What Exactly is a PhiTOP?

The PhiTOP is a 21st century dynamical object with a unique mathematical shape. It is in the broadest sense a “philosophical toy.” It can be spun like a top – which it is. Or it can sit quietly on a desk to be enjoyed as an elegant and sensual sculpture. It was designed for the delight of people of all ages from 5 to 105. It makes for an exceptional executive gift. But it also captures the imagination of young children. It is a scientific demonstration apparatus: the PhiTOP does for angular momentum what Newton’s Cradle does for linear momentum. When spun, it illustrates the difference between equilibrium and stability. In that sense, it complements Danish polymath Piet Hein’s famous static (non spinning) super-egg. The PhiTOP encourages exploration of force, mass, density, gravity friction and time. And of aspects of electromagnetism as well. It is a tactile aid to meditation and relaxation, akin to worry stones or Chinese Baoding balls. And it can even be used as an excellent drink cooler!

Download the full academic paper about it here.

A Distinguished Historical Pedigree

The PhiTOP is the final outcome of an informal experimental study that I conducted using “ellipsoidal” and “egg-shaped” objects that can be spun from an initially horizontal position, which then, surprisingly, stand erect. The rise of the “center of mass” of such spinning objects that roll and/or slide on a surface because of contact friction is a difficult mathematical physics problem. It was first tackled by several great 19th century mathematicians and physicists such as Lord Kelvin and continues to be analyzed to this day. The class of such objects studied in the 19th century included chicken and other avian eggs; wooden “darning eggs” and wooden Easter eggs; and natural stones found in riverbeds. The underlying physics is related to that of the motion of the tippe-top, a semi-spherical top which, when spun, inverts and spins in the opposite direction. The physics is also is related to that governing the motion of some stone archaeological artifacts called “celts” which, when spun in one direction, can stop and reverse direction. Now known as rattlebacks or “wobblestones,” they are made from a wide variety of materials and are the basis of a popular science toy sold under the name “space pets.” Observation of the non-intuitive motion of these kinds of objects probably dates back to antiquity, yet still provides mathematical challenges to 21st century mathematicians and physicists to provide complete analytic treatments of their motion. 

I have had a long-standing interest in spinning tops. A paper that I wrote about them can be found here, accompanied by Top-ology: The Film, a short film that I made in collaboration with my daughter, which was called "more captivating than anything Hollywood has produced in the past decade" by Gizmodo. Last year, I began experimenting anew with a variety of egg-shaped and “prolate” ellipsoidal objects made from stone and wood. Many of these natural and man-made objects do not have simple or simply defined mathematical shapes. For example, a chicken egg is thicker on one end than the other. This “ovoid” shape comes in many variations. Similarly stones – natural or worked – also display a wide variation in their exact shapes. I found that only those with a ratio of length to width of about 1.5 – 1.7 consistently displayed interesting dynamical behavior when spun. By interesting I mean, when spun from a horizontal position, they would rapidly rise, stand erect, spin at a different rate and then gracefully settle down to a static horizontal position. I found that objects with the ratio greater than about 2 to1 will not stand up and spin stably; those with the ratio less than about 1.4 display erratic and less visually pleasing and interesting behavior.

The Making of...

With this somewhat informal study employing pre-existing objects as a guide, I then set about making objects in plastic by employing rapid prototyping techniques (using 3D printers such as the MakerBot). I also used CNC (computer numerical control) machining techniques to make the objects from a variety of metals and plastics. I focused on one class of well-defined mathematical shapes: prolate ellipsoids (or prolate spheroids). These look somewhat like eggs, but are symmetric with respect to the long axis. The objects were made in a wide variety of lengths, from about ½ inch to as much as 5 inches long. Experimenting with these, again, the ratio of length to width for the most enjoyable performing ellipsoids came out about 1.6. 

1.6 turns out to be very close to one of the most famous numbers in mathematics: the golden ratio (also “golden mean” or “golden section”), usually labeled “phi” (Latinized) or “Φ” (Greek). It is the solution to the equation 1/ Φ = Φ - 1. The solution is Φ ~ 1.618… This number is associated with a wide variety of interesting topics in mathematics, physics and biology. Because this ratio is said to be aesthetically pleasing to people, it has purportedly also been employed by famous artists such as Leonardo da Vinci in their paintings and by ancient architects in some of their monuments. A prolate ellipsoid that incorporates this number as the ratio of length to width can be called a “golden ellipsoid.” It is a unique mathematical shape. Incorporating this ideal mathematical golden ellipsoid shape into a real physical object gave rise to my resulting PhiTOP. A refereed paper reporting the development of these ideas was presented at the Bridges 2015: Mathematics, Music, Art, Architecture, Culture Conference held this summer. A patent is now pending on this physical object for its utility as a toy, and the names “PhiTOP” and “ΦTOP” have been trademarked.  

A Philosophical Toy and Then Some

Playing with the PhiTOP is a joyful experience. When spun either with one hand (which takes a bit of practice!) or, more easily, with the thumb of one hand and the index finger of the other, the PhiTOP suddenly stands erect. The “WOW” response produced in people who see its motion for the first time is palpable when it first stands, going from a resting horizontal position to a spinning vertical position. The top can then spin in the upright position for 1 – 2 minutes. It then settles down in a few minutes, producing a soothing sound, and interacting with ambient light to produce a miniature light show. All told, it leads to about 6 minutes of pure joy and escape from the drudgery of one’s daily tasks. It makes for a wonderful desk toy (akin to Executive Decision Makers or Euler’s disc). 

Bring a refrigerator magnet next to a PhiTOP that is made from a non-magnetic metal such as aluminum or brass while it is spinning and it will slow down and stop. This is a simple demonstration of the magnetic induction principle that underlies Nikola Tesla’s famous 19th century Egg of Columbus demonstration that led directly to the acceptance of AC electricity over DC. The original 19th century apparatus employed a rotating 3-phase magnetic field configuration with no moving parts to spin up a copper egg shaped object. The video on the Watch It Spin page shows an aluminum PhiTOP sitting on a concave mirror, spun up by a magnetic chemical stirrer placed beneath the mirror. It is the same effect I got to test on a replica of Tesla's original apparatus in Belgrade, Serbia this past May while prototyping!

The PhiTOP also produces a marvelous optical illusion as it slows down: the gelatinous ellipse effect. Even just sitting quietly on the desk, it is an eye catching miniature sculpture. And placed in the hand, it is a meditative tactile object, which can act as a stress reliever.

The Vision for PhiTOP Future

A limited edition of PhiTOPs were fabricated as part of a successful Kickstarter campaign in two different metals using CNC machines: aluminum and brass. Each one is then hand-polished to remove end burrs to reduce surface friction and increase spin times, and to provide an aesthetically pleasing mirror-like surface finish. This is a labor-intensive process.  First edition aluminum and brass PhiTOPs are available in limited quantities by contacting us through this website.

So far, we have not yet found a way to fully automate the fabrication of the PhiTOPs so that the price of production – and hence sales price - can be brought down. One of our future goals is to come up with a method for the production of an inexpensive version of the tops for distribution in science museums and in schools so that children can also experience the joy of learning with this marvelous hands-on and eyes-on science and mathematics toy. My goal is that the PhiTOP not just become this year’s Pet Rock (which would certainly be marvelous!!!) and a splendid executive gift but also that it become an addition to tomorrow’s children's “learn through play” toolbox, which traditionally included objects like the yo-yo, slinky, jax and more traditional spinning tops - an additional tool for children’s STEAM (Science, Technology, Engineering, Art, and Mathematics) education - although anyone can indulge their own inner child!