As a physicist and keen tennis player, I would like to share an amusing “discovery” I recently made. In my office, I have about 20 used tennis balls and so decided to try building some tennis-ball “pyramids”.
As you might expect, a four-level pyramid has a triangular cross-section, with 10 balls at the bottom, followed by six in the next layer, then three and finally one ball on top (image top right). When I carefully removed the three corner balls from the bottom layer plus the upper-most ball, I ended up a with a beautiful, symmetric structure of 16 balls with three hexagonal and three triangular sides (image top left).
Interestingly, the corner balls in the second-bottom layer are kept in equilibrium, hanging over the layer below. These “exposed” balls are held in place because the balls directly above press down on them and into the two adjacent balls of the bottom layer – producing a pair of reaction forces to balance their weight. The torques are balanced too, with enough friction between the felt-covered balls to guarantee equilibrium.
Intrigued, I recreated my original 20-ball pyramid and found that when I removed all three corner balls in the lowest layer but left the single ball on the very top, I was able to take out the three corner balls in the second-bottom layer. What I ended up with was a bizarre, Christmas-tree-like structure made of 14 balls (image middle right).
It then occurred to me that the top three layers ought to remain in equilibrium even if the lowest layer were not there. So when I rebuilt the Christmas tree without that bottom layer, I created a beautiful and delicate seven-ball structure (image bottom right). The top ball is crucial for keeping the structure steady: it presses down on the three balls in the layer below, which in turn presses down on the three balls on the table. Their counter reaction keeps the middle layer steady. Again, friction is vital: without it, there would be no torque balance and the balls would roll away.
Friction is vital: without it, there would be no torque balance and the balls would roll away
Moreover, I could make this seven-ball structure even higher by adding one extra three-ball layer after another, in which each tower has (3n + 1) balls, where n is the number of triangular layers. It got increasingly hard to make the towers as they got taller. Indeed, to create the seven-storey, 19-ball structure – the tallest I’ve built so far (image bottom left) – I needed special “scaffolding” in the form of tennis-ball boxes and my hands to support the tower as it went up. I could remove the “scaffolds” only after putting the top ball on.
I can find no mention of such structures online and am sure they would have interested Martin Gardner – that great fan of “recreational” science – were he still alive. I wonder even if my “discovery” could be turned into a board game of some sort, with players required to build complex structures from such balls?
Institute of Theoretical Physics, Ilia State University, Georgia