User Case Study: Maple helps discover the mathematics-based Gömböc shape

Maple’s computational power which can calculate and explore very sensitive details was critical to the discovery

When the existence of the Gömböc shape was discovered in 2007 by Hungarian scientists Gábor Domokos and Péter Várkonyi, it solved a long standing mystery. For years mathematicians had discussed, debated and tried to prove its existence using mathematical equations. Vladimir Arnold, a Russian scientist, had conjectured its existence, but it took a decade to prove it conclusively and create the shape. The New York Times called the discovery “one of the best ideas of the year.” Like in many other mathematical developments, Maple also played a role in creating the Gömböc.

A Gömböc is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. The Gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (less than 0.1 mm per 10 cm). The most famous solution has a sharpened top as shown in the figure.

If you put a Gömböc down on a flat surface, resting on its stable equilibrium point, it will stay in the same position. "Even if you kick it a little, it will come back to its resting position at the stable equilibrium point," says Domokos, one of the inventors of Gömböc. If it is put down at a non-equilibrium point it will start rolling around in a systematic way until it has reached the stable equilibrium position. In other words, the Gömböc is self-righting¹. In fact, Wired Magazine calls it the “world’s first self-righting object.”

Gömböc has found an entry in the Guinness Book of Records as “the first homogenous self-righting shape.” The Natural History Magazine illustrates that “the secret is in the mathematics of its shape.”

Using Maple made the calculations more thorough and secure; its computational power can calculate and explore very sensitive details. So it was a trusted companion in our discovery process.

— Gábor Domokos,Gömböc

The invention of Gömböc is the culmination of a long process of mathematical research and Maple, the mathematical computation engine from Maplesoft, played an important role in its discovery. The yet-undiscovered shape was known to be a convex mono-monostatic object — a three-dimensional object, which because of its geometry had only one possible way to balance upright. Domokos and Várkonyi identified a two-parameter family of objects, all of which had the desired mono-monostatic property. However, not all of them were convex. Maple was used to identify the convex shapes and thus prove the existence of the shape. The process involved a large amount of complex, precise mathematical computation, and Maple’s symbolic computation power made it possible.

“The final geometry of Gömböc had to be determined with great accuracy, which meant the details were critical and we couldn’t afford to miss any,” said Domokos. “Maple was very useful in this regard. Using Maple made the calculations more thorough and secure; its computational power can calculate and explore very sensitive details. So it was a trusted companion in our discovery process.”

Speaking of the popularity of the Gömböc and the attention it received from the scientific and mathematical community around the globe, Domokos said, “The beauty of Gömböc is its absolute simplicity. It is so simple that high school students can understand it, but it has potentially great impact on the sciences and has several applications in nature. Yet the answer was elusive for 2000 years.”

Taking their discovery into the world of natural science, Domokos and Várkonyi discovered a unique application of Gömböc among tortoises. They conducted an extensive study of tortoises using complex three-dimensional models of the shell, created in Maple. Using these models, they discovered that, of the 200 species of turtles in nature, two species had Gömböc-shaped shells. This meant that these turtles had a unique evolutionary advantage in that they had the ability to self-right. “I saw that they were acting like Gömböcs!” exclaimed Domokos. Being on their back is a vulnerable and dangerous position for any tortoise. The male tortoise is known to turn over their rivals on their back in an attempt to render them helpless. So any tortoise that has the ability to get back on its belly, under gravity, without having to use its muscle, has a unique advantage, and more power. “It is the Gömböc shape of their shell that gives them this power. This is a classic example of evolution finding the optimal shape for survival,” observed Domokos.

Gomboc and nature from Gömböc on Vimeo.

Continuing his research, Domokos is currently involved in studying the shape of beach pebbles. His research with Gary Gibbons from Cambridge University attempts to describe the shape of pebbles, and the evolution of their shape. They are also trying to understand the interaction between pebbles in their collective evolution. Using Maple, Domokos was able to study a system of integrable differential equations. Solving the equation systems gives him unique insights that he otherwise may have missed. He is also studying the balance of friction and collisions in the abrasion process which results in what is called dominant pebble ratios, a phenomenon that makes pebbles in a certain geographical area display similar global geometrical features. Domokos is using Maple to determine the critical friction coefficients which are responsible for the emergence of dominant pebble ratios.

Understanding shapes in nature is becoming increasingly important, both on earth and off it. Recently, NASA’s Curiosity rover sent back images from the surface of Mars that showed an ancient riverbed. The size and shape of the pebbles indicate that water flowed on Mars billions of years ago, and by studying the images of the rock, scientists will be able to learn more about the speed and size of the ancient flow of water.

As Domokos continues his research into mathematical shapes and the discovery of natural and scientific phenomenon based on these shapes, Maple will continue to play a significant role in his research. “Maple is my favorite computational tool, it is simple and powerful,” says Domokos. “We are surrounded by geometrical shapes which our brains are not wired to understand. Shapes like Gömböc open a new language to understand such shapes.”

Gömböcs are available for purchase at the Gömböc web store in different material and sizes.

¹ Quoted from the article The story of the Gömböc in Plus Magazine

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Highlights

- The New York Times called the discovery “one of the best ideas of the year.”
- Identifying the shape involved a large amount of complex mathematical computation, and Maple’s symbolic computation power made it possible.
- The discovery of the Gömböc shape is resulting in interesting research findings in the world of natural science

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