As the holiday season approaches, the quest for the perfect Christmas tree becomes an annual tradition for many. While the selection process often involves considerations of size, shape, and overall aesthetics, there is a surprising amount of mathematics involved in creating the ideal festive centrepiece. “Treegonometry” is a term coined to describe the mathematical principles used to design the perfect Christmas tree. In this article, we’ll delve into the world of treegonometry, exploring the specific mathematical formulas that contribute to the creation of a flawless holiday icon.

Two University of Sheffield students, Nicole Wrightham and Alex Craig have unveiled a formula that promises to guide the meticulous decoration of your Christmas tree:

• Number of baubles = h √17 / 20

This formula, linear in the height of the tree (h), has raised eyebrows and led to intriguing discussions within the mathematical community.This is because a distribution on a linear surface in height would only be uneven as the height varies: tall trees would appear too bare, and short trees too covered! Let’s take a closer look at it in order to attempt to demystify the rationale behind it.

To achieve a more accurate representation, the initial formula, which was linear in h, must consider the baubles’ surface density (σ) in the context of the tree’s cone shape. Essentially, the parameter σ represents how decorated a tree is with baubles: high σ values will result in very covered trees, and low sigma values will result in bare trees. Afterwards, we will estimate the σ value to have a balanced tree (neither too covered nor too bare). Now, let’s try to express the formula for the number of baubles as a function of σ and height. Since the lateral surface of the cone is π r √(r²+h²), we can represent our distribution as the number of baubles equal to σ π r √(r²+h²). Let’s now try to express the dependence only on height, namely by writing the radius as a function of height. In general, there are more squat cones and more slender cones, so it would be impossible to express the radius generally in height only. However, dealing with trees rather than abstract cones, we enjoy some constancy in the angle at the vertex, which we can more or less expect. An average Christmas tree has a (half) opening angle of about 21 ± 4°, so its radius it’s r = h tan (21 ± 4°) = h (0.38 ± 0.08), hence:

• Number of baubles = σ π r √(r²+h²) = σ π h (0.38 ± 0.08) √(h² (0.38 ± 0.08)²+h²) =

= σ π h (0.38 ± 0.08) h√((0.38 ± 0.08)²+1) = σ π h² (0.38 ± 0.08)√((0.38 ± 0.08)²+1)

So we need an estimate for the value (0.38 ± 0.08)√((0.38 ± 0.08)²+1), and it is taking the mean 0.38√((0.38)²+1)=0.41

So the (mean) number of baubles is equal to: σ π h² 0.41

The improved formula suggests that the number of baubles required is proportional to 0.41 times the surface density, multiplied by π and the square of the tree’s height. But what’s the value of that density? If our tree has a height of 200cm and a radius of 73cm, according to the original formula, we should use 41 baubles, so its density is 8.4 baubles per square meter. Converting that value in cm would lead to the final “baubles formula”:

Number of baubles = (0.00108 ± 0.00025) h² (with h in cm!)

As you stand in awe of your beautifully adorned Christmas tree this holiday season, take a moment to appreciate the mathematical precision that contributes to its perfection. Treegonometry not only enhances the tree’s visual appeal but also ensures that it stands proudly, embodying the spirit of the season with a touch of mathematical elegance.

What do Treegonometry and VisitMath project have in common? Maths is everywhere, so we can learn by visiting famous monuments and interesting cities and studying what surrounds us more closely.

Merry Christmas and a Happy New Year from VisitMath!