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How to use a slide rule to decorate your Christmas tree
Nicole Wrightham and Alex Craig, two University of Sheffield students, published formulas for the amount of Christmas decoration required to decorate a Christmas tree of height h (in cm):[1]
Number of baubles = h √17 / 20
Length of the lights (cm) = 𝜋 h
Length of the tinsel cm) = 13 𝜋 h / 8
Height of the star/fairy (cm) = h / 10
It is strange that these formulas are linear in h. This already attracted some comments, and the suggestion that h√h would be better.[2,3] This suggestion was not further explained. Let's try improve this.
The baubles
The baubles
In the Dutch KRO television-broadcast 'De Rekenkamer’[4] of December 5th, 2013, Ionica Smeets discussed these formulas. Although she finally used the English formulas, she started with the assumption that the baubles were distributed homogeneously over the surface of the tree, with a surface density of σ, and that the tree is a cone with radius r.
This gives the following formula for the number of baubles:
Number of baubles = σ 𝜋 r √(r²+h²)
(σ times the surface of an open cone, because we do not decorate the bottom plane of the tree)
This formula can be calculated using the P and Q scales of a Sun Hemmi 153 slide rule,[5] that go like x². These scales should not be mistaken for the P-scale of a Darmstadt slide rule, that goes like √(1 – x²).
If there is a unique relation between r and h, we would only have to measure the height. To check this relation, I used Google image search to find pictures of real (Dutch) Christmas trees, and measured the r/h ratio. The pictures where filtered by hand on the tree being full frontal and not blocked by presents. I did not take into account the different species of Christmas trees. Averaged over 33 pictures, the (half) opening angle 𝛼 of a Christmas tree equals 21 ± 4°, so r = h tan(21 ± 4°) = h (0.38 ± 0.08).[6]
The improved formula-in-h for the number of baubles is:
Number of baubles = σ 𝜋 0.38 h √(0.382+12) h = 0.41 σ 𝜋 h²
According to the original formula, Ionica’s Christmas tree (h = 200 cm and r = 73 cm) had 41 baubles. This corresponds to a surface density σ of 8.4 baubles per square meter.
The improved baubles formula can be simplified to
Number of baubles = (0.00108 ± 0.00025) h² (with h in cm!)
Geometry of a Christmas tree, with a homogeneous distribution of baubles
The lights
The lights
The web site Eclecticsite[7] uses a formula for the length of the cord of Christmas lights that is based on cords running straight from the top to the bottom of the tree. This results in a highly inhomogeneous distribution of lights! The distance between the cords at the base of the tree is called A. The sum of the lengths of cords equals √(h² + r²) 2 𝜋 r / A.
Eclecticsite suggests A = 15 cm for a tree with h = 2.5 m and r = 80 cm (so 𝛼 = 18°). This requires 88 m of cord. If a single cord is used, part of it will run horizontally at the bottom of the three. This requires an additional length, dashed in the drawing, that equals half the circumference of the bottom of the tree.
So the length of a single cord is √(h² + r²) 2 𝜋 r /A + 𝜋 r
A better distribution of lights can be obtained by a helix. If this helix has a pitch s, measured in a strictly vertical direction, the cord can be described by the formula:
(x,y,z) = s φ/(2𝜋 h) (r cos φ, r sin φ, h)
with φ = polar angle, running from 0 to 2𝜋h/s (several cycles!)
The length Llights of the cord is given by:
The Eclecticsite distribution of lights
Helical distribution of lights
Using the previously found relation r = h tan(21°), we find:
For the first part we need the Sun Hemmi 153 P and Q-scales. For arcsinh the Gudermann Gθ-scale can be used in combination with a T-scale. These scales can all be found on a Sun Hemmi 153.[5] In practice, the arcsinh part of the sum can be neglected.
The tinsel
The tinsel
An improved tinsel formula can be based on the lights formula. This time we need a correction for the sagging of the tinsel between its supports. The original formulas give tinsel that is 13/8 as long as the cord.
The well known catenary formulas give for points (x,z) on the catenary:
z/b = cosh(x/b) with b the radius of the curve in its lowest point
The length L of a catenary between points (–x0,0) and (+ x0,0) is[8]
L = 2 b sinh (x0/b)
If this length is 13/8 times the distance between supports (i.e. the straight length), the ratio x0/b can be obtained from 13/8 (x0/b) = sinh( x0/b) .
Again, this can be calculated with the Sun Hemmi 153 slide rule: put 13 over 8 on the A en B scales and look for a pair of numbers that are placed above each other on the T and Gθ-scales and on the A and B scales. You will find: x0/b = 1.788 (the picture below proves this, but keep in mind that the scales do not line up in reality).
Finding x0/b
The tinsel sags by b(cosh(x0/b) – 1) = x0/1.788 (3.073 – 1) = 1.159 x0
This is rather large. The pictures I found show a sagging of (0.57 ± 0.27) x0 which corresponds to x0/b = 1.12 and a the tinsel length of 1.22 times the cord length, which is far less than 13/8. Using the factor 1.22 the tinsel-formula becomes:
The star or fairy
The star or fairy
The formula for the star or fairy seems to be based on aesthetics. The constant in this formula makes a slide rule obsolete.
Final remarks
Final remarks
The formulas published by the English students are easily computed on a slide rule. All calculations can be done on a Mannheim using one setting. For convenience you might want to use gauge marks. 𝜋 is already present on most slide rules, but, according to Panagiotis Venetsianos’ list of gauge marks[9] , √17/20 = 0.2062 and 13 𝜋/8 = 5.105 do not appear on any slide rule. So you'll need to add them to the rule.
The exact formulas can not be computed using one setting on a slide rule. Using a Sun Hemmi 153, you can do the calculations in a number of steps.
So next time you decorate your Christmas tree, keep your Sun Hemmi 153[5] at hand.
Thanks to Otto van Poelje for information on the differences between P scales.
Notes and Literature
Notes and Literature
Treegonometry creates perfect Christmas tree , University of Sheffield, 28 November 2012 (visited 7 December 2013). Cited by Libelle.
Mathematical formula for the perfect Christmas tree, gizmag,10 December 2012, (visited 7 December 2013)
Treegonometry uses math to perfectly decorate a Christmas tree, TechHive, 7 December 2012 (visited 10 December 2013)
KRO television broadcast ‘De Rekenkamer’ (5 December 2013)
If you do not own a Sun Hemmi 153, use the virtual one (and its manual)
The (half) opening angle 𝛼 of artificial Christmas trees, derived from 28 patents, equals 20 ± 4°. There are a lot artificial Christmas tree patents, but many of them are so unnatural that their 𝛼's can not be compared with those of real ones, and they are not included in this statistic.
Kerstboomverlichting, Eclecticsite, 19 March 2012 (visited 7 December 2013)
We simplify the problem by keeping the supports of a tinsel segment in a horizontal plane. If the tinsel was derived from the light cord, these points should be on a helix and the tinsel would not sag symmetrically.
Panagiotis Venetsianos, “Pocketbook of the Gauge Marks”, The Oughtred Society, 2006.
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