“The good, of course, is always beautiful, and the beautiful never lacks proportion.” – Plato
Our disinterest in proportions is directly proportional to our disinterest in mathematics.
Proportions are a fundamentally important concept to any comparative analysis, and whether we like it or not, we all use them to convey meaning among quantities. For example, in the most recent Penn State football game against Navy, the scores fell thirty-eight points for Penn State and thirteen points for Navy. The proportion between Penn State’s score and Navy’s score would be reported as 38:13 (read “thirty-eight to thirteen”). We as football fanatics, and as pragmatic mathematicians, can then conclude that, for every one point Navy scored, Penn State must have scored 38/13 points. (If Navy scored one point thirteen times for a total of thirteen points, then Penn State scored 38/13 points thirteen times for a total of thirty-eight points.) It is herein that lies the profound aspect of proportions: They can never change after set in stone. The proportion of the final scores—Penn State to Navy—in the game is 38:13. The proportion of the scores per each point—Penn State to Navy—is 38/13:1, which (if we multiply both sides of the proportion by thirteen) is 38:13. Notice in this example: The proportion 38:13 never changes after the game has been played, and even if we embellish, say, Penn State’s score after the game, Navy’s score will automatically change to keep the proportion 38:13 true.
So, why bother illustrate proportions as extensively as I have just done? Quite simply, a basic intuition of proportions is needed to continue the purpose of this post. Personally, I find it more beneficial to illustrate proportions in examples rather than in precise mathematical jargon, especially for individuals with math phobias. Hopefully, the example provided a connection to you as the reader that conveyed the major intuition of a proportion. If not, I will explicitly state it now: A proportion can be thought of as a ratio between two quantities (i.e. [first quantity]:[second quantity]) that never changes. With that in mind, let us move on with the discussion.
Just as I suggested last week with numbers, proportions are not given equal attention. There are some that naturally resonate throughout the universe, and there are others which are only used as examples in textbooks. To illustrate this point more fully, I want you as the reader to imagine the “ideal rectangle.” In your mind, picture the shape, and take notice of the proportion between its height and base lengths. Does the rectangle below represent the one you pictured in your mind?
If it does, you are in the vast majority. If it does not, you have an interesting taste in rectangles. (Please, see me!)
In any case, if you pictured this rectangle in your mind, you have successfully identified what ancient societies have called “the golden rectangle.” Supposedly, it is the most appealing rectangle one can ever construct. Why is that so? Think back to my instructions.
You were asked to imagine your rectangle and its dimensions (i.e. the proportion of the rectangle’s height and base lengths). How would you describe these dimensions now that you can see the rectangle? Perhaps, one could readily guess 2:1; adding two height lengths together will yield one base length. It certainly appears that this could be the case (i.e. we could fit two height lengths along the base length of the rectangle). However, that guess would be incorrect. So, how can we readily find out the proportion? Measure the height and base lengths of the rectangle, form a ratio of height length to base length (i.e. [height length]:[base length]), and divide both sides of the ratio by the height length. If you do this process satisfactorily, you should get a proportion anywhere from about 1:1.5 to 1:1.7, which means we need 1.5-1.7 height lengths to yield one base length. This may seem counter-intuitive: The “perfect rectangle” has a seemingly less-than-perfect proportion between its height length and base length!
The correct proportion, for which I will not show the derivation in this post, is roughly 1:1.61803398…, where 1.61803398… is an irrational number just like sqrt(2). We call this special irrational proportion the “golden proportion,” and we call the number 1.61803398… the “golden number.”
So, to answer my initial question, the “golden rectangle” is the most appealing rectangle because the proportion between its height length and base length obey the “golden proportion.” If that sounds redundant to you as the reader in any way, it is most likely because I am ignoring the true definition of the term “golden proportion.” For the purpose of this post, however, the term will be sufficient enough to stand on its own.
If you have made it to this point in the post, I thank you greatly. This has probably been more than enough recreational mathematics for the next few days as well as a fair amount of critical reading, but I urge you to finish this last section with an open mind. Everything prior to this point has been ground work for the true mathematical elegance of the “golden proportion” and “golden number.”
Consider any seashell with a spiral around some axis. Believe it or not, but this spiral is created by many “golden rectangles” of different sizes (as shown below).
Consider a tropical system barreling toward the eastern seaboard. The clouds associated with this system swirl in a pattern similar to that of the seashell, a pattern based, again, on “golden rectangles” (as shown below).
These are only two examples of nature’s affinity towards the “golden proportion.” Try to think of others, and post them in the comments section. You may want to consult artists’ depictions of nature because, many times, artists will use the “golden proportion” in their works. For example, if you draw a rectangle around Mona Lisa’s head, you will get a “golden rectangle.”
Next, we turn to the “golden number,” which is so special in mathematics that we denote it as the Greek, lowercase letter phi—ϕ. In high school mathematics, ϕ often takes a back seat to his cousin π, and though π has its own right to the limelight, ϕ can just as easily illuminate the elegance of mathematics.
Consider the continued fraction below.
The addition and division of the number one never end, yet this expression still converges to a certain value. Using basic algebraic techniques, one can discover that this never ending fraction will converge to ϕ. If you do not believe me, start crunching the beginnings of this continued fraction on your calculator. The further you go, the better you will approximate ϕ.
Now, subtract one from ϕ. Next, take the reciprocal of ϕ (i.e. 1/ϕ). Do you notice any similarities? You should. The number 1/ϕ equals ϕ-1. ϕ is one of only two real numbers in the entire universe to have this property! (Can you think of the other number?)
To close, I present a special sequence to you (i.e. a special list of numbers).
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
How might this list of numbers be related to ϕ?
To find out the answer (and learn more about ϕ and the “golden proportion”), check out this site.