The square-cube law holds true only for objects that are similar. In evolution, you can't make big leaps forward, but since most fantasy worlds are created by gods or people who think they're gods, I'm free to abuse Intelligent design.

Dragons have six limbs, the 2 wings are situated near the front legs, but just far enough not to interfere. Their (the wings') anatomy as of now is pretty much the same as avian wings and flight muscles.

That being said, assuming dragon bones are much stronger, thanks to some nanoscale engineering and a hint of graphene, how could the wings' pectoralis major, and the bone connecting to it, be rearranged to produce more power for the same mass?

Note: Before we veer off into the deepest insanity, I was thinking more of the "If strength is the function of muscle cross-section, can't we just shorten the fibers and increase the cross-sectional area, like a boss?" path.

  • $\begingroup$ Related to this question: worldbuilding.stackexchange.com/questions/143396/… $\endgroup$ – Liam Morris Apr 13 '19 at 18:34
  • $\begingroup$ Sure and then the wing can move a grand total of half an inch, cross section determines power, length determines how far it can move. $\endgroup$ – John Apr 14 '19 at 18:07
  • $\begingroup$ @John Are you trying to say a compromise isn't possible? $\endgroup$ – Mephistopheles Apr 15 '19 at 5:41
  • $\begingroup$ I am saying getting more power in return for shorter movement does not actually help you fly. the total weight of the muscle needed to get the same amount of lift does not change. $\endgroup$ – John Apr 15 '19 at 12:35
  • $\begingroup$ This is like asking is their any nozzle I can put on a bottle rocket that will let it lift a freight car to the moon, the available energy is just insufficient. $\endgroup$ – John Apr 15 '19 at 12:46

When I was a teen it was common for us to have a 50 cc scooter which, by law, could not exceed the speed of 50 km/h. When some of us wanted to tune up the scooter and get more out of the engine, one of the trick was to change the carburetor or the exhaust (or both). (don't try this at home, going at 110 km/h on normal roads with something designed to go at 50 is not only illegal, but also mighty stupid and a fast way to have an early funeral)

This trick would have allowed the engine to output more power with the same volume of the cylinder and the same structure.

How does this apply to your dragons? Well, you don't need to redesign the muscles/engine, just increase the metabolism of the beast, allowing it to burn more nutrients and output more energy with the same structure.

Incidentally, this is the same trick used by birds, which allow them to be able to fly.

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    $\begingroup$ 1. Waste heat, though that's someone else's pet peeve. 2. What you're saying isn't wrong, but I'm looking for some way to restructure the muscle, ya' know "strength" is the function of the cross-section. $\endgroup$ – Mephistopheles Apr 13 '19 at 15:36
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    $\begingroup$ Wow, this takes me back. I used to play with contact explosives and ballistic model rocketry using, um, well... black powder. Friends of mine solved their slow go-kart problems with a syringe full of alcohol and an open carburetor. It's a never-ending wonder that teenagers ever become adults. $\endgroup$ – JBH Apr 13 '19 at 16:05
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    $\begingroup$ @Mephistopheles Not just waste heat but also metabolites. It may be necessary to have an auxiliary circulatory system, and also auxiliary kidneys, to carry them away. $\endgroup$ – Spencer Apr 13 '19 at 17:38
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    $\begingroup$ This seems to be the reason pterosaurs could be so much larger than modern birds—the oxygen content peaked in Cretaceous period at about 1½ of current value, allowing faster metabolism. $\endgroup$ – Jan Hudec Apr 13 '19 at 21:12
  • $\begingroup$ @JanHudec: That explains a lot of things. Thanks. $\endgroup$ – Joshua Apr 14 '19 at 0:04

Your dragons are currently big pigeons: giant pectoralis for downstroke, presumably proportionately small supercoracoideus for upstroke. Background reading

But you could take advantage of graphene and the rule of cool and model your dragons on a unique bird: the hummingbird.


Most birds that hover have high aspect ratio wings that are suited to low speed flying. Hummingbirds are a unique exception – the most accomplished hoverers of all birds. Hummingbird flight is different from other bird flight in that the wing is extended throughout the whole stroke, which is a symmetrical figure of eight, with the wing producing lift on both the up- and down-stroke. Hummingbirds beat their wings at some 43 times per second, while others may be as high as 80 times per second.

Hummingbirds fly like insects. Their wings move with a sort of sculling motion, and the pectoralis and supercoracoideus are closer to symmetrical in their contributions. They generate vortices as part of their lift mechanism, which would be so cool for a dragon because it would generate dust devils close to the ground.

One could argue dragons are too big and heavy to fly like hummingbirds. I refer these naysayers above to "graphene and rule of cool" and suggest they devote their skeptical energies to the problems inherent in breathing fire.

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    $\begingroup$ Making them able to breathe fire is child's play. Though I'd prefer a long-range, precision strike breath-weapon with splash damage. $\endgroup$ – Mephistopheles Apr 13 '19 at 17:37
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    $\begingroup$ @Mephistopheles - dragon lugeys. $\endgroup$ – Willk Apr 13 '19 at 17:49
  • $\begingroup$ U wot? speak english! $\endgroup$ – Mephistopheles Apr 13 '19 at 17:51
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    $\begingroup$ The question is about circumventing square-cube law, and you suggest violating even worse scaling law instead? $\endgroup$ – Jan Hudec Apr 13 '19 at 20:34

how could the pectoralis major, and the bone connecting to it, be rearranged to produce more power for the same mass?

The pectoralis major connects to the arm bone, so I am going to offer a solution that is biologically plausible:

Due to some mutation, some dragons are born with thw front legs and the wings partially fused. This adds a lot of muscle power to each wing stroke.

Over millenia (or maybe longer spans), the dragons evolve to have only four limbs. The wings get ever more muscular, achieving your desired result.

  • $\begingroup$ I can abuse Intelligent design as much as I want, I clearly stated that. $\endgroup$ – Mephistopheles Apr 13 '19 at 20:50
  • $\begingroup$ Of course in birds the fore legs got changed to wings, and so was the case with pterosaurs. $\endgroup$ – Jan Hudec Apr 13 '19 at 21:04
  • $\begingroup$ Sorry, I guess I should've said the other pectoralis major. There's one for the foreleg and one for the wings. $\endgroup$ – Mephistopheles Apr 13 '19 at 22:01

You could look at the pterosaur called Quetzalcoatlus. It is thought to be the largest flying animal ever, with a wingspan estimated to be 10-11 m. Modern estimates put its weight at 200-250 kg. Earlier estimates put their wingspan at as much as 21 m, so we must assume that this much is possible for such a beast - and a dragon with a 20 m wingspan and a weight of (say) 400 kg would be quite impressive.

There are differing theories as to how well they flew, but a recent computer model suggests that Quetzalcoatlus was capable of flight up to 80 mph for 7 to 10 days at altitudes of 15,000 ft.

Quetzalcoatlus was built very differently from a dragon, with wings far to the back (see figure), but the important thing is that a flying of that size has existed, and hence, a dragon of similar size should be possible without bending physics. If your dragon is firebreathing, it could possibly blow fire beneath itself to create brief thermals for quick lift-me-ups.

enter image description here


No, the square cube law applies to many aspects of flight, the wings have to move enough air to counteract the weight of the animals.

As for changing the muscles it will not help, making a muscle shorter also makes the distance it can travel, that is the length change during contraction, smaller, your muscles will be super powerful but move all of half an inch. Which means your dragon can't really flap its wings. trading length for thickness doesn't gain you any flight capabilities. the total lift generated does not improve by making the muscles shorter and wider.

Making the bones stronger helps by reducing the overall weight of animal and reducing the weight of wings specifically, a lighter wing requires less force to move the same amount of air (because the muscles have to move the wing as well as the air). Because of this the gains you can get are pretty small, not insignificant but not all that great either. The wings still have to move enough air to counteract the weight of the animal, that is the real killer about the square cube law, the mass of the animal and the amount of air you need to displace to counteract it.

The power to mass relationship of muscles is a pretty hard limit, birds have done everything they can while keeping the muscles functional. You can have system to reclaim so of that energy (kangaroo tails) but the initial energy still needs to come from the muscles. As long as dragons are carrying around superfluous limbs they will be hard pressed to even reach the same size as birds.

  • $\begingroup$ Okay, time for plan B. $\endgroup$ – Mephistopheles Apr 14 '19 at 18:32
  • $\begingroup$ But before I do that, what about levers? $\endgroup$ – Mephistopheles Apr 14 '19 at 18:39
  • $\begingroup$ muscles already use leverage, The nature of muscles however means leverage can't gain you very much, leverage is a trade off between distance and power, basically the same trade off you can do with muscles. The total amount of energy and thus muscle is the same. $\endgroup$ – John Apr 14 '19 at 22:46

The main problems with a big dragon are, by increasing order of importance:

  • Muscle power density

There can be linear motors (in a broad sense, what muscles are) that are powerful enough per mass or volume. For an unusual evolution that solved the latter problems, this would be straightforward enough. As we'll see later, this is not even the limiting factor in human strength under normal circumstances.

  • Muscle strength

The muscle has to not tear itself and its supports apart (something human muscles will do if you use them at max power). Look up tetanus symptoms - or not, muscles at full power are horrific stuff. But there are materials stronger than human tissues that exist, and that could reinforce dragon tissues. Carbon nanotubes and graphene are rather popular choices, nowadays. Aggregated diamond nanorods can make for pretty nice bones as well.

  • Fuel intake

Muscles need oxygen and sugar (or whatever this organism may use). Those are carried by blood. Sugar level could be increases quite a lot with the right biochemistery, as long as it can metabolize its reserves (probably fat) fast enough to keep up. Oxygen is a bit trickier. Blood could carry more, but oxygen intake is limited by the lungs. They would need not only big ones, but probably something more efficient like a biological supercharger. Waste products like carbon dioxyde need to be removed, but solutions to intake should double as exhaust here.

  • Heat management

Probably the most critical, and often considered the biggest limiter in muscle strength for large organism, due to the square-cube law: muscle volume (and theoretically, strength, but also waste heat generation) goes up with the volume, while muscle surface (and its ability to evacuate waste heat) goes up with the surface. So the dragon will need an extremely efficient circulatory system to get rid of heat - you won't get much better than water, so you will have to pump more blood faster and through more vessels. This is both for heat and fuel intake, in fact, as fuel intake is also limited by surface.

Then, this heat will have to go somewhere. The aforementioned supercharger would help, but also the gigantic wings that a dragon needs to fly anyway, and that will make for nice radiators - wings must also grow larger due to the square-cube law. you can even have fun and have the supercharger pointed down, with its scalding hot exhaust a bit similar to a breadth weapon.

Increased perspiration through, especially through the supercharger, may be necessary. There could also be small cold nodules in the muscles to help get rid of heat during a brief, intense effort, though designing such biological system to be small and light enough may be a challenge.

There will have to be a balance between impractically large wings (to better glide) and impractically powerful muscles (for powered flight), depending on how efficient those systems are and how strong dragon tissue is. Anyway, this may either require artificial design or a very unusual evolution, but those may help grow dragons just a bit larger. I suspect the dragon would only be capable of punctual intense effort to take flight, and then mostly glide, like the extinct Quetzalcoatlus mentioned in the answer of Klaus Æ. Mogensen.


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