Math as a Muse (Part II)

In my previous post, I described a math problem that kept my mind busy during a church service a couple of weeks ago.  For a description of the problem, the link is here.  In any case, I will show the image that describes the problem again below...

As people sang their hymns, I began to think of the trigonometry of the situation - of the right angle triangles that are formed.  By the end of that hymn, I realized that actually, the line of sight to the nth column (pew) forms two right angle triangles which were geometrically similar (same internal angles).  As a result, the ratio of the two triangles' opposite and adjacent sides was the same.

Math as a Muse (Part I)

I have had an affinity for math for as long as I can remember.  Even in elementary school, when working with numbers or shapes, it always seemed like magic to me.  Now, as an adult, I find this magic hiding in unexpected places, like the relationships between notes in music, and in the geometry of architecture.  It was the latter that called out to me last week.

I was standing in a place of worship.  Admittedly, I do not spend a great deal of time in such places.  On this particular occasion, in a church, my mind was wandering, and I began examining all of the geometry around me: the slopes in the roof, the shapes of the stained glass, and the angle in which the sunlight came through them.  When my gaze returned forward, I began to carefully examine all of the equally spaced columns (known as pews) between me and the front of the church, where the Reverend stood.  Almost immediately, a fun math problem presented itself to me, and I spent the next twenty minutes analyzing it in my mind.

As shown in the figure below, I can see less and less of each column the further they are from me, as each column is obstructed by the one that precedes it.  But, what governs how much of a given column I can see?  I called the column directly in front of me n = 0, and then came 1, 2, and so on.  My aim was to find a function that described the height of each column that I could see for each column, h(n).  I decided that it depended on three other parameters: the column spacing (s), as well as the vertical and horizontal distance from my eyes to the zeroth column immediately in front of me (H and L).

Searching for visible height h as function of pew number n (mad paint skills, I know...)

Richard Feynman Comes Alive in Unorthodox Autobiography

I finished reading Richard Feynman's "Surely You're Joking, Mr. Feynman!" last week, and still find myself laughing about it today.  What could have been a conventional autobiography of the Nobel Prize winner for physics is instead a collection of quirky stories, through which one really gets to know the man.  To give you a sense of the tone of the book, Feynman mentions the Nobel Prize he won about halfway through it, as a sort of after-thought - the focus is rather on what he is truly proud of, like, for example, his ability to break into safes that contained top-secret information about the Manhattan project during the second World War.

"Slow Mo Guys" = Great Teaching Tool

"Sir, you're going too fast!" - it is a complaint I hear in my physics classes every so often.  Whether it is the case or not, it is true that there is an ideal speed for progressing through science content in a classroom setting.

Similarly, there is an ideal speed for the viewing of the countless science phenomena that occur in nature.  Often times, events like chemical reactions or travelling waves elapse over too short a time interval to be properly grasped.  This is actually the reason why many scientific phenomena that are now well understood went misunderstood for so long (and why others still go misunderstood).

Take something simple, like an apple falling from a tree.  Five centuries ago, people believed that the fall was at a constant speed, which was governed by the apple's mass (heavy apples would fall faster than light ones).  Of course, this assessment is wrong on many levels, but one can easily appreciate why such a faulty conclusion could be arrived at.  The entire fall of an apple might take one second, which is an insufficient amount of time for a person to gauge an event.

Had mankind invented the video camera a few centuries earlier than it did, enabling it to see the world in slow motion, early science would have evolved more rapidly than it did.  The apple could then be seen to displace more and more with each passing frame, invalidating the constant speed theory.

Some people today may not see how valuable adjusting the frame rate of an event is outside sports and action movies.  Fortunately, a few young people certainly do, and they are responsible for my current favourite YouTube channel: "Slow Mo Guys".