked them to tell me which line or lines of text showed evidence of design, and which showed randomness. I never got an answer.

  6. T_U_T

    before reading the post ( and reading which one is random ) My guess is that the left is completely random where the right is not

  7. Neat topic. Cracked has a pretty cool article titled “5 Ways Your Brain Is Messing With Your Head” that also deals with how our brains work (and sometimes don’t) in processing information.

  8. DAV

    Yes, the two sequences are equally probable but ONLY if you mean probability of getting the EXACT sequence as stated. However intuition isn’t that far off the mark. Out of the available sample 2^10 space it’s far more likely to get a sequence other than 5H5T. Or to put it another way, it’s more probable to get a sequence with small runs than a sequence with larger ones. Getting 5H5T should raise suspicions about the fairness of the flipping process. It’s not proof though.

    The same with randomly placed dots. A random placement spelling a common word is just as likely as any other specific random placement but you would (or should) be more suspicious if you got one.

  9. fos

    Another great post! You should be a professor or write books! I’m looking forward to book three.

  10. BJN

    Does the pattern on the right really look more random to you? I immediately chose the pattern on the left.

    I suspect that it’s not perception that’s the only issue, it’s individual interpretation (or misinterpretation) of the definition of “random”. I think many people think of “random” as more evenly sorted and that has to do with the kinds of events or groups that are frequently described as random.

    Language has a strong impact upon perception.

  11. Daniel J. Andrews

    Good examples, Phil. The picture one was easy enough as I’ve worked on clumping, randomness, uniformity etc in ecological studies (plant distributions). The HT coin one tripped me up for half a second though before the stat part of my brain kicked into gear. You are absolutely right. We can be fooled. The brain is wired to be fooled this way, and anyone who think they cannot be fooled is at risk of being fooled. If I get a chance I’m going to incorporate your examples for our university biology students.

    O/T: excellent post on imagination in science. I’ll make that must read for students (I don’t teach anymore but I can bend a few arms of profs and get them to make their students read it :-)

  12. I think many people think of “random” as more evenly sorted

    That’s exactly the case. Yet getting alternating heads and tails every time you flipped a coin would be pretty fishy, even though that’s as “evenly sorted” as you can get.

  13. Thanny

    DAV beat me to it. Our brains are more sophisticated than the coin-flip example allows. We see the sequences not exactly, but as categories. And we are perfectly correct to guess that a category including small stretches is much more likely than one including large stretches. In the short run, these intuitions are very good at cheat detection. They break down when the sample size increases.

  14. Not to burst anyone’s bubble, but before you pat your selves on the back for “correctly” identifying the sample on left as random, you should more carefully read the original post at In the Dark. The correct answer should be: “I don’t know because I haven’t statistically measured the set for randomness.”

    Both datasets are produced by stochastic processes, but the one of the left turned out to be more “random” (uncorrelated) than the other. To me, this is less about pareidolia than the imprecision of our language.

  15. DAV

    Yet getting alternating heads and tails every time you flipped a coin would be pretty fishy, even though that’s as “evenly sorted” as you can get.

    Ahhh! But is HTHTHTHTHT more or less suspicious than HHHHHTTTTT?

  16. Alex

    Someone should also mention Benford’s Law.

    Oh, looks like I just did.

  17. rrt

    I’m with some of the others here, Phil, I spotted the right as nonrandom right away, and not (I think) because I knew it was a “tricky” question. Perhaps having the random one next to it helped, but I noticed right away the total lack of clumping or overlay on the right. They are just too evenly spaced. I expect some clumping.

    Actually, knowing it was a potentially trick question, I was half-thinking both were random–just that the right was selected from many runs of the software (or whatever) as an unusually even result.

  18. I remember a meeting at work where someone said that members of a committee were “chosen at random”.

    “That’s not random,” I said. “If it was truly random, you would have selected a goat and the Crimean War to be on this committee.”

    Although by that criteria, nearly every committee would be made up of mostly hydrogen and some helium.

  19. SteveG

    Interesting how our brains work – and don’t work.

    I fell for both examples. Truly for many (me) randon implies homogeneity but that is not correct.

    I had to re-read the coins, but I got it.

    It reminds of the Monty Hall problem.

  20. T_U_T

    this is like saying, while looking at a picture of a triangle “I don’t know what shape it is because I haven’t put this picture through a contour extracting algorithm and subsequent classification”

  21. T_U_T

    blocquote fail again. I was trying to quote eric “The correct answer should be: “I don’t know because I haven’t statistically measured the set for randomness.” “

  22. They’re both suspicious, but I can roll a die and get the same number three or four times in a row. I don’t find that unusual, but if it wandered into eight or nine times in a row, or if I never got the same number twice, I’d be wondering. It seems intuitive to me that neither case is very likely. Intuition is a bad way to understand probability. :-

  23. Greg in Austin

    Phil said,

    “You have a 50/50 shot at a heads or tails on each throw, so any 10-throw sequence is just as likely as any other!”

    What really gets some people is that you are just as likely to get “HHHHHHHHHH.”

    8)

  24. Another way to look at the dot pictures that should make the answer obvious is looking in fourier space, if you’re trained for it. It’s obvious at a glance that there’s a lot more power in certain spatial frequencies in the right image, and as far as I can tell there’s no obvious scale in the left image.

  25. Todd W.

    @Greg in Austin

    Reminds me of Rosencrantz and Guildenstern Are Dead.

  26. Bas

    You should see this (if you haven’t already)

    http://www.ted.com/index.php/talks/peter_donnelly_shows_how_stats_fool_juries.html

    If you have 20 minutes to spare, I recommend watching the whole thing. But on the subject of coin tosses, skip to about the 3:45 mark.

    That one did my head in.

  27. I can’t really comment on stochastics but on the pareidolia front if you add a second randomizing factor (the example I remember was assigning a random color to each dot) we tend to perceive even more order when there’s clumping.

    That’s not necessarily a bad thing — free association being a great source of both scientific and artistic creativity. But, yeah, mistaking associations that we create (the cool part) for actual pre-existing artifacts (the pareidolia part) is kind of awkward. Well, except for the owner of the tortilla with Michael Jordan’s face on it or whatever. Because *they* get to sell it on eBay. Hopefully as “found art.” :-)

    figleaf

  28. F

    If a dot’s position was found to be too close to another, it’s position was redone until it was a minimum distance from all other dots.

    I found another random dot.

  29. NewEnglandBob

    I see Lincoln’s Gettysburg Address on the right, so I knew it was not random.

  30. “Random” does not imply “uniformly distributed”. Also “random” does not imply “independent”.

    Uniform Distribution: the probability that you’d find a dot at any particular location is the same for any location on either the left or right picture.

    Dependence/Independence: On the left picture, the probablility that you’d find a dot at any given location is independent of the locations of other dots. The same is not true of the right-hand picture.

  31. Grump

    “That’s not random,” I said. “If it was truly random, you would have selected a goat and the Crimean War to be on this committee.”

    I figure that a true generator of random numbers would only ever produce positive or negative infinity. Any other number would be inifinitely improbable.

  32. Joe Meils

    Unless, of course, you are in a Tom Stoppard play. Then the sequence is more like: HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH…. (until the very end, then it’s) HHHH…T.

  33. Bo Decker

    I’ve wondered why other people see Jesus on the tortilla while I see Abe Lincoln, Osama bin laden and a guy I thought about dating in college.

  34. If you stare at them and cross your eyes you see a 3D picture of Phil’s butt.

  35. DAV

    I think the reason the right picture of dots would be selected as more random than the left is because the clumps in the left give it features while the right has features discernible only at a higher level of consideration. Blur your eyes and you will see shat I mean. The left one has features while the right seems blank and nothing ‘pops out.’ The same can be said about the coin toss sequences. The 5H5T one has a noticeable feature while the bottom one seem smoother.

    I think most people equate randomness with featureless unless they are prone to over-analysis. This exercise points out some interesting shortcuts in our vision processing and how they affect our thought processes.

  36. Alex

    I already posted an item about Benford’s law on In the Dark . You can find it at here .

    I thought I’d add the link to avoid Bad Astronomy having to copy the article, like they did with this one.

    Peter

  37. GoatTuber

    Hey Phil, did you see the pareidoglia email I sent you yesterday?

  38. Gary Ansorge

    I don’t know about those pics, Phil. I tried every trick I know(even staring cross/eyed) and couldn’t see a anything but random dots,,,Seems god was on vacation when those were generated,,,

    I don’t know the odds but twice in my life I’ve flipped a coin and had it stand on edge,,,once was a quarter and once a nickel. It tickled the heck out of me,,,

    GAry 7

  39. It’s funny. I was just messing around with Excel creating random and psuedo-random point fields literally seconds before reading this. I do this a lot, so much so that the left picture DOES instinctively look random to me, while the left one looks too regular.

  40. A lot of good points have been made in comments already — imprecision of what we mean when we say “random”, distribution of head-flipping sequences (though it’s worth mentioning the aymptotic equipartition property), uniformity vs independence, lack of a specified distribution in your coin-flipping example (assumed IID and fair), and so on.

    So I’ll just concentrate on the pareidolia issue. (Which, BTW, doesn’t make it into the OED. You should point it out to them.)

    There is an alternate story than “humans are lousy at statistics”. The Bayesian Brain movement would assert that we are, in fact, quite good at statistics. It’s just that we do them in a Bayesian framework, in which you can account for (a) previous experience and (b) utility of a decision.

    The first says that you have lots and lots of experience with patterned things — patterns of how plants grow, day and seasonal cycles, language, writing, etc. So off the bat, you have some tendency to expect there to be patterns, simply because you’ve been exposed to so many of them. The second says that you tend to err in favor of things that matter to you. By and large, periodicities and human faces and so on matter to you much more than Poisson noise does — the former can get you killed (or laid, depending), while the latter is unlikely to. So, on evolutionary scales, it’s on average better to see patterns where there aren’t any than to miss patterns that are there.

    Put the two together and you get a lot of emphasis on detecting patterns — whether they’re there or not.

  41. TheBlackCat

    I immediately recognized the left picture as being random while the right is not. I’m pretty used to looking at plots like that. It is very similar to a spike raster, where neuronal action potentials (spikes) are represented as rows of dots where the location of the dot indicates the time at which the action potential occurred. Each row is either a separate trial or the same trial wrapped around. You learn pretty quickly to identify refractoriness (where no two action potentials can be too close together), which results in a spike raster very similar to the right figure. The left figure is what you see without refractoriness. These figures aren’t in rows, but if you assume very small rows and a low rate of spontaneous firing for the neuron you wouldn’t be able to tell the difference.

    Also “random” does not imply “independent”.

    Really? Can you give me an example of a system where events are random but the events are not independent of each other?

  42. p

    i guess i’m finally getting good intuitions, because there was immediately no question to me which dot distribution was higher entropy.

  43. JackC

    Phil – you had a totally cool bit of pareidolia at your Hayden lecture last night – I could totally pick out Tim Curry as “Darkness” at the bottom right quadrant in one of the nebula images. Of course, I don’t remember which one.

    JC

  44. Toby

    Does “placed at random” have a definite mathematical meaning? If not, it seems to me we have to have some idea of what parameters are involved before we can decide whether they are random.
    Take the above (left) image! My guess is that the location of every dot is determined by two random numbers, representing the x and y values in a Cartesian coordinate system.
    But we could choose a polar system instead, with the two numbers representing angle and radius! This would result in a very different scattering, with 10% of the dots within 1% of the total area (around origo).
    Still, a Cartesian coordinate system is in no way more “correct” than a polar one (and certainly not more intuitive).
    And, of course, there is an infinite lot of other options – so I guess it wouldn’t do testing them one at a time…
    Or, am I totally wrong about this?

  45. Pieter Kok

    Toby, in the case of the dots there are enough for the randomness to show up reliably in so-called correlation functions. For example, you can look at the average distance between dots, and the variance of the average distance, etc. Randomness implieds that these quantities are consistent with a Poisson distribution (or a Gaussian distribution in the case of continuous variables like distances).

    Your counter example is wrong, because when you change coordinate systems, you have to be careful to keep track of the distance measure (e.g., the Jacobian, when you’re integrating).

  46. Quiet Desperation

    Can you give me an example of a system where events are random but the events are not independent of each other?

    Dealing a deck of cards. A well shuffled deck can produce a random sequence, but each card dealt affects the statistics of subsequent cards.

    Maybe lightning strikes from a storm cloud? Chaotic motions builds up the charge, but each strike can affect the starting conditions for the next strike buildup.

  47. Quiet Desperation

    I immediately recognized the left picture as being random while the right is not.

    Same here. The *spacing* between the dots is clearly not random. I didn’t need to be told the algorithm used to make it.

    You know what it looks like? Those ceiling tiles you only see in older buildings now.

  48. Chris D.

    *looks at the one on the right* It’s a sailboat, isn’t it?

  49. JackC

    Pleased to see a few Stoppard fans here. Do you want to play Questions?

    JC

  50. Todd W.

    @JackC

    Do you want to play Questions?

    Umm, how do you play questions?

  51. Grump

    Is that not off-topic?

  52. Fly

    I see Von Neumann’s ghost in both:

    “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number– there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.”

    You seem to imply this “fooling” only happens to the religious and not to scientists.

  53. dr
  54. Jim

    Could be the centers of a bunch of ball bearings ‘randomly’ scattered in a box. That would kind of limit the minimum distance.

  55. slw

    What are you talking about, random? The left one is quite obviously a fox playing a cello, sheesh.

  56. drewski

    The one of the left looks way more random than the one on the right to me. One on the right is way, way too neat to be random.

  57. MadScientist

    The one on the right is random too – it’s just the probability of it occurring as a random event is incredibly small. I’m tempted to do a frequency analysis to estimate the constraints placed by the creator. Must … turn … off … geek … gene.

    This reminds me of many articles on “how to catch a student faking data”. I usually fall over laughing as I look at the analyses which reveal a decidedly “intelligent design”. Well – not too intelligent obviously if the student believes they know oh so much that they’ll get away with it. On the down side, in many (most?) cases it’s difficult to set up to check on all students’ experimental results. Sometimes the result is not so bad – it is obvious that the student has real(istic) data and that it is incorrect assumptions about the measurement errors which are revealed in the analysis (honest students often assign too large an error).

  58. TealkingGoat

    You know coin flipping is not really random. I can flip a coin and make it come up heads evertime or any other pattern I can imagine. I do it all the time when I let the kids think we are deciding something randomly with a coin flip ( I have allready decided so I control the outcome). Its a fun skill and really frustrates some people.

  59. DexX

    I must have studied too much probability – I immediately saw that the right pattern was far too evenly distributed to be truly random and had to have been generated within certain constraints to prevent the dots being too close to each other.

    Doesn’t make me smarter than any who picked the wrong one, just a different flavour of geek. :)

    The coin-flipping example always does people’s heads in, due to random probability being subject to so much fuzzy magical thinking. People imagine the universe seeks balance in all things, I think, and pushes to make things neat and tidy.

    Sorry, but the universe is a big, glorious mess, and the maid exists only in your mind.

  60. This also neatly explains “cancer clusters”

  61. The distribution of dots reminded me of an episode of NUMB3RS, where Charlie (the math genius) asked the FBI team to ‘stand around randomly’ and they pretty much just put themselves at more-or-less-equal distances.

    Though math is my weak point (let’s not even use the term Calculus… shiver!), I find the show interesting for it’s introducing me to various ideas and concepts in the science of Mathematics.

    J/P=?

  62. DAV

    Does “placed at random” have a definite mathematical meaning?

    Not really. “Random” is used as a magical incantation by frequentists but to Bayesians and other regular folk :-D it simply means “unknown”. It is further presumed that the output of “true” random number generators are uncorrelated and unbiased, hence “random” means “unpredictable”. Note that any correlation or bias would make them “predictable” — at least potentially.

    You know coin flipping is not really random. I can flip a coin and make it come up heads everytime or any other pattern I can imagine.

    Yes, so can a lot of stage magicians but it takes a skill most people haven’t bothered to acquire.

    I must have studied too much probability

    Really? Where? Most stat courses only briefly touch on the subject. To the frequentist it’s simply counting although what probability really means is at times hotly debated. No need to go over the entire history here but statistics is coming full cycle back to its roots — in large part because of the efforts of E. T. Jaynes. One of his books “Probability Theory: The Logic of Science (Vol 1)” has been released posthumously and is available through Amazon and elsewhere. A good, fairly easy read.

  63. Coleman Mulkerin

    I always try to get the point across about how rare it is for a number to win powerball by playing on people’s belief that order cannot come from disorder. When I am asked (as a clerk at a gas station) what number they should play I state that they should play the number 1 – 2 – 3 – 4 – 5 – 6. They will usually laugh the idea off and I will reply that is just as likely as any other number.

    Sadly even there conception of how low of a chance that number has to come in is probably still to high. Powerball is about a 1 in billion chance if I remember.

  64. It’s also frustratingly true that in a Lotto draw, the chance of getting 1, 2, 3, 4, 5 and 6 drawn is the same as drawing any other 6-number combo.

    Now, with the dot patterns… Is it not also true that the picture on the right is as equally likely as the one on the left (assuming a simple pixel-on/pixel-off array) but the “type” of pattern on the left is more likely than the “type” of pattern on the right? (Yes, I realise this wasn’t really the question)

    I’m now sitting here having fun with Photoshop noise patterns and looking for faces. I’ll get the kids to help tonight.

  65. Both images look random to me.

  66. Mushroom

    Interestingly, the random picture is a smaller jpg file than the non-random one (135 K vs 155K), although in general you expect random data to be less compressible than non-random data. My geek fu isn’t strong enough to suggest why this is, can anyone else?

  67. José

    @Coleman
    When I am asked (as a clerk at a gas station) what number they should play I state that they should play the number 1 – 2 – 3 – 4 – 5 – 6.

    Yeah, but there’s probably a bunch of other people who play those numbers, so if by some miracle you did win, your pot would be much smaller than if you played some obscure number set.

  68. I think Qui-Gon Jinn said it best: “whenever you gamble, my friend, eventually you will lose.”

  69. MadScientist

    @DAV: Bayesians are normal people? Uh … that’s nice. (Backs away slowly)

  70. Nigel Depledge

    I don’t know if anyone has spotted this before (not enough time just now to read through the comments, but…

    The BA said:

    The answer is they are both exactly as likely. You have a 50/50 shot at a heads or tails on each throw, so any 10-throw sequence is just as likely as any other! But we look at the second sequence and see no information in it. We assume it’s just random, and therefore more likely than a sequence where we perceive there is information, like five heads in a row followed by five tails. But each is just as likely.

    In fact, there is more information in the second sequence than in the first. According to information theory (or at least my interested layman’s understanding of it), the more easily compressed a sequence of bits is, the less information it contains.

  71. Hannu Siivonen

    TTHHTHHTTH has almost as distinctive pattern as does HHHHHTTTTT. To me TTHHTHHTTH doesn’t look any more random that HHHHHTTTTT. Basically they both have H:s and T:s inverted from the halfway to the end.

    By the way, is there a word for human tendency to find patterns? At least it’s somehow realted to pareidolia

  72. RC

    I don’t think the coin flips are a very good illustration of our misperceptions of randomness. As has been mentioned above, each particular sequence is equally likely, but the characteristics of each sequence are not. Which characteristics? In addition to expecting roughly half heads and half tails, we should also expect a sequence of coin flips to change from H to T, or from T to H, roughly half the time. In other words, if you flip a tails, roughly half the next flips will be tails and half will be heads. Note that sequence two contains exactly this characteristic–5 heads, 5 tails with 5 H-T/T-H switches. Sequence one contains 5 H and 5 T, but only 1 H-T switch. Our guts are right–sequence 2 is far more likely to represent a random sequence than sequence 1. Much more interesting in terms of misperceptions of randomness would be sequences with half heads, half tails, but 6 or 7 H-T switches. For example, HTHHTHTHTT. Note that this sequence contains 7 H-T/T-H switches, but I’m guessing many people would rate it “more likely to be random” than the sequence with 5 H-T/T-H switches. Sorry if this covers ground already covered. Best.

  73. Essentially the same pictures were at Cosmic Variance and In The Dark. Don’t
    you folks read each others’ blogs?

  74. DAV

    By the way, is there a word for human tendency to find patterns?

    Intelligence?

    Finding differences is easy. Ask any computer to compare two photographs of the same subject and you get tons of differences. Ask one to tell you how they are the same and you likely won’t get a meaningful answer. Many intelligence test provide images that are superficially different then ask which one doesn’t fit with the rest. The key to solving them is finding the underlying pattern of the group.

    Science unravelling the universe is pattern recognition on steroids. All hypothesis models are statements of pattern.

  75. Vorn

    A while ago I was writing a bit of program to distribute stars for a video game. I first tried using true randomness, and found it far too clumpy for my needs. I fought with it for rather a while before going for the retry method.

    Though I guess a repulsion/attraction solution would have worked too.

  76. Crudely Wrott

    A teacher once gave my class this quotation:

    “We live in a world of symbols and abstractions, and many a man dies by his own cliches.”

    I cannot recall who said that but I do recall how that teacher, among others, showed me a method to reduce the need for symbols and to clarify abstractions; literacy, numeracy and the scientific method.

    It is satisfying to see those teachers’ lessons confirmed whenever I or someone else finds a pattern that is actually there, not merely perceived. That’s how we learn, how we progress.

  77. Alan

    UK “Mentalist” Derren Brown did a good trick where he managed to flip 10 heads in a row before presenting the coin to the camera and showing it was not fake – all in one single uncut shot.

    Later on he revealed that he had been filming himself flip the coin for hours, only stopping once the 10 in a row sequence came up.

  78. whb03

    The “the Monty Hall problem”? Could you elaborate? I think I heard something about this once…

  79. Damon

    So in other words, deny and trick our brains into doing what they aren’t trained to do instinctually and biologically. Gotcha. Thanks, Phil.

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