The Mathematician: Much More Human than a Calculator

Posted on September 27, 2012 by Ryan Creedon

Please note: Given the amount of content I planned, I have decided to split this topic into two blogs. The purpose of this week’s blog is simply to stimulate the audience’s interest in the mathematician’s point of view. Next week’s blog will focus on specific stories of mathematicians.

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“Mathematicians, who are only mathematicians, have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise, they are inaccurate and insufferable, for they are only right when the principles are quite clear.” – Blaise Pascal

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Mathematicians.

Everyday Joes with a passion for precision who modern society depicts as elitists of intelligence.

To compensate for last blog’s rigorous mathematical exercises, I plan to leap into the history books to analyze individuals who solely devoted their time to the art of mathematics. More particularly, I plan to examine the lives of three brilliant mathematicians who fall just shy of the public ear. You, being a very bright and resourceful individual, may be familiar with one (or more) of these mathematicians, but for now, I leave them a mystery. (I will say that, if you are expecting a historical dissertation of Albert Einstein or Sir Isaac Newton, you will be disappointed. I have specifically picked lesser-known mathematicians for this undertaking.)

As mentioned, today, I feel it important to discuss why mathematicians are the way they are. In other words, why is it that some individuals in our society find it fascinating to study the extremely abstract field of mathematics? Answering this question will build an understanding of the mathematician’s point of view, which is critical if one wants to understand his story.

As I mentioned in one of my response comments last week, mathematics renders emotions that harken back to our origins. We started with counting herds of animals and have since gravitated towards weird abstractions such as ϕ or sqrt(2). Fundamentally, the subject at heart remains unchanged, but the study has evolved with our society over centuries, years, weeks, and even seconds. In fact, as you read this blog right now, it is very likely that someone somewhere in the world is making a new contribution to the study of mathematics.

Because of math’s inability to regress, we as human beings continuously expand it with each generation. To add to the subject, therefore, is a tradition, equivalent to the passing of the torch in the Olympics. Every single one of us has our hands on that torch. We all run our own part of the race when we use mathematics in our daily activities (for instance, when we balance our checkbooks). It is the mathematician, humble and filled with pride, who consciously decides to run the longest stretch in this traditional process. They are not elitists, nor are they any more intelligent than you or me. They are just normal people who are compelled by some unexplainable force to add to our heritage by prompting sustained mathematical inquiries. Fundamentally speaking, mathematicians do what they do because they have a desire to make impressions on our heritage, on our mathematics.

For reasons quite obvious, such dedication to the mathematical arts tolls a hefty price on one’s life. By studying stories of mathematicians, we can uncover their innermost struggles and come to empathize better with their extensive efforts to the study of mathematics. In doing so, we may also learn more about ourselves and our struggles in life. In that respect, when we learn more about ourselves, we come to the understanding that we are not much different from the mathematician. Our aspirations may differ, but our intentions to defend our heritage remain unscathed.

So, to close, as you make your way through the reading next week, I want you to keep in mind the mathematician’s point of view that I have described. Likewise, I want you to consider the quotation I have provided above from Blaise Pascal, a famous mathematician. What might this quote imply about mathematicians outside the field of mathematics? What do you expect from the stories I have waiting for you next week?

To get a fresher perspective of the mathematician, check out Timothy Gower’s speech On the Importance of Mathematics. (I will warn you now; it is very lengthy. However, you need not read the whole work to understand his perspective as a mathematician. It arises naturally.)

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Proportions: A Golden Staple of the World

Posted on September 18, 2012 by Ryan Creedon

“The good, of course, is always beautiful, and the beautiful never lacks proportion.” – Plato

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Proportions.

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.

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Numbers: From Counting to Weird Abstractions

Posted on September 12, 2012 by Ryan Creedon

“Not everything that can be counted counts, and not everything that counts can be counted.”  — Albert Einstein

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Numbers.

Limitless in every dimension but seemingly always limited in our appreciations.

Since prehistoric times, our ancestors have sought to describe quantities. At first, it began with simple cave illustrations: We perceived three buffalo, and so, we drew three buffalo. In correspondence, the notion of natural numbers—one, two, three, and so on—grew to their modern maturity as we discovered more about this subliminal art of counting.

Eventually, this keenness to quantity spawned several symbolic representations. The Ancient Babylonians, Egyptians, Greeks, and Romans each took it upon themselves to demonstrate proper counting methodology (i.e. to illustrate numbers in universally recognized fashions). Consequently, dots, lines, fingers, swirls, crowns, and even frogs were used to represent the “counting” numbers. Given time for proper evolution, these numbers eventually became the familiar figures we use today—1, 2, 3, etc.

Of course, numbers extend far beyond the idea of “countability.” In other words, there are some numbers out there for which we simply do not say, “There are ‘x’ number of students in the classroom,” or, “That tree has ‘y’ number of leaves.” These numbers could be, for example, -3, pi, and ½. To illustrate this point more fully, I call upon the famed Greek mathematician Pythagoras, a man who struggled with the notion of what we now call irrational numbers.

To be a touch more precise than previously stated, Pythagoras went through several internal battles because of the number sqrt(2), an irrational number very important to trigonometry. Using modern technology, we can approximate the number sqrt(2) fairly easily. (It is roughly equal to 1.41421356237.) However, despite how powerful our technology may be, even we cannot fully obtain the value sqrt(2) because its decimal component never ends (i.e. an infinite amount of numbers exist after the 7 in 1.41421356237). This concept troubled Pythagoras greatly because he could clearly visualize what sqrt(2) equaled geometrically using a theorem which bears his namesake. (See the diagonal line in the diagram.)

Notice the diagonal line, which appears finite, represents a number with a never ending decimal component.

 So, having seen sqrt(2) drawn as a line segment, I now present to you the very same question Pythagoras asked himself countless numbers of times: How is it that a number which theoretically never ends fits into a perfectly finite geometric observation?

The answer lies within the beautiful, and careful, construction of our number system. To read more about how we organize numbers (and reach a more satisfying conclusion to my question), click here.

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