# Pebbles, Mars and a maths problem that took 10 years

21 March 2017

In 2012 NASA’s Curiosity rover sent home stunning images of rounded pebbles on Mars.

It seemed plausible these pebbles acquired their rounded shape after being exposed to water, most likely an ancient river. But the theory was difficult to prove.

Until Professor Gábor Domokos of Budapest University of Technology and Economics solved a long-standing mathematical problem, one he had worked on for more than 10 years.

Professor Domokos will discuss the conundrum of pebbles, Mars and the maths behind the mystery in a lecture at the University of Auckland today.

The problem was to find a convex, homogenous solid object with only two points of static balance – one stable and one unstable – an object that was generally believed not to exist.

In his quest to find a solution, Professor Domokos once collected thousands of pebbles on a beach and checked every one but no such shape could be found.

Finally, its existence was proved in a 2006 collaboration with colleague P L Várkonyi. In addition, the pair created and subsequently manufactured such an object, which they named the Gömböc (‘sphere-like’ in Hungarian and pronounced Gombok).

It behaves like a roly-poly toy: no matter which way it is pushed or moved, it always comes back to its single – stable - point of equilibrium. But unlike the toy there is no weight inside the Gömböc.

The idea of the Gömböc proved to be way beyond a mathematical curiosity.

The shape of some high-domed terrestrial turtles approaches this shape so closely that the self-righting strategy of these animals can be based on their geometry alone.

More importantly, the Gömböc emerged as the key to understanding shape evolution in non-living Nature: as rocks, pebbles and sand-grains abrade by collisions and friction, the number of their static balance points is being reduced and so the ultimate (though unattainable) goal of their evolution is the Gömböc shape. The underlying mathematical models allow scientists to trace back the provenance of abraded river pebbles, sand grains or asteroids based solely on their current shapes.

This allowed the conclusion from the images of NASA that substantial rivers must have existed on ancient Mars.

This free lecture will be held at 5.15pm today Tuesday, 21 March in Lecture Theatre PLT1, Ground Floor of Building 303 at 38 Princes St, Auckland. Refreshments in the basement foyer follow the lecture.