### An Attractive Spinning Toy: the Phi-TOP

It is fascinating to watch a top spinning. It seems to defy gravity: while it would topple over if not spinning, it remains in a vertical position as long as it is spinning rapidly.

There are many variations on the simple top. The gyroscope has played a vital role in navigation and in guidance and control systems. Many similar rotating toys have been devised. These include rattlebacks, tippe-tops and the Euler disk. The figure below shows four examples.

(a) Simple top, (b) Rising egg, (c) Tippe-top, (d) Euler disk. [Image from website of Rod Cross.]

The dynamics of the top are treated in many standard texts on classical mechanics. A top is a rigid body with an axis of symmetry about which it rotates. There may be a single point in the body that remains stationary. There are just a few tops for which a complete solution has been obtained. These include the Euler top, the Lagrange top and the Kovalevsky top.

The Phi-Top or ${\Phi}$-Top

An interesting top is based on the so-called rising egg model. A hard-boiled egg, spun rapidly, stands up on the pointy end and remains vertical for some time, until the energy is damped out. The PhiTOP is a beautiful scientific toy based on this idea. It is a prolate spheroid in shape (see figure). The semi-axes are ${a = b < c}$. Indeed, the ratio of the long axis ${c}$ to the shorter axes is chosen, for aesthetic and also for mechanical reasons, to be the golden ratio: ${a/c \approx \phi \approx 1.618}$.

The Phi-Top.

The tendency of an ellipsoidal body to rise depends on the aspect ratio. If it is either too flat or too elongated, rising may not be observed. The golden ratio proves to be quite satisfactory. A more complete discussion of the ${\Phi}$-Top is found in the paper (PDF) presented by Kenneth Brecher at the Bridges conference in 2015.

When the ${\Phi}$-top is spun rapidly, it behaves in a surprising way. Initially, the long axis is horizontal; the top quickly turns and spins standing upright on its pointy end. It continues to spin in this “sleeping” configuration for a sustained period — often for several minutes. Eventually, frictional damping slows the body down and the centre of mass begins to descend. The motion becomes more agitated as the point of contact rotates rapidly. Ultimately, it spins smoothly with the long axis horizontal again until it gradually comes to rest.

The ${\Phi}$-Top in transit from vertical sleeping to horizontal spinning.

The figure above shows the ${\Phi}$-Top as it is transiting from a vertical to a horizontal configuration. The point of contact is moving rapidly and the top is relatively noisy. The 10-second video clip below shows the ${\Phi}$-Top in slow motion at this stage.

Video clip courtesy of Owen Lynch.

Sources

${\bullet}$ Web-site of Rod Cross.

${\bullet}$ Kenneth Brecher, 2015: The PhiTOP: A Golden Ellipsoid. Proceedings of Bridges, 2015, pp.371–374. PDF .

${\bullet}$  The Phi-TOP website. More movies here.