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The Golden Ratio 676

Posted by timothy
from the cradle-of-life dept.
raceBannon writes "The book surprised and fascinated me. I thought it was going to be solely about the Golden Ratio. Mario Livio does cover the topic but along the way he throws in some mathematical history and even touches on the idea that math may not be a universal concept spread across the galaxy." Read on for the rest of raceBannon's review.
The Golden Ratio
author Mario Livio
pages 320
publisher Broadway
rating 7/10
reviewer raceBannon
ISBN 0767908155
summary Through telling the tale of the Golden Ratio, Livio explains how this simple ratio pops up in all kinds of physical phenomenon and debunks the idea that the ratio is present in many famous man-made structures and art work. Along the way, he provides historical tidbits regarding some of the well-known and not so well-known mathematicians throughout the ages and he tells the story of some of the more famous and not so famous mathematical advances. Finally, he discusses the possibility that mathematics may represent some kind of global truth that exists throughout the cosmos.


I have to admit that it is a little spooky to me that this ratio, this irrational number (1.6180339887...), pops up in many varied natural phenomena from how sunflowers grow to the formation of spiral galaxies; not to mention that the Golden Ratio and the Fibonacci Series are related. It makes you want to think that there is a God with a plan.

The Golden Ratio is defined as follows: In a line segment ABC, if the ratio of the length AB to BC is the same as the ratio of AC to AB, then the line has been cut in extreme and mean ratio, or in a Golden Ratio called Phi.

On the flip side, Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books. He rightly points out that you can find the Golden Ratio in anything depending on where you decide to place the measuring tape. The idea that the Golden Ratio is such a symbol of universal beauty that it appears by accident in our great man-made buildings and artwork does not carry any weight. I think Livio makes his point.

He also uses the Golden Ratio as a framework to illuminate other historical tidbits about key mathematical figures, guys like Pythagoras and Euclid, who continue to affect the mathematical world to this day. I love this kind of stuff; the historical context of how and why these legends did what they did is very interesting to me. For example, I did not know that Euclid himself did not discover geometry or even make any great new contributions to the field in terms of ways to apply it. What he is famous for is organizing the information into a coherent fashion. He was a teacher of the highest order; so much so that Abraham Lincoln himself used Euclid's texts, unchanged after all those years, to learn the subject back in Lincoln's log cabin days.

The book is not all a philosophical discussion. Livio does use some simple math examples to make his points but it was at a level that I could follow. According to my college professor, I escaped College Calculus by sheer luck. Livio does provide the rigorous math examples in appendices at the end of the book (I did not bother with these).

Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe. To be honest, I have always assumed that it was. After all, I am a Trekkie and this concept goes unstated throughout all four TV series. The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me. I have to admit; I need to ponder that one for a while.

I recommend this book. If you like the history of science, your high school algebra class is just a little more than a dark fog in your memory, and you get a charge out of scientific mysteries, you will not be disappointed.


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The Golden Ratio

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  • The Da Vinci Code (Score:2, Informative)

    by fee^ (94129)
    On the fictional side of this type of thing, those of you into this kinda stuff (like me) should read Dan Brown's 'The Da Vinci Code'. I've read that and 'Angels and Demons'. Both fantastic reads. More Info Here [danbrown.com]
    • Re:The Da Vinci Code (Score:2, Informative)

      by Kenja (541830)
      The movie "Pi" (as in the symbol) also touches on the subject via the Kabala and Jewish numerology.
    • by Anonymous Coward
      Fictional is absolutely correct, sir. Entertaining, perhaps even compelling, but in the end purely a creation of someone else's imagination. Those who accuse "The Passion of the Christ" of being anti-Semitic should also be willing to point their guns at Dan Brown for being anti-Catholic.
    • by wfolta (603698)
      The Da Vinci Code is gripping fiction, but it's not in the same class as The Golden Ratio.

      The Golden Ratio is carefully and deeply researched. The Da Vinci Codes is allegedly based on research, but the "research" behind it is recycling tired old conspiracy theories.

      From his statements online and in his forward, methinks Dan Brown is trying to have it both ways: claim it's based on fact but use the plausible deniability of it being a fictional work. It is a gripping read, don't misunderstand me. But you ha
    • by Anonymous Coward
      Dan Brown is the greatest writer of all time.

      Provided we ignore EVERY OTHER WRITER EVER.
  • by Gorimek (61128) on Friday February 06, 2004 @01:38PM (#8203007) Homepage
    The concept of math isn't even spread very far on this planet.
  • by Anonymous Coward on Friday February 06, 2004 @01:40PM (#8203047)
    He who has the gold, makes the ratio.
  • by s20451 (410424) on Friday February 06, 2004 @01:41PM (#8203055) Journal
    Didn't read the book.

    If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal. Hence, if a different civilization has different mathematics, they have different physical laws as well.

    This is basically a postmodern viewpoint, that reality is socially constructed. This viewpoint has been largely derided by the scientific community, and has failed to replace science because it hasn't really offered a compelling alternative. The only way I can see it being true is if other civilizations don't "exist" in the universe as humans do.

    Do a google search for Alan Sokal for a scientist's viewpoint of postmodern scientific criticism.
    • by kfg (145172) on Friday February 06, 2004 @01:52PM (#8203220)
      I offer this argument to those who state "You create your own reality."

      I kick them in the shin.

      Then say, "Why did you do that?"

      KFG
      • "Do what?"
      • I answer: "Clearly, I am a masochist."
      • a la "I refute it thus?" [samueljohnson.com] ... except Samuel Johnson's method was a lot more friendly than yours!
    • by Raindance (680694) * <johnsonmx.gmail@com> on Friday February 06, 2004 @01:53PM (#8203232) Homepage Journal
      Well, I wouldn't say that "if a different civilization has different mathematics, they have different physical laws as well."

      Rather, I'd set mathematics and logic equal (there's a respected tradition that does, see Bertrand Russell, Principia Mathematica). Then, to say that mathematics isn't the same across the universe, one would say that logic isn't the same across the universe.

      Now, "Logic isn't universal" is a damn meaningless statement. It'd translate into "Logic cannot describe [timespace-area/context] X." Which is, of course, a logical assertion about X.

      I think either the reviewer's portrayal of the argument or the argument itself is bogus.

      RD
      • by NonSequor (230139) on Friday February 06, 2004 @03:40PM (#8204818) Journal
        The fundamentals of logic are assumed to be true and it's very unlikely that any alien civilization would not use the same principles or at least principles equivalent to them.

        However, it would be possible to derive mathematical systems very different from our own. It all depends on what one takes to be fundamental concepts. For example, we define functions in terms of sets, but we could also define sets in terms of functions.

        We're not even certain that some of our own axioms are true. For example, the axiom of choice says that given any set of disjoint non-empty sets, there exists a set that contains exactly one element from each set. While most people will say that this seems to be a reasonable statement, if it is true, a number of counterintuitive statements are also true.

        None of these things change the universe, only the way the universe is modeled. One might be able to come to some new conclusions and possibly even a few contradictory conclusions using a different form of mathematics, but all in all mathematics effectively is universal since there is no reason a mathematician from earth couldn't learn to understand alien mathematics.
    • by dabadab (126782) on Friday February 06, 2004 @02:09PM (#8203421)
      Mathematics clearly may not be universal as it is a purely human thing - it's not like physics or biology where you have to follow existing things. There are lots of things in mathematics that is/was completely "made up" (Boolean algebra springs to mind) - in the end, math is just as human as poetry.

      Aliens does not have different rules of physics - but they probably have different models about it,but that should not come as surprise as we, humans had lots of different models of physics (and nature in general) throughout our history (Newton's mechanic view, quantum mechanics' probability approach, etc) - but it does not mean that Heidelberg did not exist as Newton did.

      Sokal was basically trolling (to demonstrate that "postmodern scientefic criticism" is BS) - and probably that's what you do too.
      • Boolean Algebra was indeed "made up" but just happened by an amazing coincidence to be a perfect model for creating digital circuits. Or is it a coincidence? Math is a game of sorts, you start with a set of rules and see what develops from them. The trick is, if you choose the initial rules carefully, you get results with real-world use. Minkowski space was a math oddity for years until Einstein realized it was a perfect model for General Relativity. It appears that an even odder model called Geometric
      • by Listen Up (107011) on Friday February 06, 2004 @04:01PM (#8205179)
        Wow, the shear ignorance in this entire article and book write-up is amazing. Not to truly upset anyone, but everyone here on Slashdot also appears to have a high school alegbra or entry level college mathematics background.

        To start with, Mathematics is not just as human as poetry. Where do you get that idea? Yes, pure mathematics (which is my passion in life) is essentially pure thought. BUT, nothing in mathematics is just 'made up'. All mathematics is based on fundamental, logical axioms (truths), and if anything were to violate those axioms, or the completely logical conclusions drawn therefrom, it would not be mathematics. You can think of mathematics as a grand puzzle, with each discovered piece and each mathematical truth found spelling out a larger picture. You can create bogus logic, bogus mathematical problems but it does not make it true mathematics.

        You are also confusing human representation with mathematics in your other statements. On a fundamental level, a law is a law, mathematically/physically/logically/universally. The universe is not ruled by human imagination (i.e. completely imaginary human created friend(s) as in religions) and therefore the system to understand our universe has to follow the same sets of rules as the universe (even rules involving possible pure chaos, as in some areas of quantum theory). Without mathematics, our universe and all that lies within it could only be understood on a physical observation level. Mathematics is the language of the universe, it is the language of physics.

        For a slightly deeper explanation, let me explain that Mathematics does not involve physical representations as you were taught in HS and earlier. For example, the number 1 as opposed to a capital S to complete addition (which is a logical law) means that 1+1=2 is the same as S+S=* because the system is beyond the physical characters used to represent the logic. The logic would not be different in an alien society. The laws of the universe do not change, therefore the same logic would be implemented. Using a 1 or an S would make no difference.

        There is sooooo much more, but just reading this story and people's posts makes me sad on a certain level. One of the oldest "truths" in the world...The person who is always least understood is not an artist, it is a mathematician.

        • by efflux (587195) on Friday February 06, 2004 @06:50PM (#8207512)
          I'm going to take the time to respond to this post at some length, because I feel it raises certain issues that need to be addressed. I hope to address them in a manner conmensurate with their depth. Please bear with this analysis of your arguments.

          Wow, the shear ignorance in this entire article and book write-up is amazing. Not to truly upset anyone, but everyone here on Slashdot also appears to have a high school alegbra or entry level college mathematics background.

          Personally, I was rather surprised at the book review myself. I have found, however, many fine folk here on slashdot who appear to have a solid theoretical background in mathematics, perhaps they are hiding out on this one.

          To start with, Mathematics is not just as human as poetry. Where do you get that idea? Yes, pure mathematics (which is my passion in life) is essentially pure thought. BUT, nothing in mathematics is just 'made up'. All mathematics is based on fundamental, logical axioms (truths), and if anything were to violate those axioms, or the completely logical conclusions drawn therefrom, it would not be mathematics.

          I find this is an odd thing for a self-styled mathematician to say. First, I would not label axioms as "truths" but rather as "putative truths". That is their purpose (to be putative). They are styled from not only as "seeming to be true" or appealing to intuition, but also in their ability to form a basis of thought, their simplicity, a small number is required for important results (as in defining a system), and their seeming irreducibility. Axioms are usually formed "post hoc", with an idea of the desired results in mind. Don't you find it disconcerting that in Topology the definition for an open set in a metric is a union of open balls, but an open set in a topologcial system is definied simply as that which exists in the topology? It's a function of how strong of results you have, and later developments are "force fit" into prior studies. Much like topological systems are "force fit" into Metric systems by how they are to be constructed, thus claiming itself an abstraction of the latter.

          Furthermore, you claim that any results that contradict the "fundamental" axioms, is not properly called mathematics? You are aware of the results of Godel and his Incompleteness Theorem? It's more than the latest rave, it has results that bear directly on what you are oh-so-devoutly avering. In this light, what is mathematics, when it is either incomplete or contradictory? And quite often it is contradictory? Which set of axioms is the more fundamental one you wish to choose? And how do you choose it? These are problematic questions, with unforthcoming answers. Modern mathematics is rattled as it has been faced with it's own, ultimate, incompetence. It describes nothing but itself.

          You can think of mathematics as a grand puzzle, with each discovered piece and each mathematical truth found spelling out a larger picture. You can create bogus logic, bogus mathematical problems but it does not make it true mathematics.

          Disregarding your mixed metaphor "spelling out a larger picture", this doesn't make much sense to me. I guess I don't understand how you are differentiating bogus mathematics and true mathematics.

          You are also confusing human representation with mathematics in your other statements. On a fundamental level, a law is a law, mathematically/physically/logically/universally.

          On what fundamental level? What is a law? How is one to determine what is a law? And what is derived from a law? Or what is contradictory to a law? I do agree that the grandparent poster did mix things up when he contrasted mathematics to physics et al.

          The universe is not ruled by human imagination (i.e. completely imaginary human created friend(s) as in religions) and therefore the system to understand o

    • by Ralph Wiggam (22354) * on Friday February 06, 2004 @02:09PM (#8203430) Homepage
      Different civilizations use different math systems, but they're all related by basic concepts. Addition is addition, no matter how you write it down. Any two math systems can be translated back and forth.

      -B
      • by Karhgath (312043) on Friday February 06, 2004 @04:17PM (#8205431)
        Different civilizations use different math systems, but they're all related by basic concepts. Addition is addition, no matter how you write it down. Any two math systems can be translated back and forth.

        WRONG.

        Let's take 2 valid mathematical system: Classical Mathematics(CM) and Intuitionist Mathematics(IM).

        One thing that is provable in one system might not be provable in the other, or could even be wrong.

        For example, if we take the mathematical subset of Logic we have Classical Logic(CL) and Intuitionist Logic(IT).

        In CL, NOT(NOT(P)) |= P.
        It is easy to see why.
        Same with A OR NOT(A).

        However, for IL, something is only True if and only if it's provable.

        So, NOT(NOT(P)) |= P becomes:
        If there is no proof that a proof of P is impossible, then P is provable. This is invalid. The absence of a counter-example doesn't prove the fact.

        So we see that NOT(NOT(P)) doesn't imply that there is a proof of P.

        Same for A or NOT(A), because we cannot assume that it's always possible to either prove A or it's negation.

        One of the fundamental differences in the 2 math systems is that, in IM, it requires a constructive proof.

        So, in IM, you cannot prove something like that:

        Proof
        (...)
        Case1: A = X then (...)
        Case2: A != X then (...)
        (...)

        This doesn't work, for the same reason as A or NOT A, you need to prove one or another, so you need to prove that A = X or that A != X.

        Ok, the point is, these are 2 working, acceptable and valid mathematical systems, but they cannot be swapped, because CM != IM.

        So, NO, two math systems CANNOT be translated back and forth. This is but the tip of the iceberg.
    • by CGP314 (672613) <CGP@ColinGregory ... t ['Pal' in gap]> on Friday February 06, 2004 @02:32PM (#8203764) Homepage
      Even if the laws of mathematics are not universal, that doen't mean that the laws of the universe aren't... er... universal. There isn't any law that says `Nature follows math'. As far as we know, math is a useful tool that seems to be able to predict what will happen - but that doesn't mean that math is the cogs and wheels behind the scenes.

      -Colin [colingregorypalmer.net]
    • by Mr. Slippery (47854) <`ten.suomafni' `ta' `smt'> on Friday February 06, 2004 @03:26PM (#8204589) Homepage
      If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal.

      You're assuming a relationship between mathematics and the "laws of nature" that isn't there. As Einstein put it, As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality."

      Mathematics is as socially constructed as any other form of language. It is based on axioms and defintions, not observation of reality. We select those axioms and definitions in a way to be useful to us, just as we select for those lingustic constructs that are useful. But this selection is based on our desire to communicate with others - it is a social construct. Once upon a time if you asked mathematicians what nubmer, when squared, gave negative one, they'd say there was no such number; now, any bright middle school kids know it's i.

      "Reality" is also to a large degree socially constructed, since all can ever speak of is our observations, which are socially conditioned. You see what you expect to see or are trained to see. (You don't see the fnords [mit.edu], or Sombody Else's Problem [wikipedia.org], while the hypothetical planet Vulcan [csicop.org] (the one inside the orbit of Mercury, not Mr. Spock's home) was observed several times, as were Blondlot's N-rays [skepdic.com].) This is why double-blind protocols are used - though if everyone involved has an expectation, that doesn't help.

      What we think of as "reality" is just a model that we mostly share. The electron, for example, is not a component of human experience but a component of a model that unifies and predicts many observations. That is a very good and useful model, but it is entirely conceivable that some extra-terrestrial civilization has (or some future human civilization will have) a model that is just as useful but doesn't contain anything like electrons. (Just like Chinese Medicine has a "patterne-thinking" model of the human being that is radically different than and incompatible with the reductionist model, yet is extremely useful.) What would such an electron-free model look like? I can't tell you, I'm too conditioned by the electron model.

      Remember: for any set of observations, there are an infinite number of hypothesis to fit them. There's no end to the curves you can plot through any finite set of data points. We see the points and call them a line, but it ain't necessarily so [principiadiscordia.com]. The best we can do is eliminate lines that don't go anywhere near the points.

    • by photo storm (610997) on Friday February 06, 2004 @03:41PM (#8204831)
      You bring out a very subtle fallacy, and one that is tied to philosophical issues regarding mathematics. I bit of history is in order:

      The fundamental question is this: is, or isn't, mathematics an extension of logic? A smart man named Frege (read about him here [st-and.ac.uk]) said, yes, it is. He showed a way to connect formal logic with set theory, which is the basis for mathematics as we know it.

      There was only one problem: Russell's Paradox. Bertrand Russell [st-and.ac.uk] showed that, using Frege's axioms that defined set theory, we have a contradiction - Russell's Paradox. And as any student of logic knows, a contradiction can be used to prove anything at all, which means that mathematics as Frege defined it was not viable.

      To make a very long and very interesting story short, Russell (with Alfred Whitehead) attempted to create a foundation for mathematics that would not give rise to Russell's paradox - the Principia Mathematica. And everyone thought the world was cool.

      Then, in the 1930s, Kurt Godel [st-and.ac.uk] came along and smashed a hole in Russell's approach by showing that, given a sufficiently powerful formal system, one will always find unprovable truths and irrefutible falsehoods. So mathematics was, by that line of reasoning, incomplete.

      This leaves the door open to a variety of critiques, the most relevant of which is that it is automatically not universal. After all, how could it be - there are things missing! We can't prove everything that is true, and we can't disprove everything that is false!

      Godel's argument tells us that we are unable to describe the universal laws of nature using non-universal and incomplete mathematics. That dosen't make mathematics useless - it just places a limit on what we can or cannot do. For instance, we cannot use deductive mathematics to describe the laws of nature in their entirety, because we know that any effort to be complete is doomed to failure - by Godel's theorems.

      Also, there are some specific areas of mathematics that lead to direct examples of non-universal, but nonetheless consistent interpertations of nature. Take, for instance, Euclidean and differential geometry. Euclidean geometry is the geometry of flat planes, whereas differential geometry describes abstract mathematical notions. It was once thought that Euclidean geometry is "sufficient", and that it is the simplest way of representing spacial relationships. However, as it turns out, differential geometry is actually much more simpler when it comes to dealing with, say, the theory of relativity - even though it is not intuitively connected to our perception of the universe.

      So in short, we have two different "geometries", each of which can, supposedly, explain spacial representation. Both are valid, but one is much more useful. Neither is universal. And yet, there is no contradiction.

      I don't know about anyone else, but I think this stuff is interesting.

  • by kzinti (9651) on Friday February 06, 2004 @01:41PM (#8203059) Homepage Journal
    Something I like about the golden ratio is that it is the number that is exactly 1.0 greater than its reciprocal. This makes it easy to remember the exact value: just solve

    x = 1 + 1/x

    You'll get a quadratic with the solutions (1 +/- sqrt(5))/2, or 1.618... and -0.618...
  • by liquidpele (663430) on Friday February 06, 2004 @01:41PM (#8203060) Journal

    How could it not be?
    Math is just a way of describing objects, forces, and interactions... how could you describe them differently??
    • by greatmazinger (747174) on Friday February 06, 2004 @01:46PM (#8203124)
      Math is just a way of describing objects, forces, and interactions..

      Ummm, no. That's not math. That's physics. Math is more abstract and one can do math without associating any of the concepts with "reality". One you use math to model reality, it becomes science and engineering.

    • We should be able to apply our mathematics to everything. However, if our thought process is drastically different, they ("aliens" if you will) may not recognize it as a corollary to their ideas. They may not even have a use for it.

      For instance, math doesn't really need to add. The concept can be completely explained with the concepts of negative numbers and subtraction.

      To get a real handle on the concept of different mathematics models, take the extremely difficult class of Abstract Algebra. It's cal
    • by Timmeh (555676) on Friday February 06, 2004 @02:09PM (#8203432)
      HEY! Don't push your modernist science bullshit on me, PATRIARCH. My goddess awakening mentor told me about you so-called intellectuals. Using 'symbols' and 'information' is just another form of OPPRESSION.
  • Movie (Score:5, Informative)

    by savagedome (742194) on Friday February 06, 2004 @01:42PM (#8203072)
    The movie PI [imdb.com] is also a very compelling watch for those who are interested.

    • Re:Movie (Score:5, Funny)

      by ath0mic (519762) on Friday February 06, 2004 @01:49PM (#8203181)
      ...which I think also holds the record for the "longest movie title in history."

      :)
    • Pi the movie (Score:5, Interesting)

      by cryptochrome (303529) on Friday February 06, 2004 @02:02PM (#8203356) Journal
      I literally watched that movie 2 nights ago. Spooooooky....

      Not bad (aside from one glaringly obviousl mathematical error). The thing that I mulled over the most was the proposition that a large integer could be a number of fundamental significance. In the movie it was 216 digits long. I had always figured all the really fundamental numbers were irrational. After thinking about it and looking up on the internet it seems there are actually only 6: pi, e, i, 1, 0, and phi (and arguably, -1). And the first five can be directly related with the equation:

      e^(pi*i) + 1 = 0

      phi is not directly related to the others in such a manner (In the movie the god number is somehow tied to both pi and phi). Although pi and phi both happen to be ratios that are also irrational. But to get back to my original point, the suggestion that any number of a truly fundamental significance besides 0 and 1 would be not only rational but an integer seems improbable.
      • Re:Pi the movie (Score:5, Interesting)

        by kallisti (20737) <rmidthun@yahoo.com> on Friday February 06, 2004 @02:38PM (#8203867) Homepage
        suggestion that any number of a truly fundamental significance besides 0 and 1 would be not only rational but an integer seems improbable


        What about the Monster [wolfram.com]?
        This is the largest "simple" group which doesn't fit into any group category. What this means is rather hard to explain in simple terms, but this group has lots of mysterious connections to other maths. The order is 2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 *29 *31*41*47*59*71.

  • What reasons would there be for an alien to not understand or accept that one plus one equals two. Any being capable of human-equivalent level of thought would be able to count objects. Whether they did in this in base-2 or base-3 or base-10 or base-12, it doesn't matter because all these bases can be reconciled to each other.

    Could there be some areas of mathematics that humans have discovered that has not been discovered by an alien race? Sure. Prior to Newton there was no calculus and so Kepler had to discover the period of planetary orbits using geometry and algebra. But this does not mean that Kepler would not have used calculus if it had been available to him, only that such a concept had not yet been thought of.

    But counting and simple addition and subtraction are mathematical operations that are mastered even by animals. It is fairly condescending to assume that aliens could not even fathom those levels of mathematics.
    • by Sique (173459) on Friday February 06, 2004 @01:53PM (#8203233) Homepage
      With just adding and subtracting you get something called 'constructive Mathematics'. It's a subset of Mathematics, and it's missing some important axioms necessary for quite usual things like calculus (nonexistance of a supreme for any limited set of numbers) and algebra (no selection axiom, thus you can't prove that infinite dimensional vector fields have a base).
    • quote: "It is fairly condescending to assume that aliens could not even fathom those levels of mathematics."

      And it's fairly narrow-minded to assume that another life form in the galaxy has a frame of perception that's even remotely similar to ours. This is more than just saying "what if they see in infra-red!", but rather to say that we feel mathematics is the de facto language of the universe because it (as with terrestrial life on Earth) doesn't yet have a competitor.
      • I can take this a step farther. The human brain is set up to recognize patterns. In the same way that all computers come down to mathmetics and binary (no matter what it is you fed into it or what output you desire) the human brain breaks down into pattern recognition.

        I see no reason to assume that an alien race has brains that function like that. They could function in any number of ways, I think they all could have an effect on how those creatures approached mathmetics.

      • This is more than just saying "what if they see in infra-red!",

        The reason we see in "visible light" is because that is the brightest radiation given off by most stars, especially our nearest one.

        If an alien's local star gives off most of its radiation in a slightly different part of the spectrum, still mostly visible light, but let's suppose shifted more towards blue, then the alien's visual systems will be evolved to see that the best.
    • Seeing how Archimedes did his thing with volumes [thewalters.org] simply through deduction and observation, I find it hard to believe that others in the universe wouldn't have the same basic powers of observation and logic. Archimedes almost discovered integral calculus through what amounts to puzzling out how to calculate volumes of strange objects. Its a leap of logic and happened in the absence of advanced mathematics as we know them. I suppose its possible that our way of doing math is very different, however at the low
      • No, but quantum behavior does. It is there, but not there, and ove there too... Also the problem is the info comes in quantas. If a truly alien race first dicovered this, and constructed their physics -> math based on this, then the concepts of integers for them is extremely alien.
        • No, but quantum behavior does. It is there, but not there, and ove there too... Also the problem is the info comes in quantas. If a truly alien race first dicovered this, and constructed their physics -> math based on this, then the concepts of integers for them is extremely alien.

          An interesting idea; under what kinds of conditions might it be possible? An electron may be both here and there, but a sheep (to pick a particular object) is either here or there. Technically, I suppose that this sheep

    • by arbour42 (731167) on Friday February 06, 2004 @02:03PM (#8203367)
      Prior to Newton there was no calculus

      In a fascinating book, a Hindu scholar and monk, Sri Tirthaji, discovered in the Hindu Veda scriptures the basis for our math system. There he found shortcuts for most all our math work - easy ways to do difficult long divisions in a matter of seconds, quadratic formulas, PI to over 32 digits, the Pythagorean theorem (much before the Greeks), derivatives, calculus.

      Our math is actually from the Vedas, and the Arabs got it from them, and then spread it through the Western world. The Vedas are at least several thousand years old.

      The book is called Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas [amazon.com] and can be found at amazon or used book stores.

      It's one of the major works of genius of science. The first time i read it, it was shocking how advanced it was, and simple! Any division such as 1.748362 / 59487 can be done long handed (pencil and paper) in a minute.

      Our math system, how it was discovered or invented, who knows and how far back, is absolutely brilliant.
      • I read the book in question a couple of years back. Just would like to say this:- you'd like to distinguish between arithmetic and mathematics, as also, perhaps, between actual scientific history and propaganda.

        Vedic 'math' is mostly arithmetic; it's about how to multiply numbers faster (cool method that; helped me throughout most of school) and, like you said, long form division. Even in that, I doubt it was from the Vedas themselves; I remember reading about those 'tricks' (using the term in a broad sens
  • Thumbs up (Score:2, Offtopic)

    by haxor.dk (463614)
    I own the book, bought it a year ago myself. A good read.

    If you're looking for something a bit along the same lines, but sprinkled with history, religion and conspiracy, I can recommend "the Da Vinci Code" by Dan Brown.
  • by Wind_Walker (83965) on Friday February 06, 2004 @01:46PM (#8203121) Homepage Journal
    Regardless of what radix used (10-based, hex, octal, etc) "6" of something is still "6". Call it whatever you want, some alien name. There are still 6 items there.

    By the same nature, prime numbers are always prime. There exist a certain number of things (5, 7, 11, etc) and cannot be evenly divided. Period. We call them prime numbers, and we use our base-10 radix. Aliens could call them Borgolsmocks in their base-182, but the fact still remains that a Borgolsmock cannot be divided evenly.

    And I firmly believe that no intelligence would survive for long without a knowledge of mathematics. Counting the days for crop rotation, the ability to evenly divide food among the tribe, and communication of the number of animals in a herd... mathematics will be generated in the evolution of any intelligent species.

    And it is truly universal.

    • by Lemmeoutada Collecti (588075) <obereon.gmail@com> on Friday February 06, 2004 @02:09PM (#8203423) Homepage Journal
      Borgolsmock is divisible using a thersian constant. To assume that just because your single dimensional mathematics cannot divide a borgolsmock does not similarly limit n-dimensional. Why the mere act of division is a limitation of linearly conecived time, a limitation we have never faced.

      Aside from which, where you see a single item, I percieve an infinite semi recursive series. There are more than one apple in that one apple. There are an infinite (using your limited numbering) number of apples. That apple you call 'one' in fact contains the entirety of it's temporal measurement, which is a bounded infinite series. So now tell me about this concept you call singularity?

      • And maybe if you inverted the warp plasma through the deflector dish, you could create an inverse tachyon field that could disrupt that thersian constant! Make it so, Number One!

        You see, I was limiting myself to thinking of... what's the word... reality. I was talking about plausable scenarios of intelligent extra-terrestrial life. You've obviously spent too much time reading Sci-Fi novels and not enough time in what we like to call the "real world" - and it's not that shitty MTV show you watch.

  • A god with a plan? (Score:5, Interesting)

    by mindstrm (20013) on Friday February 06, 2004 @01:46PM (#8203129)
    Why does this make you think there is a supreme being, with a plan? Just beause things work out?

    The balance and beauty of nature and all that?

    OF COURSE there is a pattern, and things work out. Look at evolution.

    You take a puddle in the middle of nowhere.. it has an ecosystem in it with a perfectly balanced population (too many, it dries up, too few, they reproduce...). Would these little creatures say "Oh wow! Look how there is JUST enough water for each of us! There must be a GOD!".... silly, right?

    Nature seems balanced in the world, becuase that world produced nature... they are intertwined, infinitely.

    Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

    The fibonacci series and the golden ratio are related? Sure are.
    (The ratio of successive numbers in the fib. series approaches the golden ratio as you go upwards)

    But it's not so weird, is it? A sunflower.. the way it grows, it builds on itself.. in a spiral... naturally following this series.

    Is it some grand creator that made it that way, or is it just the plain, obvious way for something to grow?

    What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

    Until then, i'ts just nature.
    • What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

      If God had intentionally inserted all these frequently recurring constants and ratios everywhere

    • Since I'm taking a class on general relativity this semester, I'll weigh in with a quote of Albert Einstein.

      What I'm really interested in is wheter God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all.

      In other words, maybe nature is what it is because God created it that way, or maybe it is what it is because it has to be.

    • Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

      Indeed, there are simple examples of this, for example the fraction 1/3 can not be expressed in decimal notation except as an infinitely repeating value (0.3333...), however, expressed in base three it is simply 0.1.

      This leads to why you shouldn't use floating point numbers on a computer to balance you
    • by scorp1us (235526) on Friday February 06, 2004 @02:23PM (#8203583) Journal
      I tend to agree with you. But I think that the use of "God with plan" actually weakens the argument.

      Assuming God is all powerful, as is the usual definition of God, then God would not need to follow any plan. Things would just be. 1+1=2, 1+2=3, etc until you try to do math with a number that God had not created yet. Then thigns would break down.

      Of course, that is the plan - to keep things consistant so they scale and continue to work. x1+1=x2 ... x(n-1)++1=xn gives a number line that is infaninte in size. Things since all things are mathematical in nature, and constructed from the same elements of the universe, everything shares a common mathmatical foundation. When that is the case, of course a few of the same numbers pop up everywhere. Pi, e, and the Golden Ratio to name a few.

      Once you realize that God is slave to math and rules, then you must comclude that math is more powerful and absolute than God. Therefore your old notion of a traditional God should be superceeded byt the ultimate one - mathematics.

      When you pray, you pray that the maths of the universe work out in your favor. Since we mathmatically backtrace events, we know that God has not suspended reality, but you have mathmaticaly evaluated the likely outcomes and calulated the propability of your favored action to be within the realm of mathematical rendering. So you pray. Had it been clear cut you would not have wasted your time.

      Math is the CPU in wich the universe runs.
    • One small point (Score:3, Insightful)

      by Atario (673917)

      Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

      Well...not holy, but yes they are significant. They can't be expressed as a fraction in any counting system. (Unless, of course, you cheat and decide to "count in units of" the exact irrational you're trying to express, in which case it's going to be "1" -- you've just replaced one symbol (pi, e, ph

    • by zapp (201236)
      And who is to say that God didn't design the system such that it balances itself in such a way?

      People like you seem to take for granted that the universe just exists. That spacial dimensions and time all sort of slid together to happen to be this way. Matter just so happens to work in such a way that that puddle can exist (not just one puddle, but any infinite number of similar but totally different puddles) and sustain an entire eco system.

      We can simulate the entire puddle and ecosystem in a computer wit
  • by gpinzone (531794) on Friday February 06, 2004 @01:46PM (#8203133) Homepage Journal
    Donald in Mathmagic Land. It was a great little video Disney produced back in 1959 with Donald Duck. The narrator goes off the topic at times, but the overall animated descriptions of the golden ratio and its related golden values were awesome. Unfortunately, this Disney short is not available on VHS or DVD currently. Look to eBay to find a long lost copy of it.
  • Furniture design (Score:5, Interesting)

    by hulap0pr (562242) on Friday February 06, 2004 @01:47PM (#8203134) Homepage
    The golden ratio concept is a big part of furniture design. Case pieces (boxes, bureaus, etc...) appear more balanced and pleasing to the eye when the golden ratio is followed. Go home and measure your highboy...
  • How hard was it to write this without mispelling ratio as ration?
  • universal math? (Score:5, Interesting)

    by mblase (200735) on Friday February 06, 2004 @01:48PM (#8203159)
    The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me.

    Yes and no. Mathematics is just a way of modelling things abstractly. Even things like counting from one to ten is a model for concrete objects, and provides a way of, say, making sure the number of cows you have today is the same as the number of cows you had yesterday. At the higher level, mathematics lets you model shapes, motion, acceleration, and gravitational collapse of entire stars.

    The most common types of mathematics we use include decimal arithmetic (trading with money), algebra (solving for unknown quantities), and geometry (simplifying the world into abstract shapes). Hundreds of other branches of mathematics exist to model different things in different ways, and none of them are "right" -- they all are optimized for particular problem-solving.

    With that in mind, I find it inconceivable that advanced civilizations on other planets would not have some kind of mathematics, and at least share the natural numbers with us (not necessarily base ten, though). If all you're doing is raising food for your family, then even arithmetic may be more than you need to bother with. But anything that involves abstract problem-solving, measurement, and/or exchange of goods for trade is going to need some kind of math. The models they use may bear no resemblance to the ones we use, but that doesn't mean it's not math.
  • I'd be fascinated to hear more about this. I want to get the book but I'm impatient and want to discuss it now! :)

    I would think that math in some was is universal, in the sense that every sentient creature has to figure out a method of counting. Some creatures count in base 10, others base six, maybe base 12. Other creates could figure out a counting base we haven't thought of yet. However, if they have a method of counting and measuring, I'm sure we'd have a method of translating their mathmatical models to our own, without too much trouble.

    Perhaps the definition of math here is different than mine? Thoughts?
    • The flaw in that line of thinking, which many on /. are making, is assuming that what we percieve singularly is similarly percieved by another species. Let's take a thought walk for a moment... First, we percieve an object, say a book, at a single position in time/space. While we can percieve the entirety of it in space, our perceptions cannot percieve simultaneously the entirety of it's temporal measurement. Therefore, we see on book in the now.

      Now another lifeform comes along, one which can percieve the entirety of the book in time/space. They percieve not only a different book than we are capable of, but further, they may percieve each temporal book as a seperate item, just as we percieve spacially translated objects as seperate. So where we see a single book, they see an infinite number of books. We can only assume that their method of counting would differ from ours, or that we would be unable to correlate ours to theirs because we can not percieve the many, only the one.

      Assuming that another specias percieves the universe the way we do is the height of hubris, and the largest flaw in alien contact scenarios. Our mathematical beauties when percieved on a larger scale may be no more than a mere curiosity, instead of the vaunted unchanging laws.

      Just a thought.
  • "Captain, I believe there is a 1.6180339887 percent probability that any security officers beamed down to the planet will survive."
  • Definition FYI (Score:3, Interesting)

    by Prince Vegeta SSJ4 (718736) on Friday February 06, 2004 @01:48PM (#8203166)
    Golden Ratio:

    The number 1.618..., which is half the sum of one plus the square root of five (1+SQRT(5))/2. This number was known in ancient times, and has many interesting properties in many fields. In Fibonacci series, the higher one goes in the series, the closer the ratio between a number and it's predecessor comes to the Golden Ratio.

    From "The Technical Analysis of Stocks, Options & Futures" - William F. Eng

    Geez, I never thought my Trading and /. would come together. Then again it is delving into the Uber Math Geek world - lol

  • This is more of a historical book than anything else. Since it is called the golden ratio, I expected more about "the golden ratio" that what it offered. I wish I had read the Amazon customer reviews before I wasted my money on this junk.
  • by bravehamster (44836) on Friday February 06, 2004 @01:50PM (#8203183) Homepage Journal
    Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe. To be honest, I have always assumed that it was. ... The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me.

    From what I understand, the vast majority of realistic first-contact scenarios postulated involve using mathematics as a common ground to bridge the language barrier. 1 + 1 equals 2 in every language on earth (except New Age holistic 1 + 1 = 3 crap). It makes sense and it works everywhere. It would be awfully damn hard to build a spaceship without mathetmatics, let alone trying to calculate launch trajectories or transfer orbits. Unless they had such an intuitive grasp of higher level mathematics that they don't even consider it worth talking about, I don't see how any species that had no concept of math could ever rise above the level of pointy sticks and sharpened rocks. And even then you'd probably want to keep track of how many rocks you had to make sure Lurg over there didn't *borrow* a few.

  • by dtfinch (661405) * on Friday February 06, 2004 @01:53PM (#8203231) Journal
    He probably just caught on to the idea that if you say something outrageous enough in your books, like math is wrong, people will buy them.
  • Phi (Score:5, Funny)

    by Rupert (28001) on Friday February 06, 2004 @01:53PM (#8203240) Homepage Journal
    I hate it when people use extreme amounts of decimal precision when talking about irrational numbers. Really, is 1.6180339887 (or 1.6180339887498948482045868343656) much more informative than 1.618? If you're going to do calculations with it, use the exact value:

    1/2 * (sqrt(5) + 1)

    and sort out the irrational bits at the end, rather than introduce rounding errors at the beginning.

    That's just a rationalisation, of course. My real reason for complaining about decimals is that it feels wrong. 1.6180339887 does not look like a profound number. It's like the number is a beautiful woman, and the decimal representation is the pornographic pictures she posed for when she was young and needed the money. Yes, it looks like her, and it may even be useful. But the real thing is *so* much better.
    • I think you've just stumbled on a way to get more guys to study math...
    • Re:Phi (Score:3, Informative)

      by pomakis (323200)

      I hate it when people use extreme amounts of decimal precision when talking about irrational numbers. Really, is 1.6180339887 (or 1.6180339887498948482045868343656) much more informative than 1.618? If you're going to do calculations with it, use the exact value:

      1/2 * (sqrt(5) + 1)

      and sort out the irrational bits at the end, rather than introduce rounding errors at the beginning.

      This is sound advice for hand calculations, but of course it doesn't apply for floating-point calculations performed on a

  • math and humanity (Score:3, Insightful)

    by jstoner (85407) on Friday February 06, 2004 @01:56PM (#8203265) Homepage
    It's an interesting question: how far could a civilization get without math? IANA historian, but it seems to me the more sophisticated a (human) civilization, the better its mathematics. The Aztecs did develop a fair amount of math completely independently of Eurasian civilizations.

    Could a race become spacefaring without math? Could they develop the radio communications we could use to detect them? I suppose they could if the circumstances of their environment or adaptation (Low-gravity, bio-radio communications) allowed it.

    But how would you arrive at the necessary conclusions without an abstracted intellectual framework like math? Maybe progress would just be slower.

    Hmmm... makes you wonder what we're still missing.
  • Try Journey Through Genius [barnesandnoble.com] by Will Dunham.

    It covers a sampling of many of the great theorems and proofs of mathematics in a form that anyone with high school math can follow, as well as giving interesting insights into the personalities of the mathematicians (where this can be known). Most of them were, um eccentric. It is nice to know that Euler at least was well adjusted, if you couldn't exactly call him normal.

    Euclid is represented twice here: once for his proof of the Pythagorean theorm and once for
  • by Master Switch (15115) on Friday February 06, 2004 @01:56PM (#8203276) Homepage
    While the syntax we use for mathmatics is culturally defined, the content beneath them is not. We humans discover, not invent mathmatical constructs. As much as we would like to think we create, we do not. We iterate and find the best fit solutions.

  • Debunking constants (Score:5, Interesting)

    by hcg50a (690062) on Friday February 06, 2004 @01:59PM (#8203301) Journal
    Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books.

    This doesn't sound exactly right.

    I think it may be the case that writers have attributed the use of phi in art when there was no such intentional use by the artist.

    But the very nature of phi makes it unlikely not to appear in certain contexts.

    Same with pi.

    The thing I love about math is that it has utterly nothing to do with reality or the universe or anything at all.

    Typically, however, physicists make assumptions that match, more or less closely, to what is happening in the real world, so the conclusions from such assumptions match, more or less closely, to what is actually happening in the real world.

    But there is no reason why some utterly alien intelligence can't make a set of assumptions that match their reality, which would be utterly alien to us, yet still valid, and still recognizable by mathematicians, if not physicists.

    This is the giant flaw at the end of the book Contact, by Carl Sagan. Ellie discovers a message in the constant pi, placed there by an intelligence. If this were a constant of physics, that would imply the existence of some incredibly advanced intelligence that engineered the universe to contain a constant with precisely that value. This is somewhat plausible, and I believe it was Sagan's intent.

    But he picked pi, which actually has nothing at all to do with this or any other universe.

    What kind of incredibly advanced intelligence can possibly engineer that? I can only think of One.
    • This was actually the most tedious part of Livio's book. His argument, essentially, is that if a) the artist does not explicitly say that he used the Golden Ratio, or b) the Golden Ratio appears in the work to an accuracy of the third decimal, then the Golden Ratio was not used.

      Aside from conveniently (and fallaciously) proving a negative, the first condition is highly unlikely to be satisfied in any premodern work, and the second condition borders on the absurd. In one case, he takes a discrepancy of less
  • 1.61803399 (Score:2, Informative)

    by karmaflux (148909)
    More here. [vashti.net]
  • by Den_onda_kotten (616799) on Friday February 06, 2004 @02:09PM (#8203424)
    There is a really great page that explains the relation between the Golden ratio and the Fibonacci numbers here [surrey.ac.uk]
    The fibonacci number is the series 1,1,2,3,5,8... where every number is the sum of the two numbers before it. What does this have to do with the golden ratio? Everything! Just check it out, you'll be amazed.
  • by Xoder (664531) <slashdot.xoder@fastmail@fm> on Friday February 06, 2004 @02:13PM (#8203456) Homepage
    I see a lot of nay-sayers in this thread talking about "How could some alien not understand that one of one thing and two more of that same thing make three?"

    You are assuming that everyone has a concept of cardinality. Realistically, people don't have much of one beyond the number six (yes, there are outlyers for whom eight objects in a group is eight objects not one-two-three-four-five-six-seven-eight objects). If a being had no concept of cardinality, that would make many things more difficult, but many others much easier. This organism would not think of a system as the sum of its parts, but rather as a cohesive whole (or rather the cohesive whole). It is likely that they would be philosophical geniuses compared to us. There are creatures of this type toward the end of Calculating God by Robert J. Sawyer (See your favorite bookseller and/or your local library), and their possible existance is not implausable.
  • by HarveyBirdman (627248) on Friday February 06, 2004 @02:32PM (#8203772) Journal
    10 is not a given. Other number systems used in ages past include base 5 and base 20. The Babylonians used base 60. I believe that's where we get our time system from (60 seconds per minute, 60 second per hour).

    Two other interesting books: Zero: The Biography of a Dangerous Idea by Charles Seife.

    Trigonometric Delights by Eli Maor.

    Both books cover the a lot of historical ground in mathematics.

  • by Talinom (243100) on Friday February 06, 2004 @02:34PM (#8203796) Homepage Journal
    The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me.

    So that is why all of those UFOs are crashing all over the place.
  • Fibonacci (Score:5, Interesting)

    by Lewie (3743) on Friday February 06, 2004 @02:52PM (#8204067)
    This book was a great light read, the math is not difficult and some of the classic paintings and such were really cool to see.

    The most interesting part of the book for me was the correlation between Fibonacci and the Golden ratio. As I read it, as you ascend the Fibonacci sequence the ratio between the current number and the one before it converges on the golden ratio. F20 divided by F19 is as near the golden ratio to as many decimal places as any of us have use for, probably.

    An interesting "party trick" was also mentioned that I remember vividly. Take any two numbers and add them, then take the new number and the larger of the first two and add them, then take the new sum and the old sum and add, ala Fibonacci. Continue for twenty or so iterations and the 20th number divided by the 19th will be damn close to the golden ratio. This is, I think, because any such construction is a linear multiple of the base Fibonacci set (see prev. paragraph). When you divide, the common multiple falls off and you still get Phi. I thought that was pretty cool. :)

  • by Charles Dodgeson (248492) <jeffrey@goldmark.org> on Friday February 06, 2004 @06:43PM (#8207427) Homepage Journal
    I read the book a few month ago, and I largely agree with the review, but I have a few more criticisms as well.

    First let me highlight one of the really nice points that the author makes (with many well-researched examples in the book). Recently created myths about things long ago can easily be mistaken has ancient stories. It was interesting to learn that the Renaissance fascination in art and architecture was basically a 19th century invention. For me, the most interesting thing about the book is its debunking of similar historical myths, always working to show what grain of truth their might be to them.

    One minor gripe I have is in the context of the praise above. While debunking historical myths, the book reinforces the myth that Einstein's theory of Special Relativity was primarily motived by the Michelson-Morley experiments.

    For me, the both the most interesting thing and the most disappointing thing about the book is that the history of the Golden Ratio isn't all that interesting. What turns out to be most interesting is the history of the myths about the Golden Ratio.

    This is not to say that the Golden Ratio isn't interesting itself. It's relation to fractals, repeated fractions and parallel curves is interesting, but I guess I would have preferred a "happy ending" where it would play something likes its reputed role in psychology/aesthetics. Of course it is hardly the fault of the author that it doesn't have such an ending

Faith may be defined briefly as an illogical belief in the occurence of the improbable. - H. L. Mencken

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