Bottles in a Wine Rack:
Proofs and Generalizations

By N. Bowler

First, we must note that the arguments I gave earlier are better suited to rhombi and indeed will always work if the construction is performed using rhombi rather than circles.

However, as was pointed out, there is still some good behaviour even in the problem cases. Indeed, we have the following conjecture:

which I shall now prove.

In order to do this we must analyse the possible types of bad behaviour. We know by the equalities mentioned in my earlier post that for all m and k there exist k' and k'' such that r(m, k) = r(0, k') and l(m, k) = -r(0, k''). We also know that the circles in the base layer do not overlap so r(0, k) < p/3 for all k. So we know that l(m, k) - r(m', k') < 2p/3 for all m, k, m' and k'. Thus none of the rhombi is too sharp, and any problem rhombus must be too flat. For there to be such a rhombus we must have

for some k₁ and k₂.

Now assume for a contradiction that k₁ ≠ k₂. Let d(k) be the distance from the centre of the kth circle to that of the (k + 1)^st circle in the bottom row. Thus d(k) = 2·cos(r(0, k)). Also, d(0, k) ≥ 1 for all k and:

so letting r₁ = r(0, k₁) and r₂ = r(0, k₂) we have:

Now this is impossible, as is most easily seen in a diagram. Nevertheless, I shall give an algebraic proof. Without loss of generality, assume r₁ < r₂. For fixed R = r₁ + r₂, the rate of change of C = cos(r₁) + cos(r₂) with respect to r₁ is -sin(r₁) + sin(r₂) > 0. So C is minimised when r₁ is 0. But then C = cos(R) + 1 > 3/2 which is the desired contradiction.

So in fact k₁ = k₂, and problems arise iff there is some (necessarily unique) K with r(0, K) < p/6. In this case, we have one problem of type (3) since

and one of type (4) since

In fact, if the bottles are placed in the manner employed by the applet you wrote, and we then consider the values this gives us for the l and r functions, we obtain the same equalities as before except that:

Working out the consequences of this for the top layer in the same way as before shows that the (n - K)^th and (n + 1 - K)^th bottles in the top layer will be higher than the others and level with one another, as conjectured. It also shows that all the bottles to the left of these two will be level with one another, and similarly for those on the right. However, the height of the bottles on the left may differ from that of those on the right. These extra predictions are easily seen to be true upon experiment.

(The applet below helps visualize the geometric properties underlying the foregoing algebraic derivations. The inner circles at the bottom can be dragged left or right and the ones in the next row - those with the red center - can be clicked upon. See what happens.)

The proofs and derivations of the results below follow exactly the same pattern as the above, with only slight modifications of the dull algebra. So I shall simply mention the results without proof.

Bottles in a badly broken wine rack:

Suppose that the wine rack is so broken that the sides, although straight lines, are both slanted, and are not necessarily slanted to the same angle. Then if we have good behaviour (which I will define better below, and which is almost equivalent to layer separation) up to row 2N-1, the bottles in row 2N-1 will be collinear, so that we may add a lid which is tangent to them all. Furthermore, the lid, the base and the two sides of the rack will form the sides of a cyclic quadrilateral.

The nature of good behaviour

First I shall set up some convenient notation. Let r(k) = r(0, k) for any k. Recall that d(k) = 2·cos(r(k)) so that these are natural quantities to work with.

It is then necessary and sufficient to have:

These conditions may be simplified, but have to be simplified differently in three different cases; that the sides both slope inwards or both slope outwards or one slopes in and one out. Note that if the sides are vertical, they simplify to precisely the condition given previously.

• A Circle-Stacking Theorem
• A Property of Rhombi
• Bottles in a Slanted Rack
• Bottles in a Wine Rack
• More Bottles in a Wine Rack
• Proofs and Generalizations

Copyright © 1996-2008 Alexander Bogomolny