Traditionally, my wife Greta and I put out our Christmas decorations around Thanksgiving Day each year. This is typically because it is the only time between Thanksgiving and Christmas that I am not busy finishing up my fall semester (e.g., end of semester tests/homeworks/quizzes, final exams, grade submissions, and December graduation), and Greta and I are not as busy with holiday travelling and Christmas shopping.
Last weekend, we put out out decorations, and we encountered a problem that has regularly bothered me while decorating our Christmas tree.
We like lots of lights on our tree, but the challenge each year for me is to space these lights uniformly. Typically, we wrap the strands of lights around the tree starting from either the top and work down or the bottom and work up. By the time we have finished covering our tree, we typically have a few extra lights that we place in strategic places or not quite enough and are forced to tweak the spacing slightly to ensure the tree is covered.
In the past, this tweaking has yielding an acceptable distribution of lights (even though the spacing is not uniform), but this year we were too greedy with the lights at bottom of the tree, and therefore we were in need of too many lights at the top to “tweak” the spacing for complete coverage. We were forced to either pull the lights off and start again or purchase another strand of lights. We decided to purchase another strand, but the problem of uniform spacing still troubled me. As a mathematician and a problem-solver, I still wanted to know the optimal spacing of the lights so that the distribution is uniform and the precise number of lights are exactly used without tweaking them.
Here’s how I came up with my Christmas Tree Lights Spacing calculator:
The problem-solving process began by modelling a Christmas tree with the shape of an inverted cone.
By modelling the tree as a cone, the most natural way to wrap the lights around the tree is by using a curve known as a “conical helix”.
This curve is like a standard helix (e.g., a spring or Q*bert snake) except that the curve tapers as it progresses upward. Mathematically, the curve can be described parametrically by:
where r is the initial radius of the conical helix, h is the height of the conical helix, and ω is the angular/rotational frequency of the curve, where 0≤t≤1. Calculus can be used to find the length of this curve (i.e., arclength), and the following figure shows a screenshot of determining the vertical spacing between successive rotations of the curve given the diameter (which is twice the radius), height, and arclength. This is equivalent to knowing the bottom diameter of a Christmas tree, height of the tree, and the length of the strand(s) of lights.
For example, our Christmas tree is 3½ feet wide (diameter) at its base and 8 feet tall. When using 75 feet of lights (i.e., three 25-feet strands), the vertical spacing between rotations is about 7 inches. This ensures that if we begin wrapping the lights in a conical helix pattern beginning at the bottom of the tree and vertically space the lights by about 7 inches between successive rotations, the strand(s) of lights will terminate precisely at the top of the tree.
In order to determine how to wrap your tree, use my Christmas Tree Lights Spacing calculator. Simply provide the calculator with the diameter of your tree at its base, the height of your tree, and the length of your strand(s) of lights. The calculator then provides you the vertical spacing between rotations, and, as a bonus, it also provides you with the number of rotations as well as a picture demonstrating the result.