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Hollywood Nonsense: Is it possible to overcome the square-cube law?

  1. Jul 24, 2013 #1
    Hollywood Nonsense: Is it possible to "overcome" the square-cube law?

    I'll admit up front: my knowledge of physics is sub-fundamental. If any questions I ask seem braindead ignorant, know that it comes from a genuine lack of understanding. Apologies in advance.

    For those who don't frequent the movie theaters, Pacific Rim is a movie that was just released. It features giant humanoid machines squaring off against giant, mostly-humanoid creatures. If you can get past the utter absurdity of such things, it is immensely entertaining and I highly recommend it.

    One aspect of physics that I am vaguely familiar with is the square-cube law, which in this situation would render such giant creatures impossible: their immense weight would crush their bones. The internal structure of any vertebrate can't function at that kind of macrolevel. The machines, on the other hand, made me curious.

    Let's say that, through some bizarre sequence of events, it became absolutely imperative that humanity create a giant* humanoid machine capable of human-like movements. The square-cube law would still apply, obviously, but is there any combination of materials or any structural design that could effectively overcome such a limitation? For the purpose of this question, there is no limit to available resources. What you need, you have. Would such an endeavor be like Sisyphus pushing his boulder up the hill, or is it at least remotely possible?

    *"Giant" in this case being roughly 80 meters tall and weighing about 2,000 short tons.
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  3. Jul 25, 2013 #2


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    The fundamental mathematics that shows the surface area of a sphere rises as the square of the radius and that the volume and mass, rise as to the cube of the radius will always be true.

    A change of materials technology might move the limit to size but the fundamental square and cube relationship will hold as much for the new material as they did for the old.
  4. Jul 25, 2013 #3
    I understand that the law itself can't be nullified. By "overcome the square-cube law", I mean "designed in such a fashion that the machine could function as intended in spite of the stresses provided by the square-cube law". Would it be even remotely possible to do such a thing, or are the extreme stresses just too big of an obstacle?
  5. Jul 25, 2013 #4
    For now I would suggest using graphene as the primary building material. In principle, graphene could be cut into any manner of shapes and rearranged into various structures that would be very strong. I'm currently picturing soap-bubble shaped arrangements of graphene, but I imagine that the strength of the material would weaken with higher curvature (flat graphene is a hexagonal lattice, so adding curvature might require certain places where the atoms form pentagons or heptagons). Since graphene has some pretty weird electronic properties, it might be possible to implement some of the circuitry in the skeleton, further reducing the weight of the robot.

    The weight of the robot's skeleton depends somewhat on the arrangement of graphene used, but we might get a reasonable estimate assuming that the skeleton is made up of round graphene spheres of a certain radius ρ, which characterizes the "density" of the bones (like a bird's bones, the graphene bones will be hollow to reduce weight). It will also depend a lot on the anatomy of the robot itself, but for an upper bound we can assume that the robot is spherical (Harte 1985). One square meter of graphene weighs less than 1 mg, meaning that a 100 m tall skeleton will probably weigh less than about 3/ρ kilograms (here ρ is measured in meters).

    The choice of ρ depends on what sorts of applications you want for the robot. Note that when optimizing the efficiency of the robot, the bone density will change depending on the location. Clearly it's also important to think about the anatomy of the robot. For simplicity, it's probably best to assume the robot has a four-fold symmetry among its limbs, with a small cockpit (or, more realistically, brain) in the center, from which the limbs protrude. The mass would be distributed equally among the limbs, which should be able to support the robot, allow the robot to jump and climb (or in a fight, punch and grapple). Obviously the robot must also be strong enough to carry the machinery needed to do all of this. I don't have time to continue for the time being, but there are a few questions that may be worth investigating:

    What is a lower bound for the "wiggle room" of each limb, and at the joints? Does it matter how far out along the limb you are?

    How would you propel the robot (electric motor vs. combustible fuel vs. nuclear/hypothetical fusion energy), and how much would the machinery weigh?
  6. Jul 25, 2013 #5


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    Your question is like the application of an irresistable force to an immovable mountain.
    If you have super materials that can handle super stresses you can only build super things. Never infinite.
    The square and cube laws apply equally well to the relative size of things you build with the new materials.
  7. Jul 25, 2013 #6
    I think the question is not whether it is possible to overcome the "square-cube law", but whether the technology depicted in "Pacific Rim" can in principle be produced with known materials.
  8. Jul 25, 2013 #7
    I wouldn't go that far. The robots in Pacific Rim are able to engage in hand-to-hand combat similar to an unencumbered human, and the neural interface they use is either pure fantasy or centuries ahead of our current technology. I'm thinking of something significantly less ambitious: something with more traditional controls and that can perform basic movements like walking, leaning, etc. A proof of concept, if you will.

    The movie makes no effort whatsoever to be scientifically accurate. This is just curiosity on my part.
    Last edited: Jul 25, 2013
  9. Jul 25, 2013 #8
    An elephant and dog drawn to the same size via differing scaling show that the elephant legs are thicker in proportion to the height. Giant robots are feasible but scaling would be required so that limbs have sufficient cross sectional area to support the huge amount of weight. Based on what I know about anatomy, that is how I would approach it.

  10. Jul 25, 2013 #9

    Andrew Mason

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    If scientific accuracy is not a problem then the square-cube law is not a problem. These machines just need to incorporate anti-gravity technology.

  11. Jul 25, 2013 #10
  12. Jul 25, 2013 #11


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    That is certainly capable of hunting down and eating things...
    See slide 14/28

    You could double it's linear dimensions but;
    it would cost 2^2 = 4 times as much to paint and
    it would weigh 2^3 = 8 times as more and
    it would cost 2^3 = 8 times as much to build.
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