O Tannebaum

Today is the birthday of mathematician, Mary Cartwright, born 1900 in Aynho, Northamptonshire, England. Cartwright was the first woman mathematician to be elected to the British Royal Society.

Today's quote is from The Littlest Christmas Tree: A Tale of Growing and Becoming by Janie Jasin. "Thank you, Dear Creator, for Life. Thank you for Dreams. Thank you for Ideas and Thoughts and Feelings. Most of all, thank you for choosing me to grow - just for today - and to know the wonder of Your World and its many Possibilities."

A tradition that has been established in our family is the assembly and decorating of our artificial Christmas tree on the day after Thanksgiving. Early in our marriage when my wife and I were hanging the lights on the tree, I noticed I struggled with the project. My placement of the lights as we circumnavigated the string about the tree just wasn't quite right. I had too many lights in the back of the tree or the plugin wasn't situated in accordance to our electrical outlet. As my children grew older, I happily abdicated my role of string assistant to more capable and eager hands.

This year my eldest son, Jacob, asserted his birthright, erected the tree, and hung the lights. I assisted. I had been on administrative leave for about 25 years and I thought now is the time for redemption. I thought, "I can do this!" I also searched for another moment to atone for those instances that I was less than the parent I should have been. Jake is my practice child. What I mean is that I practiced being a parent on him.  There are several classes of students that graduated from Lake City Lincoln High School that I was allowed to practice on prior to my first teaching job. I did not have that convenience with Jake. I made mistakes. Mistakes that I still regret to this day. Our relationship has become more than father and son. We have become friends. He has become my confidant. I value his opinion. I look forward to our time together. I look forward to moments of redemption.

He had progressed to the finality of the tree construction when I began this annual event. Quickly a philosophical discussion emerged between my wife and son as to where to start the string of lights. 

"At the bottom," my wife insisted, "closer to the power source. The plug should be on the bottom right."

"At the top," countered my son, "next to the star. It can guide me to that source."

 My wife relented. Sometimes perseverance can appear as stubbornness.  Jacob and I weaved the lights counterclockwise about the tree. This weave appeared as a dance - a dance of a couple in which one is trying to lead and the other, not confidant in the placement the of step or the rhythm, is trying to follow. After a few rotations, a pirouette, and a few string entanglements, Jacob stopped me. "Dad, you're hanging them too low and you need to wind them around the tree so the strings rise and fall. They should make a wave", he instructed. I grew silent. "Dad, are you ok? Are you upset?", he inquired.

"No, I'm ok", I responded. He paused and placed the bubbler in the center of the tree. He regrouped, "Are you thinking about math?"

I smiled slyly. He sighed.

When we were done, he began hanging his ornaments. The ornaments that he was carefully placing on the tree marked his life. My wife and I choose ornaments for each member of the family that signify an event that we view as significant for that year. These ornaments have accumulated for the past 29 years.They can represent moments as exhilarating as birth and marriage or as humbling as rowing a boat across a lake after the motor stopped running.  Each member has their spot on the tree where their bauble is placed. These assortment of symbols, delicately removed from their containers and carefully hung above and below the periodic function of string of lights that winds itself, top to bottom or bottom to top, (depending on your view) around the tree.

His phrase "rise and fall of the strings" resonated in my mind. I have never thought about the mathematics in a Christmas tree. Years ago, I taught seventh graders. During this time of year, I taught prime factorization I would croak the tune "O Tannebaum" and insert the words "O Factor tree, O Factor tree, How beautiful are your factors!". The tree is obviously conical in shape and and the Greeks had described parabola, circle, ellipse, and hyperbola as planes intersecting a cone, but those words, "rise and fall" kept haunting me.

I searched for mathematics in the seemingly infinite Google search engine and found Treegonometry, a series of mathematical formulas that would compute the "perfect" amount of trimmings for ideal Christmas tree. These formulas were derived by students at the University of Sheffield in the United Kingdom. The formulas are as follows:

1) Number of ornaments = Square Root of (37) ÷ 20 x (height of tree in cm)

2) Length of tinsel (cm) = 13 x pi ÷ 8 x (height of tree in cm)

3) Length of lights (cm) = pi x (height of tree in cm)

4) Height of star/angel (cm) = (height of tree in cm) ÷ 10

A six foot tree would 37 ornaments, 919 cm of tinsel, 565 cm of lights, and a star that is 18 cm tall.

I realized in my blog, The Circle of Geometry, I had described the helix as my best symbol of life and there was helix of lights that wound about the tree but this helix, as life, is more complex.

Above, I have provided a top view of the lights and a three dimensional view of the lights traversing the tree. The top view is a spiral. The second view is a conical compression spring often referred to as a tapered spring. I have seen these springs in faucets, holding in the gaskets. The advantage of these types of springs, is the stability they provide when placed on structures such as battery contacts or push buttons. Does the spring start at the bottom or the top?

In my son's description of "rise and fall" and "make a wave", he was expressing the sinusoidal curve which can be shown by stretching a spring.

Unlike the helix, I described in a previous blog, this tapered spring is a better symbol of life, with one addition. I would include "the rise and fall" of the spring as it rotates about its axis. The "ups and downs" that we encounter each day add to the overall wisdom we gain as each year passes.

The tapering of the spring represents the acceleration of time. When my children were born, my wife and I were excited about the brief celebrations of solid food, walking, no diapers, and eventually, no daycare. The time before each of those events were anticipated and calculated.

As a child, the days before the holiday break and ultimately, Christmas, could be excruciating long. I marked the days on my calendar and again, those days were anticipated and calculated.

My first year of teaching was, in my mind, the longest of my career. I made the mistake of marking each day of school and highlighted my last day of the year. Calculated? Yes. Anticipated? Very much so.

Now, I don't measure the day. I do not look forward to the last day of school. I don't calculate and I avoid anticipation.

My parents are in the twilight of their lives. They have often remarked about how time has quickly passed. As the tapered spring reaches the apex, time is rapidly revolving.

I believe the wavy tapered spring is life. As each year turns, and each experience raises the rotation, the circumference of the circle becomes smaller. Time shortens. Opportunities to mend torn moments pass by. Children become adults. The time spent with them becomes rarer and more treasured.