How much tinsel do you need to decorate a Christmas tree?
Our idealised tree will be a cone with opening angle . The tinsel lies on the surface of the cone, and I’ll state that the curve described by the tinsel can be expressed parametrically by the equations
where the axis coincides with the tree’s axis of rotational symmetry, and corresponds to the tip of the tree. This vector describes the locus of a conical spiral, for which the distance between successive loops is constant.
To make sure this spiral lies on the surface of the cone, we have to make sure the constants and are chosen appropriately. To do so, it’s useful to define the tinsel’s radial distance from the tree’s axis of symmetry:
Then if ,
So after wrapping around the tree once, the tinsel has moved outwards by an amount and downwards by an amount . I hope you’ll agree that this means the angle made by the tinsel’s spiral path with the axis satisfies
Hence for the tinsel to lie on the surface of the tree, we require that and satisfy
which we can also express as
This is the distance between successive loops, as measured along the surface of the tree. It’s a bit like the tinsel’s ‘wavelength’, which is why I’ve called the quantity .
For now we’ll continue to work with the parameters and , and switch to the physical quantities and when it’s appropriate.
To measure the length of the tinsel, we use an integral of small line elements:
The length of each element can be written in terms of the differentials using the equation
which can conveniently be expressed using the parameter :
Using the equation for , we can work out the necessary derivatives:
Substituting these in gives
Now would be a good time to change to the parameters we introduced before:
So we need to solve the integral
By exploiting the identity
the integrand can be simplified using the substitution
The next few steps require the ‘double-angle’ formulae
allowing us to say
which is equivalent to
Let’s try a few values. For a Christmas tree with opening angle 35 degrees adorned with 5 turns of tinsel separated by a length of 40 cm, the total length of tinsel needed is
That’s just for a modest tree! Try some for yourself. For very long lengths of tinsel, the expression above can be approximated as
since the logarithmic term grows much more slowly than the quadratic term.