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...)

I will not reveal the solution to this problem in this post.  Posting the question and then solving it immediately would absolve you of the fun of considering it on your own.  The basis of teaching comes down to asking good questions more than it does answering them, and this is, after all, an educational science blog.  My next post (Part II) will include a complete solution and analysis.

Before calculus, linear algebra, and statistics dominated the minds of mathematicians, geometry was the biggest show in town.  Four centuries ago, it was pretty well the only show in town.  And, it was a sufficient tool for pretty well the only science in town, that of astronomy.  As geometry was the only tool used to study the motion of planets, astronomers were unbelievably skilled in geometry, more so than most mathematicians today.  However, calculus proved to be a far more powerful tool for studying the motion of orbital bodies - so much so that Newton invented this branch of mathematics in order to do so.

In a sense, mastery of geometry and its use in scientific pursuits is something of a lost art.  These days, geometry is called on mostly by actual artists rather than mathematicians.  Still, geometry, with its alluring lines and patterns fills the background of our lives; it is fun to focus in on it every so often.

There is so much beauty in mathematics as a whole.  I hope that you amuse yourself with the geometrical problem that amused me last week.

*** For the detailed solution to this problem, click here.