Gömböc – Finding Consilience

Where math meets nature.

   Scientific discovery can speed the heart and excite the mind. And, when that discovery transcends the purely theoretical and joins hands with the empirical, we stand up and applaud. All stand!
    All living things are the result of the non-random process we know as evolution.  From the molecular to the morphological, selective pressures have acted to yield the entirety of our planet’s biomass. Achieving a true understanding of any living thing remains a challenge.  The approach that is required, we know, is consilience, the recognition of a fundamental unity of all knowledge.  The term was first coined by William Whewell (who, by the way, also gave us the word scientist) in 1840 when he put it thus: “The consilience of inductions takes place when an induction, obtained from one class of facts, coincides with an induction obtained from another different class. Thus consilience is a test of the truth of the theory in which it occurs.” I have for you here a tale of consilience.
    The year was 1995, the place Hamburg and the occasion, the International Conference on Industrial and Applied Mathematics where the great Russian mathematician Professor Vladimir Igorevich Arnold had given a spellbinding talk whose central theme was the number four.  Attending the meeting was Gabor Domokos, currently professor and head of the Dept. of Mechanics, Materials and Structures at the Technical University of Budapest.  As a Fulbright Fellow visiting America in the late 1980s, Domokos had developed an interest in the Euclidian problem concerning the minimum number of points of equilibrium of convex 2-dimensional shapes (for practical purposes, a shape cut from a piece of plywood) and eventually published his proof that every homogeneous convex shape has a minimum of 4 points of equilibrium, 2 stable and 2 unstable.  Arnold’s plenary lecture on the number four piqued Domokos’ interest.  Although a lunch with the great Professor yielded little more than frustration (Read Domokos’ highly entertaining account of his lunch with Arnold here.), Domokos’ later chance conversation with Arnold proved to be seminal. It is during this conversation that Arnold conjectured that there must exist a class of 3-dimentional objects that are mono-monostatic, that is to say objects that have only one stable and one unstable point of equilibrium.  Domokos decided to take up the challenge.
    Together with his graduate student Peter Várkonyi, currently an assistant professor also at the Technical University of Budapest, Domokos spent the next ten years searching for proof of the existence of the elusive mono-monostatic body.  Although most of their quest involved complex mathematics way beyond my understanding, more mundane techniques played a role as well.  For example, while he was on his honeymoon in Greece, Domokos tested and classified 2,000 beach pebbles. "Why he is still married, that is another thing," said Várkonyi in an interview.
  Eventually they came up with the answer, the first known homogenous mono-monostatic object and solid proof that Arnold’s conjecture was correct.  Enter the Gömböc.

    First, a note on the name: Gömb is Hungarian for sphere, gömböc meaning sphere-like (Not to be mistaken with the folkloric and very scary kis gömböc, a round creature in the loft that remained from a killed pig, which swallows everyone one after the other who goes to see what happened to the previous ones.) Seen simply as a design object, the Gömböc is a thing of beauty. In an interview with Bob Macdonald on CBC’s Quirks and Quarks, Domokos, in his rich Hungarian accent, quipped, “...well, some people claim it’s beautiful; I can’t tell because I’m an engineer.” Having the appearance of a pinched sphere, it looks at once knapped and organic.

     Beauty aside, the amazing thing about the Gömböc is that it is mono-monostatic, the first example of a previously unknown class of such shapes.  “Wrong!” you may be thinking.  “What about the WEEBLE toy?”  Well, the WEEBLE was/is indeed self-righting and like the Gömböc has one stable and one unstable equilibrium point.  However, it accomplishes this acrobatic feat not by virtue of its shape but rather by virtue of its inhomogeneity.  That’s cheating, of course and we’ll have none of it.

    Proving Arnold’s conjecture, in my mind, would have been impressive enough.  However, the pair, upon seeing their first Gömböc prototype, soon realized that self-righting must be a useful property and that there must exist mono-monostatic-like shapes in nature.  Notice that I said mono-monostatic-like and not mono-monostatic; indeed, the grim reality is that the harsh conditions that exist outside the engineer’s brain would quickly lead to surface perturbations on any naturally occurring mono-monostatic body thus robbing it of its self-righting property.  In addition, living things are not homogeneous.  However, Domokos and Várkonyi surmised that for certain terrestrial creatures such as tortoises and beetles, there would be a definite survival advantage to having a shape that has a tendency to self-right.
    I asked Domokos to explain how he first made the connection between the Gömböc and biology.  He answered in an email:

    “The turtle thing came up on a hike here in Budapest, in the summer of 2006. We found a beetle lying on its back and unable to turn over. I helped it to roll over and afterwards we realized that by having a Gomboc shape the beetle would not have needed help. Since beetles are so tiny, I started to look at turtles on the web and I soon found the pictures about the Indian Star Tortoise which does look like a Gomboc.
    We were just about to publish our paper in the Mathematical Intelligencer. At that point we had no hard evidence of the turtle connection but it sounded so attractive that the editor-in-chief agreed to put it on the cover. This was a high-risk operation for a serious journal and also for us.
    After that followed a year of very detailed measurements in the Budapest Zoo, Hungarian Museum of Natural History and most Pet Shops in Budapest, digitalizing shells and observing turtles, building computer models and thinking.
    Then we wrote the biology paper which was rejected 5(!) times subsequently, finally, with some luck it made it into one of the best biology journals (Proceedings of the Royal Society B) and after it appeared it made big news in the scientific community and the media.
    This is the turtle-story in brief.”

  That self-righting in terrestrial turtles should impart a survival advantage now appears self-evident.  A turtle that ends up on its back either as a result of an accident of terrain or in combat with a competing male is in serious trouble.  Unless it can right itself, the turtle is vulnerable to the elements and to predators.  Most turtles are able to use their long necks and legs to actively right themselves.  However, high-domed species such as the Indian Star Tortoise (Geochelone elegans) have relatively short necks and legs making active righting much more difficult.  The shape of these shells, we know is an evolutionary response to a variety of survival pressures including thermoregulation, protection against predators and camouflage.  Thanks to the work of Domokos and Várkonyi , we can now add righting to the list.  We now know that around 10 of the world's 200 known species of sea and land turtles possess some of the self-righting properties of the Gömböc.  Domokos: “We discovered it with mathematics but evolution got there first.”  Pure mathematics meets evolutionary biology; consilience.
    Their finding landed them on the front cover of the Mathematical Intelligencer, the very same distinguished publication that had featured Ernő Rubik’s cube on its front cover 30 years previously.  What is it about Hungarians and brilliant little plastic objects?
    For the Gömböc to be truly mono-monostatic, its shape must conform to the mathematical model within a very close tolerance. At present, it is manufactured by ultra-thin-layer, high-resolution 3-dimensional printing.  A Gömböc” requires a 0.1 mm tolerance in all dimensions for it to be mono-monostatic. The lengthy printing proceeds in layers of 16 microns (0,016 mm) thickness.  Embedded within the acrylic-based photopolymer object are the serial number and the logo and in order to maintain true homogeneity, these markings must be of the same density as the Gömböc.
    You can actually buy a Gömböc through the website . The pricing formula, though, could only have been conceived by an engineer:  In EUROS, 900 + 200,000/serial number.  Gömböc number 1 (you figure out the price) was given to Professor Arnold for his 70th birthday.  Lover of beautiful objects that I am, I bought one of course.  And, on impulse, I bought a second one for a person whose intellect I greatly admire and who, unlike myself, will be able to appreciate the mathematical significance of the Gömböc on a much higher plane. I think he'll be pleasantly surprised.

Joseph Froncioni

Comments

[Bunyip]

Ah when maths predicts something that is found in nature - or nature predicts something that is found in Maths. This is why I loved Pure maths over Applied Maths - because Pure maths would be the one that linked with nature and Applied maths made too many approximations to fit.

Other examples - the shape of fast fish and of bird wings - maths can show why they are the best shape for the purpose but evolution developed it first. Also occurs with neural networks - or trying to find the most efficent path from a to b in a branched system - our conventional plumbing and electronic structures - for sewers and computer systems is not nearly as efficient as the structures shown by blood vessels and nerve networks. And there's lots more.

[Patrice]

Hi Doc,

My ankle is well. It twinges sometimes, even when I am not insistent on teetering about in heels, but that is the price I have chosen to pay for my idea of fashion.

Now, about Gömböc – Finding Consilience. I have read, and reread it. I have even begun to think I understand it. I went to the website and thought of ordering one of the pretty little thingys. But decided I’d rather have a couple pairs or three of shoes instead. I have dreamt about the damn thing.

Please! Please! Please! Write about something new. If not I will remove this Intramedullary Rod myself and whack you over the head until you do.

In other words, I miss your witty writing.

Toodles,
Trice.

[AndyF]

I'm outting myself here, but I'm the lucky recipient of the Gomboc. :-)

I think I understand what it is mathematically, but what struck me most about it is how beautiful it is!

I've yet to find a suitable surface to display it, but I'll be building one soon.

Thanks, Joe, for your generosity and curiosity! :-)

Cheers!

Andy

[PGA]

Hi,

I've been reading your blog for a while and I decided to post a comment now to 1. thank you for taking the time and writing these articles (I really appreciate those 0,1% internet users, who actually contribute for quality content of our “web 2.0”) and 2. tell you how interesting I find your blog. I got here some months ago through a link to the "running shoes" article, and then found myself fascinated by the other themes you write about.

I'm a molecular biotechnologist graduate student, so I share your views (and concerns) about evolution. Currently I’m writing my thesis about biomolecular mechanics and dynamics and I found this blog entry fantastic! I had no idea about the existence of the Gömböc!

Greetings from Germany!