A few days back, the University of Sheffield published a press release concerning a set of mathematical formulas which could be used to create the “perfectly decorated” Christmas tree.  Since then, the story has been picked up by numerous gullible media outlets.  (Admittedly, few of these are major media sources, but a number are tech or “geek” related online sources or blogs.)  The UK-based retailer Debenhams — which has 240 stores in 28 countries — requested the creation of these formulas and is apparently encouraging their use in stores.  The calculations have been given the dubious moniker “Treegonometry.”

When I first stumbled on this story, I was intrigued by the use of mathematics to solve a common problem: how many decorations do you need to buy for a Christmas tree of a given size?  But the more I looked at the University of Sheffield page, the more disturbed I became.  The formulas may be the creation of a couple 20-year-old students in a math club, but they are being effectively endorsed by a university, a major retailer, and a number of news sources.  Yet anyone with an 8th-grade knowledge of math should realize that what is being presented is absolute nonsense.  The students are clearly either far less intelligent than we should expect from university math students, or they have attempted to perpetrate a subtle hoax (which has now been spread by a university press office and some media sources).

Before we even get to the formulas themselves, I should note that even the press release itself has ridiculous errors.  The University of Sheffield includes some “Christmas Tree Facts” on its website, including the following:

The world’s tallest Christmas tree would, at 2,600ft tall, need more than 16,000 baubles, over 4,000 meters of tinsel, almost 2,500 meters of lights, and an 80 meter tall star.

Aside from basic fact checking, does anyone at this university’s press office have a clue about what numbers mean?  A 2600-foot Christmas tree would be nearly a half of a mile high, almost twice the height of the Empire State Building!  For the record, the tallest giant sequoias in California are less than 400 feet tall.

What this “Christmas Tree Fact” is referring to is not a tree at all, but rather a giant lighting decoration on the side of a mountain near the Italian city of Gubbio.  True, the lights go up the mountain in a shape that looks like a sort of “drawing” of a Christmas tree, but that hardly counts as an actual “tree.”  And it makes no sense to apply a formula made for a 3-dimensional tree to an effectively 2-dimensional mountain surface decoration.

Unfortunately, the errors in the “facts” are only the beginning of the problem.  Here are the formulas given for decorating a tree:

The first odd thing about the formulas is their complexity.  What is the need for square roots of numbers?  Why use √17/20 in the first formula, when you could just use 0.2?  The exact value of √17/20 is an irrational number, 0.206155…, but the results given by the formulas cannot be accurate to more than a digit or two (as will be explained below), so what is the reason for this complicated looking use of square roots and fractions?  Similarly, why 13×π/8, when one could just say 5 or 5.1?  (Again, the exact value is the irrational number 5.105088….)  The use of pi at least seems to be more relevant than the arbitrary inclusion of a square root of 17, since most trees have a roughly circular shape when viewed from above.

But if these formulas are meant to be used by average people, why not make them as easy-to-use as possible?  It’s much easier for someone to plug in 0.2 × height into a calculator, or even 0.206 × height, rather than taking a square root of some random number.  What does the square root of 17 have to do with the shape of trees?  Its value is about 4.1.  Is there some reason that approximation is inadequate here?

Now if this were the only problem with the formulas, the students would at best be guilty of not understanding significant digits or failing to make their formulas user-friendly.  It certainly is possible that they performed some calculations that ended up with a square root or some factors of pi in deriving these expressions.  On the other hand, these mathematical expressions might also be an attempt to hide the simplicity of a rather mundane model.  Which looks more “complicated” (and therefore requiring the expertise of university math students): the above expressions or the following ones, which are reasonable approximations to them?

  • Number of baubles = tree height ÷ 5
  • Length of tinsel = 5 × tree height
  • Length of lights = 3 × tree height
  • Height of star = tree height ÷ 10

All those square roots and fractions and factors of pi don’t appear to do much.  Now, you might say that they could provide additional accuracy to the expressions.  Maybe there is a reason why the length of the lights is closer to 3.14159… than to 3.  But that level of accuracy would require a rather accurate model of the tree shape.

And yet the tree shape is not modeled well at all by these expressions.  The only measurement is the height of the tree.  Even if the wacky constants used are roughly accurate, the relevant variable for amount of lights, ornaments, etc. is the surface area.  To estimate that, you’d need at least one (and preferably two or more) radius/diameter/circumference measurements to take into account the shape of the tree (conical, full with a bulge in the middle, very thin, etc.).

But the biggest issue is that the formulas say that amount of lights, etc. should vary linearly with the height of the tree.  That is, if you have a tree that is twice as tall, their formulas would require twice as many lights, twice as many ornaments, etc.  However, the surface area of the tree will increase roughly with the square of the height (assuming that the general shape of the tree remains the same).  That is, for a tree that is twice as tall, the surface area will be about 4 times as much; for a tree three times as tall, the surface are will be about 9 times as much.

Let’s say, for example, that the Sheffield formula is accurate for a tree that is 9 feet tall.  If we then tried to use the same formula for a 3-foot desktop tree, we’d end up tripling the density of lights, ornaments, and tinsel on the tree’s surface.  It would clearly look overdecorated, particularly given that a smaller tree would likely be easier to see through at spots, thereby increasing the apparent density of lights in particular.  On the other hand, if the Sheffield formula were calibrated to a 3-foot desktop tree, it would recommend rather sparse decorations for a taller tree in a living room.

I do imagine the numbers might stay roughly accurate for a giant outdoor tree compared to an indoor one, since you’d use larger lights and larger ornaments on a giant outdoor tree, and therefore would need fewer of them.  But if you’re trying to decorate a 3-foot tree on your desk vs. a 9-foot tree in your living room with similar ornaments, these formulas will produce vastly different results, even with trees of the same shape.

I think the absolute minimum for a reasonably accurate formula would be three measurements: (1) height, (2) radius at the widest point of the tree, and (3) the height of the widest point.  Or, alternatively, (1) height, (2) radius halfway up, and (3) radius at lowest large branches.  A set of three measurements would allow an extrapolation of surface area, which would be an estimate of “fullness.”  Of course, there will still be trees with odd shapes, and different kinds of trees may require different lighting (e.g., thin pine needles don’t hide lights as much as a thick spruce).  But at least such a formula could aim for consistency in a variety of scenarios.

One might challenge the validity of the question posed to the Sheffield students.  Is it reasonable to come up with a formula for the number of ornaments and the length of strings of lights without knowing anything about the size of the ornaments, the brightness of each light, how close the lights are on the string, etc.?  In a way, the problem is difficult to solve without standard “components” for decorating, not to mention the inherent subjectivity in the “proper” amount of decorations.

Nevertheless, when such formulas are announced with authority by a university and a large commercial chain, they could at least give some rough guidelines that make some sense.  As we’ve seen, however, the proposed “solutions” to this problem not only lack appropriate information to solve it (i.e., the shape of the tree, specifically the surface area), but also make little sense even if we assume a consistent tree shape.

My guess is that university math students are not that dumb.  They probably started by making some very rough approximations: “the star on top should be about 1/10 of the height of the tree.”  Relating the height of one prominent ornament to the height of the tree makes some sense.  But then, rather than considering surface area, they just began to estimate other decorations for a “normal” height tree: “the amount of tinsel might take roughly 5 times of the height of a ‘normal’ tree to cover it, but we need fewer lights, maybe only 3 times the height of the tree.”  And then someone thought: “Hey, 3 is close to pi, and pi is related to round things, so we could just use pi for the light formula!”  Pretty soon, there were fractions and more uses of pi and even square roots, making it all look very cumbersome.

I can’t say for sure that this is what happened, but it seems rather unlikely that the students would use complex methods resulting in these odd irrational numbers and yet not consider the need to incorporate surface area factors rather than simply height.  It also is mildly suspicious that the three complex-looking formulas effectively amount to multiplying or dividing by 5 or 3.

While I wouldn’t expect any editor in the mainstream media or the University of Sheffield press office to do the detailed analysis I’ve done here, I would hope that anyone with any basic numerical intuition would question formulas that claim to scale to various sizes yet clearly don’t contain enough information to perform the necessary calculation.  It should be intuitively obvious that formulas that only take height into account wouldn’t recognize the difference between a thin tree and a full and bushy one.  The former might not even need half of the decorations of the latter to appear full of ornaments and lights.  And once that question comes up, the next one presents itself: how does this formula actually scale?  If you just begin to plug numbers in, they immediately seem rather crazy for very small or very large trees.

For example, in the theoretical “2,600-foot tree” referenced above, it is claimed that only 16,000 ornaments would be required.  That’s an average of only 5 ornaments per foot of tree height — do 5 ornaments really seem sufficient to encircle a foot-high band around a 2,600-foot tree, except near the very, very top??  Similarly, the formulas predict that if I had a tiny rosemary bush that is 1-foot tall on my desk, I would need about 3 feet of lights to decorate it, which seems rather excessive, no?

Just a minimum of common sense should have shown the university, Debenhams, and any other media that picked up this story that it makes no sense.  But because the formulas contained complicated-looking square roots and Greek letters, it must be the product of deep thought and analysis, right?

To the students at the University of Sheffield: congratulations.  You’ve not only got a lot of attention for a well-played hoax, but you’ve also demonstrated the complete and utter mathematical illiteracy of a huge number of people.  This is an important lesson.