Question
Dear Mitch,
I was wondering if you know anything about the "Golden Ratio" of mathematics, because in one of the high school classes I teach, a student brought up the fact that we are now in the month of April, which was the month Leonardo da Vinci was born in 1452. I was impressed that she knew that piece of trivia, but my real surprise came when another student brought up the fact that da Vinci, like many other painters, sculpters, and architects, had a preoccupation with the irrational number called Phi, and that mathematicians in particular have been studyiing it for a long time, and even though it's not as well-known to the average person as another irrational number called "pi" (as in pi x r-squared), Phi is the one that has historically been used and relied upon more than pi by artists, sculpters, architects, and designers. The student, whose father happens to be an architect, also pointed out that according to what he's heard his father discuss and debate so many times over the years, Phi seems to have such a mysterious power in so many areas of our lives that many people believe it is the secret number behind every structure in nature ... ranging from the rate that most animals reproduce ( which, incidentally, led the mathematician, Fibonacci, to discover his own important series) to the inflence it has long been believed to be a common element in the layout of each person's face as well as each snowflake that has ever fallen from the sky.
Well, I probably don't have to tell you this, but by now I'm sure you correctly assumed that this particular group of students I teach is not only an honors class filled with some of the most voracious appetites for learning that a teacher could ask for, but this year I believe I've got the most inquisitive and motivated class I have had the pleasure of teaching in the almost 20 years I've been a classroom instructor.
In any case, the reason I'm writing to you is this: Even though I would hate to burst the students' "bubble" about the famous Golden Ratio, I find it hard to resist the temptation to present an alternative view, which is supported by an almost equal number of mathematicians, architects, and art historians, who feel that this "Golden Ratio" is not quite all that it is cracked up to be in the real world. I have found myself equally persuaded by the skeptics' articles, which, from the early days of the study of phi, were also penned by highly credentialed and respected members of the fields I mentioned that propose (and even find ways to show ) that this magic and mysterious ratio is, well, overused and overrated in people's analyses of buildings, paintings, etc. One particular writer/mathematician whose name is momentarily escaping me, makes a very convincing argument that a lot of the ancient buildings and paintings long-assumed to be designed with strict adherence to this "Golden Ratio" are, in fact, not authentically present with as high a level of fidelity to 'Phi's magic number relationship' as many people would like to believe. Rather, these critics argue, the 'Golden Ratio' fans are sort of coerced into thier committment through the way that some authors, critics, and art historians make clever decisions that subtly present authentic-looking lines to emphasize their point. However, upon closer inspection, one can see that many of these lines and angles are remeasured and redrawn with greater precision than their photographs and polished images would naturally show before the images in the books are sent off for reproduction and publishing. And, of course, anyone wishing hard enough to find certain angles and lines to support their theory that they can 'round some numbers up', and 'round other numbers 'down' in an effort to end up with the 'evidentiary lines' boldly drawn and then printed wide enough to cover the underlying differences and inaccuracies between the reality of a piece and its adherence to the 'Golden Ratio' theory.
BOY!, I don't think I have EVER written such a long question in my life, but about halfway through I realized that in order to be fair to both sides I would need to devote enough energy to convey at least the gist of each side of the matter which sprung up in my class.
So, any response whatsoever from you would be a great help when I open up the class for the discussion that I know will excite many of the students at least as much as any of the planned lessons I will be delivering to them in the coming weeks .
Thank you for your patience in reading this almost non-ending dissertation of a note and (I hope) you will be able to respond to us, even if you do so with a quick three or four sentence approach -- as I am drawing a blank and find myself actually hoping for something to inspire me with some kind of springboard-thought that would get the students to shoot their hands up and begin to take sides for their classmates to analyze and debate.
So, whatever thoughts you have, whether you agree with them, disagree with them, or find yourself on the perverbial fence, please feel free to send them our way!
Warmest wishes for the lovely season on its way, and, once again, thank you for the time you might spend on this subject, and an even bigger thank you for all the fun problems, solutions, ideas and tricks you've shared over the last three years for students from Kindergarden through senior year of high school. I don't think you can imagine how appreciative many of the students (as well as faculty members) appreciate the tips, tricks, strategies and hints you've provided for students who (I think) would be even more stressed out regarding their upcoming SAT exam than they seem after copying down some of your foolproof methods. So, on behalf of those juniors and seniors, who tend not to be as fanatical as I wish they were about sending Thank You notes, THANK YOU.
Mr. Commandeer
North Carolina
Answer
Dear Mr. Commandeer,
WOW!
I must agree with you: you certainIy are in the top ten questions we've received since starting three years ago, and, although I cannot check for sure in the very near future while I am struggling to keep up with the letters and questions we answer but don't bother posting, I would venture to guess that you might so far hold the record for the longest query we have ever received!
I also would like to point out that I personally enjoyed reading your question, in part because I too have long-been fascinated by the 'Golden Ratio' theorem of Phi.
If everything goes as planned, I should have my response posted by early tomorrow morning. If not, then I will see to it that it is posted by tomorrow evening, though it may not make its way onto the screen until such a late part of the pm that you'd be more likely to be satisfied with a Sunday morning visit to the Adler-n-subtract.com site than if you devote too much time and energy staying up to see what appears here Saturday evening!
Until then...
Okay, I'm back. Now, readers of this particular question and answer need to realize that many, many books -- that's right, entire books -- have been written about the subject of this fascinating irrational number (or ratio) called phi. So, obviously, this is not an appropriate place to even attempt to get into some of the properties it seems to have, nor is this an appropriate place to try to give its long history, nor even why that history still has gaping holes during certain periods in which the mathematicians who were most active chose to devote their energies to other topics. Additionally, to anyone who is interested in persuing the topic in greater depth, it certainly is not hard to find such disussions and writings, even here on the Internet. Still, I promised to share some of the common notions about the number, which is certainly one of the most interesting numbers we have within our system of mathematics, so here goes:
First, the basic concept of the number was first advanced by the Greek geometer Euclid, but the actual name we use when referring to it wasn't invented until the early 1900's by an American mathematician named Mark Barr.
Secondly, people often miss interesting writings on the subject because it is actually known by varius names such as the "golden section" and the "golden ratio".
Third, although many books written for the general public state that phi is equal to 1.618, this is not really the whole story. That verrsion of phi is a rounding of its real value, which is an infinite nonrepeating decimal that looks like this: 1.6180339887....
The difference between the rounded version and the real, infinitely long nonrepeating decimal version is more important than a lot of students and laypeople might intuit. Why? Because the whole point of phi (just like pi, actually, the ol' 3.14... guy) is that a good part of its mystery and elusivenes actually comes out of the fact that there seems to be no end to its depiction, nor any known pattern. That's what makes a number like pi and phi "irrational". Recall, for a number to be "rational", it must be expressible as a ratio (or fraction) of two integers. A lot of people think that if a number is an infinitely repeating decimal, then it is necessarily "irrational", but that is not the case. Take, for example, the fraction 1/3. Well, it can, as you see, be expressed as the ratio of one integer to another (1 to 3). But in decimal form it is .3333333333333... forever. Note: 'rational' comes from the word 'ratio'.
Now, as I alluded to last time, there is an extremely close connection between phi and the Fibonnacci sequence. Remember that one? It goes like this: 1,1,2,3,5,8, and so on, with each number being the sum of the two numbers directly preceding it. Fibonacci studied rabbits (or some rodent, but I'm pretty sure I recall that they were rabbits), and he discovered that their population increased exactly like the series that bares his name. It has also been noted that the quotients of successive terms that are adjacent, going from right to left, like this: 2/1, 3/2, 5/3, etc. come closer and closer to phi, which you can quickly see for yourself by pulling out a simple calculator and giving it a try. Obviously, that interesting fact is high among the features that makes phi so intriguing.
Still, that is neither the simplest or most common feature of phi's properties, nor is it among the ones that architects and artists often try to work into their pieces. No, the simple aspect on which they focus is the fact that phi is easily utilized as a ratio that many, many artists and architects feel is uniquely pleasing to the human eye and mind; they also tend to fixate on one very interesting but not inexplicable feature that I have not yet mentioned, which is this: If you take a line segment (or piece of wod or string, etc.,), and call one end A and the other end C, and then place a third point between them (call it B, and place it so that the AC segment of the rope -- or whatever you used -- divided by the BC portion of it, you will find you can find a point for that B which will make AC/BC = the length of BC/ AB. In other words, the larger section can have the same relationship in proportion to the smaller section as the entire line has to the larger of the two sections.
And then there are the thousands of examples of this ratio in nature, such as this: A person's height divided by the distance from his or her belly button to the ground equals phi. And that, of course, is the kind of tidbit that gives the number (or ratio) its 'divine' nature. But there's a litle teeny-tiny problem with that: It's not a particularly true statement. Everybody is different, and some people have belly buttons a little higher than their height would lead one to calculate, while otherrs have it a little lower. Still, it is true that there are many, many places in nature where the phi ratio does come remarkably close to the ratio of things that grow. And this is not true of many of the other topics we see on chalkboards in classrooms as we go through the typical American mathematics curriculum. One example often noted is the measurements involving the way leaves spiral around a plant stem, or the way a flower's seeds arrange themselves as they blossom. But without getting into too much detail here, if you think about it, it makes perfect sense that plants, trees, and the like, would benefit from a garanteed pattern that is irrational. Otherwise, for example, if leaves sprung out of a plant's stem in a simple pattern that was ordered by a rational constant, such as in four or five inch increments, the leaves would be springing out in places directly above other leaves, shadowing them from the sunlight they need for photosynthesis and depriving their underlings of raindrops that might be neccesary to keep the exterior moist in regions that require moist protection from a blazing sun. Etc..
Lastly for now, many articles about phi seem to point out the fidelity with which ancient architects employed its proportions in the design of the pyramids as well as buildings as important as the parthenon. This is possible, of course, and architects, like members of any other field, are individuals. So there is a wide range of proportions employed in various works created during any period, and I, for one, once forced myself to stare at the parthenon from the best position I could locate to absorb the overall form as well as its geometric constituents, and then, after sitting down with a pencil and a piece of paper a tourist had been kind enough to leave on the ground when he had finished the candy bar it had sheltered, I was disappointed to find that if I was forced to arrive at the Phi ratio, I would have to select particular points of the structure to begin and end my calculations, and those points were either ones I selected arbitrarily (but not SO arbitrarily that I wouldn't be able to fudge the final results with some heavy-handed rounding of the measurements), or I would have to do a few of the steps of my calculations carefully but incorrectly in order to land on a figure that I could smudge up and round into place enough to please the small crowd that had gathered around me to drip ice cream on my head.
There is, I suppose, a number of morals that both mathematicians and laypeople could take away from the above true account of my experience. But if I had to select just one as the part that I would feel most comfortable attaching to my name, it would be this: When sitting down to calculate, it's not a bad idea to be clear about expressing your likes and dislikes to anyone approaching with a dripping cone. Every culture has its do's and don't's, but there has to be something universal about the limits regarding how long a child should be permitted to drip his snack in the hair of another before a parent rushes in to stop the action.
I hope I managed to shed even a little bit of light on something, somehow, and I hope you enjoyed reading about the mathematically fascinating ratio we call phi.
Until next time, be well mathematically,
Mitch