## Monday, 13 May 2013

### Egg shells to turtle shells.

Over the past few weeks we have looked into the production of self righting objects. We started by considering weebles that have heavy bottoms, we then took a detour and looked at the stability of egg shapes and how we could make any shape have more stable points. Finally, we saw the work of Gábor Domokos and Péter Várkonyi and their discovery of the gömböc.

No matter how you initially orient the gömböc it will always wobble and rotate itself to finish standing upright. Importantly, the gömböc is made of only one material, so its density is uniform. Mathematically, the gömböc is known as a mono-monostatic body. This simply means that it has exactly one stable and one unstable equilibrium point.
 Figure 1. A procelain gömböc. Hand painted by Ms. Pálma Babos [1].
This is all very nice and the gömböc makes a beautiful little toy to play with, but is it useful for anything other than a paper weight? Amazingly the answer appears to be yes. Embarrassingly, nature had solved the problem a long time ago in the form of the turtle shell.

Many turtles have quite flat shells and use their necks to turn themselves over using a so named “break dance” technique as seen in Figure 2. However, for turtles with much taller shells this is not possible because their neck is not long enough. Thus, a couple of turtle species have shells shaped like a gömböc in order to allow them to roll over easily. The movie below shows Gábor actually putting some turtles on their back and watching them right themselves.
 Figure 2. Self righting turtles. On the left a relatively flat turtle uses its neck. On the right, the turtle uses a gömböc shaped shell.
 Figure 3. As the shell gets higher relative to its width the number of equilibria change.
Perhaps the most impressive link between the mathematics of the gömböc and the turtle shells is that alterations of the gömböc shape can be linked to different species. Importantly, as the gömböc height is varied the number of stable and unstable equilibria change. This is also seen on the shells. This is shown in the Figure 3.

So what have we learned over the last few weeks? For me, it is good maths will always lead to interesting outcomes and that nature is a far better mathematician that we will ever know.

For more information on gömböcs and their connection to turtles take a look at www.gomboc.eu or the original research paper can be found here.

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[1] The Herend porcelain gömböc is a high-end, beautiful piece of art but, sadly, it does not work. Porcelain technology is simply not good enough. Picture taken from http://www.gomboc-shop.com/

Self righting turtle photos courtesy of Gábor Domokos and Timea Szabó.
Turtles: