Banner
    How Many Limbs Should Humans Have?
    By Mark Changizi | May 10th 2010 10:51 AM | 37 comments | Print | E-mail | Track Comments
    About Mark

    Mark Changizi is Director of Human Cognition at 2AI, and the author of The Vision Revolution (Benbella 2009) and Harnessed: How...

    View Mark's Profile
    In War of the Worlds, giant alien robots emerge out of the ground and begin vaporizing large numbers of actors. There’s a lot to like in those scenes, but there are three things I could not stand.

    Like those three legs they walked around on.   Not their fragile-appearing spindly-ness,but their actual three-ness.

    war of the worlds
    Please get Tom Cruise first, please get Tom Cruise first, please get ...

    There should be more legs. Around six of them, in particular.

    “Look,” you might reply, “it’s an alien ship, and who knows what kinds of principles they’ve uncovered.”

    Of course, that’s possible. But another way to look at it is that we Earthlings come in a large variety of body and limb plans, and yet we don’t find the three-limb design anywhere. Perhaps that’s a good argument that aliens wouldn’t build a ship with three legs.

    What do we Earthlings do for limb design?

    We tend to follow a law, one that may cut across all animal phyla, a law I first published in the Journal of Theoretical Biology in 2001 [ http://www.changizi.com/limb.zip ], and elaborated upon in my first book The Brain from 25,000 Feet [ see final section in http://www.changizi.com/ChangiziBrain25000Chapter1.pdf ].

    This “limb law” relates an animal’s number of limbs to the length of those limbs (relative to the body’s size).

    When an animal’s limbs are very long relative to its body size, I argued that the optimal reaching-out solution (that uses the least amount of “wire,” or limb material) is to have about six limbs. (This applies to animals with limbs that are approximately radially directed around a perimeter. For animals whose limbs directions are uniformly spread over a spherical surface, the expected number of limbs in this case would be about 12.)


    As the animal’s limbs shorten relative to body size, the expected number of limbs rises, with tremendous numbers of limbs when the limbs are very short. (By the way, a snake is consistent with infinitely many infinitely-short limbs – i.e., no limbs.)

    More generally, the law predicts that an animal’s number of limbs is inversely proportional to relative limb length. And, more specifically, the law predicts a particular proportionality constant, so that “six” is the solution in the case of really long limbs.

    Letting L be the limb length and R the radius of the animal’s body, then k = L / (L + R) is the relative limb length, or “limb ratio”.

    The number of limbs, N, is expected to vary approximately as 
    N ≈ 2π/k 6.28 k-1

    The figure below (from my first book) shows how the number of limbs in fact relates to limb ratio, for 190 species across seven animal phyla (Annelida, Arthropoda, Cnidaria, Echinodermata, Mollusca, Vertebrata, and Tardigrada).

    limb length ratio to limb number

    The predicted trend is shown with the solid line, consistent with the N 6.28k-1 equation we saw just above. 

    The actual trend is shown with the dotted line, leading to an empirical equation of N 6.24k-1.17 … or very close to the prediction.

    To get a better impression of the Limb Law that Earthlings appear to follow, check out this little dynamic visual program by Eric Bolz, allowing you to vary limb length and watch how the number of limbs varies: http://www.changizi.com/limb.html The right vertical axis allows you to modulate the limb ratio and watch the number of limbs change. The bottom axis allows you to make longer or shorter creatures. The left vertical axis just allows you to resize the creature on the page.

    The alien ships from War of the Worlds should have – given their long limb length and assuming they should be treated as approximately pointing around a perimeter – around six limbs. Not three.

    That’s why they look so silly. They’re outside of the sweet spot in design space for limbs.

    In my next piece, I’ll discuss how this limb idea tells us why we have 10 fingers, and perhaps, therefore, why we have a base-10 number system.

    Comments

    “Look,” you might reply, “it’s an alien ship, and who knows what kinds of principles they’ve uncovered.”

    Of course, that’s possible. But another way to look at it is that we Earthlings come in a large variety of body and limb plans, and yet we don’t find the three-limb design anywhere. Perhaps that’s a good argument that aliens wouldn’t build a ship with three legs.


    You're right, Mark when you say that we do not find three-limb designs in nature, but we do in robotics! It is based on the pendulum-like motion of the bipedal walking of human beings. So, you could say that the design is based on a principle of motion found in nature! Have a look:

    Your article is quite interesting, btw. ;-)

    If the video doesn't work, then just use this link: Dennis Hong: My seven species of robot


    Hank
     that we do not find three-limb designs in nature, but we do in robotics!
    I would contend it's because robotics people are all HG Wells junkies rather than it being an optimum configuration based on human walking - it is fine for standing, of course.

    The Three Laws of Robotics also show up everywhere, though they are completely outside human nature.   :)
    I would contend it's because robotics people are all HG Wells junkies rather than it being an optimum configuration based on human walking - it is fine for standing, of course.
    Oh, I would have to agree on that point, Hank. I see no real-world application for this three-legged robot. I think they just did it to see it they could do it. : )
    Mark Changizi

    Nice. Although I suppose I'll now argue, then, that Earthly creatures will kick three-legged robots' butts this side of the edge of the universe.
    Hank
    I think Gullivar Jones and John Carter showed that nigh on 100 years ago.  As that worthy science tome The League of Extraordinary Gentlemen II clearly showed, it was the Mollusks on the run from those warlords of Mars that caused the rumpus here on Earth - and those critters were smart enough to build robots and spaceships.  So just robots alone would go down easy.
    LOL....Oh, most definitely, Mark! : )
    we already have a base-10 number system, but americans are too afraid to learn it even though its the simplest number system in the world.

    Gerhard Adam
    I don't think you mean "number system", but rather you mean the "measuring system" (i.e. metric). 

    I think you're also overstating that Americans are "afraid" of the metric system.  I suspect it's more likely that there's no compelling reason to change. 

    Despite it's ease of use, Celius leaves a bit to be desired, since it's not as granular as the Farenheit system in terms of what humans can actually sense regarding temperature variations.  So perhaps all those metric users shouldn't be afraid to learn the Farenheit system and then we can split the difference.
    Mundus vult decipi
    Mark Changizi
    I agree. English units tend to be "human", even though metric is supposedly "rational." Rational to whom? A computer? We're human, and the coding system best for us depends on our brains.  The nice thing with Fahrenheit is that 0 to 100 pretty much spans the natural range of temperatures. For spatial measures, we have inch which is finger-segment-length, foot which is foot length, yard which is arm length-ish, and so on. (This is not my argument, btw -- I read someone else who made this case more fully. Can't recall where, now.)
    Hank
    Americans may never switch - it was an arbitrary unit created by a corrupt government in France who wanted to overthrow English domination and we don't like that, even if most people in the US do not remember why.  The French spend a lot of their time trying to be an alternative in an anglo world and coming up short and doing a bunch of work to create a 'more accurate' system and getting the measurements wrong is also typically French.

    That same year they instituted a 10-day week too.  Why isn't anyone talking about how much better a decimal week is?    Because it has no human benefit and scientists didn't adopt it.   I once discussed how we had science to thank for freedom.  It seems we can thank science for Friday nights too.
    logicman
    the coding system best for us depends on our brains

    I think of temperatures in degrees C. or K.  I could never intuitively grasp the Farengesundheit system.

    Does that make me a logic genius, or merely abnormal?

    Hank:  quit knocking the French!  We English do it so much better. :-)
    you said it yourself "earth based lifeforms". We don't know what aliens will look like or act like, hence the word "ALIEN"
    but that is still a good argument.

    Mark Changizi
    Although we don't know what aliens will look like, by understanding the design principles governing Earthlings, we can often guess which Earthly features are likely to be more universal, and which are likely to be peculiar to Earth.

    For example, here's a piece I wrote on this, concerning alien vision.
    Then again, there’s the “manufacture” factor to consider, in that most Earth animals seemingly might just as well have been moulded using two-part injection moulding, giving us a symmetry that probably dominates the limb quantity forcing it to be even where possible. Except starfish.

    Mark Changizi
    Maybe. But I'd argue that we have roughly five "limbs" that reach out: two feet, two arms, and head. And some of us primates have a prehensile tail (a lone unpaired added limb).
    logicman
    Can I please blow up Hollywood?   Pretty please?

    H.G. Wells' War of the Worlds is set in England.  England.  You know, that tiny island off the coast of Europe?  The one that the Pilgrim Fathers couldn't get out of quick enough?  That's the fella!

    The book is based heavily in real scientific principles.  No magic-lightning-strikes-twice-and-turns-into-a-Martian nonsense.

    Why three legs?  In the book, but not in the movie versions (oh, how I hate Hollywood) the Martians move in a pattern of triangulation.  Like surveyors.  In Wells' day surveyors would use tripods.  Today they use coffee bars while their computers and satellites do all the work.

    What made H.G. Wells associate Martians with surveying methods?   I suggest it was Percival Lowell's 'canals of Mars'.  Lowell's sketches bear more than a passing resemblance to surveyors' triangulation networks.



    Why would H.G. Wells be thinking of surveying?  Surveying in England was always associated with the military.  Our Ordnance Survey maps were originally produced to assist in the siting of guns with which to dissuade visits by French tourists.

    Wells would have been aware of the surveying activities in the U.K. and India.  At that time, before the advent of the aeroplane, creating an accurate map from basic geometrical principles was an astounding acheivement.  I have no doubt that the newspapers would have been full of it - as indeed they so often are today.

    But how natural is a tripod to a biological organism?

    In Wells' day, no self-respecting gentleman would be seen dead without a shooting stick.




    q.e.d.
    Larry Arnold
    That is an interesting point, as old as the riddle of the sphinx, "What goes on four legs in the morning, on two legs at noon, and on three legs in the evening?"

    I have reached the three legs stage. That apart I am a photographer and familiar with tripods, a tripod is the most stable arrangement for remaining even on an uneven surface, same goes for three legged stools.
    logicman
    a tripod is the most stable arrangement for remaining even on an uneven surface
    I think that is why the shooting stick was invented - it goes rather well with a brandy flask.


    Quocunque Jeceris Stabit

    Isle of Man coat of arms.

    Mark Changizi

    Ah, but don't forget the "wobbly table" theorem...
    http://www.maa.org/devlin/devlin_02_07.html
    I'm not quite at the "three legs stage" yet, but at age 54 it's only a matter of time, Laurence! LOL ;-)


    Oedipus explains the riddle of the Sphinx, by Jean Auguste Dominique Ingres, c. 1805

    Mark Changizi
    Brilliant!
    LauraHult
    In the book, but not in the movie versions (oh, how I hate Hollywood)
    the Martians move in a pattern of triangulation.  Like surveyors.
    On the contrary, the diagram shown in the 1953 movie clearly showed a triangular pattern of attack:

    "This much is certain. It is vital to prevent the Martian machines from linking up. Once they do, they adopt an extraordinary military tactic. They form a crescent. They anchor it at one end, sweep on, until they've cleared a quadrant. Then they anchor the opposite end, and reverse direction. They slash across country like scythes, wiping out everything that's trying to get away from them."
    - General Mann

    The good General said "crescent", but the diagram clearly displayed a triangle pattern.
    logicman
    I stand1 corrected, Laura.  It's so long since I saw the earlier version.

    Somewhere, in one of my many old books - circa 1930 - is a diagram showing a survey method using a chain pinned at one end and moved in an arc.  The free end is then pinned and the other end moved in an arc.  That triangulation diagram is a close match to one of Lowell's 'canal' maps.

    [1] - To be perfectly honest, I am sitting down. :-)
    Gerhard Adam
    ... a tripod is the most stable arrangement for remaining even on an uneven surface, same goes for three legged stools. 
    That may be true for a stationary object, but it wouldn't be for a ambulatory one.  In effect,  with four legs, as one moves, the other three provide the stability to remain upright until everything can get repositioned.

    For a two legged being (like humans), they are already stable at two legs, so the use of a third support can be used to help maintain balance, but it doesn't really participate in motion as much as it simply provides a stable point to center the weight.
    Mundus vult decipi
    Mark Changizi
    Of course, my hypothesis is much, uh, crasser than any of those you all are suggesting. "Reach out cheaply" is all mine says, in a sense, with no mention of what the "reaching out" is for. The exact number of limbs (for any limb ratio) will depend on the details, although I claim that my simple formula will tend to get it roughly right.
    logicman
    Tripod ambulation could be feasible if implemented as a three-way bipedalism.  Just select a pair of legs, use the third as a balancing tail and amble off in any one of three directions.  A three-legged robot could extend a redundant leg as a tail for balance whilst reaching out in the opposite direction with one of its (three?) arms.  That would be a fun robot to build.  It might even serve some useful purpose, which would of course completely spoil the fun.  :-)
    Mark Changizi
    Hank
    Patrick said "It might even serve some useful purpose" ... 

    Boxing animals is a useful purpose.  And fun to watch!
    I wouldn't want to be around a robot designed like this....especially with those powerful hind quarters coming at my chest!
    Gerhard Adam
    Even though I know you've stated that you consider all the appendages (i.e. head, arms, legs, tail, etc.).  If we consider each separately, couldn't we also be pretty accurate by arguing that each represents either zero(0), one(1) or a number of pairs described by the Fibonacci sequence?

    So we could argue that we would expect to see 1 pair, 2 pair, or 3 pairs of legs (or some number of pairs), but never an odd pair of legs (with the stipulation that the only odd value would be one).
    Mundus vult decipi
    I think the whole point of the three-legged robot design is to better understand the locomotion of bipedal walking, with perhaps the long-term goal of building a bipedal robot that can walk more like a human than present day prototypes. It may even be possible one day from what is learned from this research to design a bipedal robot that can not only walk like a human, but actually run. I could be wrong, but that's the only practical real-world application I can see coming from this type of robotic research.
    Mark Changizi
    The entire goal is to build robots fast enough that we can't catch 'em.  :)
    LOL Mark! ;-)
    Mark Changizi
    Oh, and here's some limb cuteness "data" from graphjam...
    Love the graph, Mark! lol : )
    logicman
    Number of legs vs. cuteness.

    Interesting.

    Gaaaah! Kill it!!  Kill It !!!

    Scientists perform experiment to determine most effective way to incapacitate rogue robot.
    Water pistol, or nail gun?
    LauraHult


    Ah, the joy of four limbs!  Yes, but is s/he huggable, squishy "cute"?