On the Shapes of Water Fountains and Times Tables

Stephen Eberhart
Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings (2003)
Pages 361–366


According to the well-known Galileian approximation (near Earth's surface, ignoring air resistance), the trajectory of a freely falling body such as a ball or drop of water isa parabola; but what of the many drops of water springing from a fountain (under equal pressure but at different angles to the horizontal) - what describes its over-all profile? Apparently unrelated, the usual shape of a times table is a semi-square since the commutivity of ab = ba makes the other half of the square redundant; are there different naturally-motivated shapes in which to display such facts of elementary arithmetic? In particular, what if the multiplication is in a finite modular system? Surprisingly, the two questions turn out to be related and lead to Mobius strips in special cases.