The Academic Juggler

 

PICTURE THAT!

By Arthur Lewbel

Most photographs of juggling show objects scattered more or less randomly in the air, but occasionally something odd happens. In a picture of an ordinary cascade objects may be neatly ar­ranged in pairs in the air, or clubs may be all horizontal or all vertical at the same time. Why does this happen?

 

Start with a three ball cascade, numbering the balls in the order that they are originally thrown. When ball two is at the top of its are, ball three is being thrown and ball one is being caught. If a picture is taken when ball three is just as close to one hand (on the way up) as ball one is to the other hand (on the way down), the resulting photo will have a single ball on top and two in a horizontal pair near or in the hands.

A similar effect can happen with any number of balls.  Consider a five ball cascade that was started with 3 balls in the right hand and two in the left. When ball three (on its way to the left from the right) is at the top of its are, ball one is being caught in the left hand and ball five is leaving the right, so balls one and five form a horizontal pair near or in the hands. In between are balls two and four. They will also be in a horizontal row, with ball two heading down to the right and ball four going up from the left. Balls two and four are both the same height because they are both equally distant from ball three in time, and hence are both equally distant from the top of the pattern.

 

In an N ball (or any object) cascade for any odd N, when one ball is right at the top of its flight, the remaining balls are arranged in (N-1)12 horizontal pairs. In each pair, one object is going up, the other down, and the pairs themselves alternate so that if one pair has the object on the right going up, then the next pair up has the object on the left going up.

 

 

This pairing of objects can be seen in the accompanying photo of Demetrius AIcarese attempting seven clubs. Notice also the vertical distances between these pairs of clubs. The higher up one looks, the closer is one pair of clubs to the next pair up (or to the single club on top). This occurs because adjacent throws are equally  spaced from each other in time. When an object is thrown upward it slows down, stops at the top of its flight, then speeds up on the way down until it is caught. So, to be equally spaced in time the objects need to be closer together vertically near the top of the flight than near the bottom.

 

This explains pairing, but what about all clubs being vertical at the same time, like in the photo of Larry Merlo? A mathematically detailed explanation can be found in ''The Physics of Juggling," by Bengt Magnusson and Bruce Tiemann (The Physics Teacher, November, 1989), but the basic idea is simple.