Monday, 18 February 2013

Finding the impossible

Over the past few weeks we have been looking into stable and unstable objects. Our aim is to try and find a shape that is a natural weeble. Specifically, we want a solid shape that has constant density throughout (i.e. it is made of only one material) and this shape must have only one stable equilibrium point and one unstable equilibrium point.
Figure 1. Gábor Domokos and Péter Várkonyi.
Finding this shape has not been an easy task and became an obsession for a couple of mathematicians: Gábor Domokos and Péter Várkonyi, from the Budapest University of Technology and Economics. Early this year I was very fortunate to meet Gábor and learn, firsthand, the lengths that he went to, to prove the existence of such a shape.

Figure 2. Pebble beach on Rhodes.
Gábor worked on this problem for a long time, making headway to a final goal, but a proof of existence or non-existence eluded him. Thus, he did what any good scientist would do; he did an experiment. Whilst on holiday in Rhodes he collected 2000 pebbles from a beach and, along with his wife, set about catergorising the pebbles by their number of stable and unstable equilibria. Although he found many interesting shapes (and attracted the attention of a couple of police officer as he was returning the pebbles to the beach) he did not find the one he was looking for.

Figure 3. Part of Gábor's stone collection, ordered according to (number of stable equilibria, number of unstable equilibria).
During their work they found many clues pointing to the properties that the object must have (if it existed). For example, they discovered that such an object could neither be too flat object, nor too thin. If it was two flat, like a piece of paper, then it would be would have two stable equilibria, one on each of the flat sides. Similarly, a thin object, like a baguette, will generally have two unstable equilibrium points at the tips. This suggested that if the object existed and was perfectly smooth then it would have to be very close to a sphere.

Using this information they started modifying the sphere and, finally, they did it. They found a three-dimensional object that had exactly one stable equilibrium point and one unstable equilibrium point. In other words they had theoretically shown that it was possible to create a weeble using only one material (you did not need to weigh the end). However, because they wanted a smooth solution their initial shape was incredibly close to the sphere. This meant that although they had proven the principle it would not be physically possible to make such a shape as the error tolerance in the construction was so low.

Building on this, they decided to relax their desire of smoothness and so the shape could have corners. This allowed them much more freedom and eventually they produced a shape that was visually very different from the sphere (see Figure 4), but still maintained the property of having only one stable and one unstable point.

Figure 4. The computed solution on the left and a fabricated version on the right.
Due to the mathematics showing that the smooth object should be close to a sphere the shape was called a gömböc (pronounced gumbuts), which is a diminutive of the word gömb ("sphere" in Hungarian).
Below is footage, taken by Alain Goriely, of a gömböc that Gábor brought to Oxford.

The final post on this subject will be in two weeks and we will take a look at one of the most amazing natural uses of the gömböc.

Gábor Domokos and Péter Várkonyi pictures:
Gömböc pictures courtesy of Gábor Domokos.

Monday, 4 February 2013

The egg of Columbus

Figure 1. An egg standing to attention.
Last time I demonstrated that if a weeble of uniform density existed, then it could not be effectively 2D. This then led to considerations of the stability of 3D shapes in order to try and find the elusive shape with exactly one stable equilibrium point and one unstable equilibrium point. Since we can construct shapes with lots of stable points what we would really like to do is find a find of reducing this number. However, today, we take a sleight detour and consider the opposite problem of making an unstable point stable!

We start with a challenge (supposedly) set by Christopher Columbus. Whilst dining one night, a nobleman approached him and suggested that finding the Americas was not that impressive because anyone sailing out that way could not have missed them. In reply Columbus challenged the nobleman to take a normal egg and place it so it stood upright. Now, there are many ways of doing this, but most use some other piece of apparatus (such as alternating current), but Columbus used only the egg.

When the nobleman gave up Columbus simply took the egg and tapped it gently breaking the top of the shell, making it flatter. The moral of the story is normally anyone can solve a problem once they have seen a solution. However, for us, the story shows how to make unstable points stable.
Figure 2. Stabilising the unstable. By flattening the top the egg becomes stable upside down.
What is more amazing is that you can actually rigorously construct a mathematical algorithm that does exactly this flattening operation. Thus, if a uniform density weeble does exist then following this algorithm will allow us to construct a solid with any number of stable equilbria.

In two weeks we return to the trail of finding this illusive shape.