A hexapod muscle study

Some time ago I wrote about the Great Hexapod Revolution, which I might also have called the Quite Considerable Hexapod Revision. At any rate, that particular revolution or revision has conceptually been completed for quite some time, so I am busy revising old paintings as well as coming up with new ones. The latter are not really necessary as The Book is basically done.

Yes, The Book is done.

Basically.

The 'done' part means that there are easily enough double pages to fill a book, so rather than producing more I am shifting emphasis towards finding a publisher. I do not expect immediate success, which explains 'basically': I might meanwhile just as well keep on thinking about Furahan creatures.

Let's review the hexapod revision. Their skeleton still reflects that of early representatives of the clade. There is no vertebral column, here defined as many short similar bones placed end to end running from front to back in the vertical plane dividing left and right halves of the animal (technically, the sagittal plane). The most bare bones version of that skeleton (sorry for that one) would remind you of a foldable ladder. As the clade started with animals without legs, it would be strange to use the name Hexapods ('six-leggers') for all of them. I chose the new name 'Scalata' instead, based on the Latin word 'scala': ladder. The word 'Scalata' is technically correct while 'scalates' is suitable for colloquial use. Hexapods then become a subgroup, consisting of scalates with legs. 

Click to enlarge; copyright Gert van Dijk

The next big step was deciding the shape of the legs. I built on the zigzag principle, in which successive major leg segments bend one way at the topmost joint, the other way at the second joint, and so on. By reversing direction, no joint is ever really far away from a line perpendicular from the hip down. Being close to that line reduces the force needed to keep the joints in those positions, meaning muscle power. have a look at these posts here and here. The image above was taken from these earlier posts and explains that principle.

The least force to keep the segments in place is needed when all segments are stacked vertically, making the leg into a column. That is a fine way to conserve energy but does not produce athletic animals. Vertical leg bones are typically found in large non-athletic animals: think of elephants and sauropods. In smaller animals all segments can be closer to the horizontal than the vertical position, because fighting gravity costs relatively much less (for scaling effects, see here and here). The actual position of the leg bones will depend on mass and athleticism.
             
I suggested in earlier posts (here and here) that it wouldn’t really make a difference whether the legs started by being angled forwards ('zig') or backwards ('zag') at the hip joint. Mammals are peculiar in having their front legs start with a zag and hind legs with a zig. This is a consequence of how they re-engineered their original sprawling posture: front legs rotated backwards, with elbows pointing back, and hind legs forwards, with knees pointing forwards.

Should that reversal be seen as a natural 'law' or as an evolutionary coincidence that became locked in place? I could not think of any physical reason for this pattern and so had freedom to decide what to do with scalate legs. All three pairs of legs underwent the same rotation, which is simple and keeps them out of each other's way. The top segments all point backwards.   

 

Click to enlarge; copyright Gert van Dijk

Another decision was how to join the legs to the scala. Should the hip joint allow movements in all directions, or should they restrict movement in one or more directions? Should the joint be so 'open' that all positions need to be controlled by expensive muscle activity, or do we let bones and ligaments take up some of the stresses? I decided to give the joint surface a 'roof' in the hip to push against, transferring weight. The image above shows three possible patterns: in A, the bone sits directly underneath the spherical joint, allowing three-axial rotations and simple weight-bearing. In B, the joint does the same, but the shaft of the bone is shifted a bit to the side, allowing room for gut, eggs, or whatever. In C, the joint restricts rotations around the axis running down the bone and the bone extends a bit past the joint. That sturdy upwards spur can be used to attach muscles to; that's the hexapod hip joint.  

The main propulsion force involves swinging the upper leg segments thighs backwards: retroflexion. The thigh has a limited range of motion, from an angled pointing just a bit forwards to a much larger backwards angle. To work over that range, hexapods have one large muscle starting behind the joint and attaching to the hip bone below the joint, exactly like human buttock muscles. But another big muscle originates in front of the hip and inserts on the spur above the joint. These two muscles act in concert to pull the leg back: they are 'agonists'.

That range of motion has consequences for where muscles can produce the most force. The force exerted by muscle fibres is most effective if these fibres make a right angle with a line from the insertion site to the axis. If that angle is not 90 degrees, only the component of the force that is at a right angle is useful to rotate the bone; the remainder just presses the bone into the joint or pulls it out of it. When the bone rotates, the effective force component changes with the rotation angle. You would want to place a muscle in such a way that most muscle fibres do useful work over most of the movement range. 

 

This animation shows a muscle placed to the front (at left) of the bone, inserting at the spur above the axis of rotation. The bone rotates through its working range, which is shown three times. The four  panels show fibres at different sites of origin. The red lines show the parts of the force that do the actual rotation. It is obvious that fibres starting high above the spur are not much use and can even pull in the wrong direction. The most useful fibres start at the level of the joint or lower, so this is where the muscle should be. I have not shown the other muscle, the one pulling on the thigh below the axis while starting behind the joint (at right). The principle is the same, but now the most useful part lies at the top.      

As I like Latin anatomical nomenclature for its simplicity, I named these muscles ('simple' is here based on the premise that the names are in another language you have to learn anyway). For the front legs, the front muscle is the musculus retractor artus primi anterior, and the hind one is the m. retractor artus primi posterior. For the middle and hind legs, replace primi with secundi or tertii.
 
There are of course muscles that work in the other direction, the 'antagonists', but these are weaker and run the other way, lying underneath the big 'retroflexion' muscles.

Click to enlarge; copyright Gert van Dijk

Click to enlarge; copyright Gert van Dijk

The two images above shows the result of some experimental ZBrush sculpting. Here you see a general hexapod with some main muscles shown. I still find ZBrush extremely non-intuitive, but am very slowly feeling my way around it.

This animal is probably the size of a horse. Note that the middle legs are sturdier than the others. That is because that is where most of the mass is! The middle legs are also wider apart, to allow room for a possibly sizeable gut and also for the front and hind legs. Of course,  the scheme underwent substantial changes in particular with those predators that freed their front limbs from locomotion ('centaurism') and turned them into weapons. maybe I’ll show those anatomical changes too, one day.           

Avatar 2 The Way Of Water, or do skimwings tip the scale?

Many readers will by now have seen the second film in the Avatar series, 'The Way of Water' (TWOW). I felt that the story of the film resembled that of the first film a bit too much, but never mind that; this is not a blog about cinema, but about speculative biology. Luckily, TWOW offers new species to enjoy, to watch again and to think about. I used the book ‘The Art of Avatar TWOW' by Tara Bennett to write this post. There will probably be two posts on TWOW; this one will be about the skimwing, chosen because it has a lot to offer from a biomechanical point of view. It  will be a long post and there are equations at the end, so you have been warned...

That book contains quotes and explanations that confirm a conclusion I had drawn from viewing the first film, and that is that the shapes and form of the life forms on Pandora  are primarily governed by audience appeal, with biological plausibility taking a definite second place. I deplored that second place when I wrote my post about the first film, posted 13 years ago to the day. I still do, but now accept that the people in charge of making films think this is what the audience wants. They may be right; but I, and I guess many readers of this blog, are not typical in this regard. We like our science well-done, not rare. Relegating plausibility to second place is acceptable as long as film makers do not claim that the life forms they present are biologically sound. I got the impression that they did make that claim for the first film, but for TWOW the book acknowledges that the director's opinion of audience appeal came first. 

Click to enlarge; copyright 20th century studios

The goal: can we answer the question whether skimwings can 'taxi' as shown?  

 The skimwing is basically a gigantic flying fish: it has a long slender body and two fins that double as wings. Like flying fish, the skimwing is fast enough to partially leave the water, with just the tail in the water to propel it. Flying fish use this stage to accelerate and leave the water altogether, but skimwings do not do that; they just taxi along.
    Could the skimwing as shown really taxi in the way flying fish do? That is not an easy question because of all the factors that are involved. Let's consider the problems.
 

1. While taxiing, the animal's tail provides thrust propelling the animal forwards, but the tail may also produce some upwards force. If so, the wings need to provide less lift than if all the upwards force is due to lift. In this post, I will assume that the tail only provides forward thrust, so the wings are responsible for all the upwards force. But I will get back to this matter in the end. 

2. As for lift, we will use conventional equations. See this post for an introduction. We are not on Earth, so lift is altered by the higher atmospheric density on Pandora, where the story takes place (1.2 times that of Earth).

3. The animal's weight is also other than it would be on Earth because the gravity constant of Pandora is lower than the one of Earth (it is said to be 0.8 of that of Earth). 

4. A more difficult effect to deal with is the 'ground effect', which means that flying objects (animals and aircraft) experience extra lift if they fly close to the ground. The literature makes it clear that flying fish use the ground effect, and the book states that skimwings do so too. 

 

Click to enlarge; copyright 20th century studios

5. Finally, there is a literally enormous difference between skimwings and flying fish: flying fish are only about 35 cm long and I estimate skimwing length to be 12.2 to 13.5 meter. I derived that estimate using the image above, in which the skimwing appears to be 4.5 times as long as an adult Na'vi. The internet tells me that Na’vi are 2.7 to 3 meters tall. I used a skimwing length of 12.5 m as a reasonable estimate.      

I am not an aeronautical engineer, so what follows should be seen as nothing more than a layperson's attempt to understand how all the above factors might work together.  But first, what do we know about flying fish?     

 

Click to enlarge; from Wikipedia

Flying fish can swim, 'taxi' and glide
 
Flying fish are often said to take to the air to escape predators (Socha 2015). That may well be true but has not been proven; other animals jump out of the water for a variety of reasons. Squid may glide above water to escape predators too but also to catch prey (Socha 2015);  whales apparently breach to impress other whales, while dolphins may do so to achieve greater overall speed. As jumping out of the water must take a lot of energy this seems a strange way to save energy, but moving through air obviously offers less resistance than through water, so that gain may offset the additional energy needed to leave the water. 

There is another odd effect going on here, and that is that swimming just below the surface of the water costs more than swimming in deeper water. Why? Well, all swimmers push water out of the way to make room for themselves, but in shallow water some of that displaced water moves upwards, which costs more energy than if the water only moves sideways. That additional energy cost may just tip the balance, making leaving the water more efficient than pushing all that water up (Socha 2015).
    
Taxiing
Flying fish accelerate under water, break though the water's surface, and then accelerate some more with only their tail underwater, from 10 to 20 meters per second (Socha 2015). That's from 36 to 72 km/h. Calculations show that they need 350 Watt to swim underwater at 10 m/s, but only 36 Watts while taxiing; taxiing is therefore quite efficient (Deng 2019). So far so good!
    Flying fish then leave the water and glide, held aloft by their two or four wings (some species use both pectoral and pelvic wings, others just the pectoral wings). Although some papers describe the flight path as relatively flat (Fish 1989, Socha 2015), meaning at a constant height above the surface, this can only be an approximation. The reason is that flying fish have no propulsion while in the air, so the glide is either completely passive or else lift is helped in some way. In a true passive glide losing height is physically unavoidable (Socha 2015).
     This suggests that their lift is indeed helped. Updraughts can help (Fish 1989) and so can the ground effect. In fact, the relatively flat trajectory itself suggests that the ground effect does help (Socha 2015). Wind tunnel experiments with stuffed flying fish (Really? Yes, really) showed that the ground effect reduced drag (let's say that ‘drag’ is the force impeding forwards motion) by 14%. As lift stayed the same, the so-called lift-to-drag ratio was improved, which is not bad at all (Park 2010). Remember that number of 14%, because we will need it later.
     As flying fish know their physics they do come down to the water and may repeat the procedure: taxi, glide, land, etc. They may cover distances of some 400 meters in 30 seconds (Park 2010), suggesting a mean speed of 48 km/h.       
    
Gliding
The wings proved to be like bird wings, in particular as regards wing loading. 'Wing loading' is calculated as the area of the wings divided by weight of the animal, so it tells you how much kg a square m of wing carries. A low value makes flying easier. The wings of flying fish  are designed for high lift and low drag (Fish 1989). The four-winged species have lower wing loading, pointing to increased lift at low speeds, than the two-winged species.

Scaling flying animals

Now it gets more complicated. But not that much, so hang on! I have discussed scaling winged animals before, but I will repeat the main thoughts here. What we will do is to take Earth’s flying fish, scale them up, export them to Pandora to accommodate the different atmosphere and gravity, calculate their weight there, and then see whether they can produce enough lift to keep that weight aloft in Pandoran air.
   
Lift is proportional to just three relevant parameters (there are two more: angle of attack and a constant, but if we keep these the same throughout we can ignore them).

• density of the air (D), in kg per cubic meter
• area (A) of the wing, in square meters
• square of speed (S^2), in meter per second (I cannot use superscript, which is why I used '^2' to indicate a squared speed)

LIFT = D x A x S^2

If a flying animal flies at a stable height and does not sink, the amount of lift it generates must equal its weight. We can calculate weight W as the product of the gravity constant G and the mass of the animal M.

WEIGHT = G x M

So remember this for stable flight:

LIFT = WEIGHT

Now we need another look at scaling; (see here for flying and here and here for an introduction). Suppose we take an animal with length L and make its length, height and width all twice as large as before (that's called 'isometric scaling'). Its length becomes 2L. However, the area of the wings is the product of length and width of the wings, and as each became twice as long, the area becomes four times (2x2) as large. However, the volume of the animal has three dimensions, so that becomes eight times (2x2x2) as large. The mass corresponds to the volume and also becomes eight times as large. The lesson here is that mass increases more than area, and that is a problem. 

Click to enlarge; copyright Gert van Dijk

Weight becomes eight times as large, but the four times larger wing area will only get you four times the lift. To fly stably, lift must equal weight, so we must find a way to achieve eight times the original amount of lift. In the scheme, above, the first way to do so is labelled 'Enlarged flying fish 1'. In that option, squared velocity was made twice its original value. To do that, velocity itself needs to become 1.4 times larger (1.4 is about the square root of 2). Hence, the larger animal has to fly faster if its wings stay in proportion with the body. But to land and take off, the animal needs to be able to fly at low speeds too: you cannot fly fast all the time.

Is flying faster the only solution? No, we can also choose to have the enlarged animal fly at the same speed as its predecessor ('Enlarged flying fish 2'). We still need to achieve eight times the lift, so we need to make the wing area eight times larger. If we make the length and width of the wing each 2.83 times larger (2.83 is about the square root of 8) we get that. Mind you, this solution will reach a dead end at some point because the enlarged wings will also add weight, which needs to be lifted, etc. I discussed how much weight such larger wings will add to the animal in an earlier post (here); it is dramatic! 
         
To conclude, we have a choice of achieving more lift for the larger animal by either enlarging wing area or flying faster; either solution will reach a limit at some point. On Earth, larger birds combine larger wing area as well as higher speed to achieve that higher lift. That combination can only be an evolutionary compromise between the costs of high speed (difficulty in taking off and landing) and large wings (more weight). There must be a physical limit where flight is no longer feasible for an animal. Does scaling up the skimwing tip the scales? (Sorry for that one, but I could not help myself) 

The case of the skimwing

The length of a skimwing is at 12.5 meter 35.7 times larger than that of a flying fish. If we multiply the dimensions of a flying fish by 37.5, its wing area becomes 1275 times larger and its mass becomes about 45,500 times larger. Wow!

We now need to do an analysis as explained above, but the animal gets to be not twice the size, but 37.5 times. The results of this thought experiment are in the appendix, for those who want numbers. If we choose to scale the animal isometrically, meaning that the proportions of the animal stay the same, then it must taxi not at 20 m/s (72 km/h) as the flying fish manages to do, but at a staggering 351 km/h. That is wholly unrealistic; for one, the Na'vi sitting on top would be blown off…

How about the other approach, meaning making the wings larger? Well, isometric scaling made the wing length and width each 35.7 times larger. It turns out that we need to make each 174 times larger instead! I did not bother calculating how much weight that would add.

Save the skimwing!

But the animal might derive part of the upwards force from beating its tail. True, but there is a good reason why tail walking dolphins are not large. It is, once again, scaling: the force needed to push against the water depends on the cross section of the muscles, meaning area, and needs to equal the animal's weight. It is the exact same problem as before: force increases with the square of length and weight with the third power. Turning to tail walking instead of gliding trades one unsurmountable scaling effect for another.

But how about the ground effect? Well, in flying fish that reduced drag by 14%. While that is not the same as increasing lift, it may be treated that way. The appendix shows that a gratis 14% increase in lift still do not result in a viable animal, and neither did halving the mass of the skimwing: that won’t fly (sorry for that one too).

Conclusion
 
What a pity. It seems that the designers did not realise how much that third power puts a brake on scaling up animals. Admittedly, those effects are not immediately obvious, but it's not rocket science either. The simple conclusion is that 12.5-meter-long taxiing flying fish are too large to work, even under Pandora's favourable gravity and atmosphere.

Does it matter? The science in TWOW seems fairly typical for how Hollywood treats science, meaning with rather limited respect for plausibility or accuracy. That's not good news; actually, it's not news at all. However, there is good news: the film shows a very profound love for the natural world. If that helps make people care about nature, I'm all for it. 

As far as that love for nature is concerned, consider this: the skimwing differs in only one important aspect from flying fish: its size, and it is exactly that difference that makes the skimwing impossible. But everything else that makes Pandoran skimwings fascinating was already fascinating about flying fish, right here on Earth.     

Acknowledgement

Abbydon made insightful comments on a first draft of this post. He also added remarks about wave effects that would make using skimwings as public transportation rather unreliable.   

 
Selected references
FE Fish. Wing design and scaling of flying fish with regard to flight performance. J Zool Lond 1990; 221: 391-403
Park H, Choi H. Aerodynamic characteristics of flying fish in gliding flight. J Exp Biol 2010; 213: 3269-3279
Deng J, Wang S, Zhang L. Why does a flying fish taxi on sea surface before taking off? A hydrodynamic interpretation. https://doi.org/10.1101/765560
Socha JJ, Jafari F, Munk Y, Vyrnes G. How animals glide: form trajectory to morphology. Can J Zool 93: 901–924 (2015) dx.doi.org/10.1139/cjz-2014-0013  
Sullivan TN, Meyers MA, Arzt E. Scaling of bird wings and feathers for efficient flight. Sci. Adv. 2019;5: eaat4269 

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Appendix

 

Abbreviations and parameters

            FF                    Flying Fish

            SW                   SkimWing

            Length             L

            Area (wings)     A

            Volume             V

            Mass                M

            Weight             W

            Speed               S

            Density of air    D

            (local values for Pandoran SW and Earth FF)

                                    D_SW = 1.2 ∙ D_FF

            Gravity constant G (local values for SW and FF)

                                    G_SW = 0.8 ∙ G_FF

            Lift                   LIFT

 

----------------------------------------------------------------------

 

Take the length of the animal of the FF as base value of 1; the SW is 35.7 times longer.

            L_FF = 1

            L_SW = 35.7 ∙ L_FF

This results in the following for area and volume for SW:

            A_SW = (35.7)² ∙ A_FF         or         A_SW = 1275 ∙ A_FF

            V_SW = (35.7)³ ∙ V_FF        or         V_SW = 45500 ∙ V_FF

For weight we get:

            W_SW = V_SW ∙ G_SW                   Pandoran values

            W_FF = V_FF ∙ G_FF                        Earth values

For lift we get:

            LIFT_FF = D_FF ∙ S_FF² ∙ A_FF

            LIFT_SW = D_SW ∙ S_SW² ∙ A_SW

We can now start filling in values for Pandora using Earth parameters, not altering wing area A and speed S on Pandora yet:

            for lift:              LIFT_SW = (1.2 ∙ D_FF) ∙ S_SW² ∙ A_SW                     

            for weight         W_SW = V_SW ∙ G_SW

                                    W_SW = V_SW ∙ (0.8 ∙ G_FF)

                                    W_SW = (45500 ∙ V_FF )∙ (0.8 ∙ G_FF)

                                    W_SW = 36400 ∙ V_FF ∙ G_FF

                                    W_SW = 36400 ∙W_FF

If the flight is stable, weight must equal lift, so it follows that:

            LIFT_SW                                     =          36400 ∙ LIFT_FF

            (1.2 ∙ D_FF) ∙ S_SW² ∙ A_SW     =                 36400 ∙  D_FF ∙ S_FF² ∙ A_FF

             S_SW² ∙ A_SW                                                        =                 30333 ∙  S_FF² ∙ A_FF

 

We have SW parameters on the left and Earth FF parameters on the right and can play with this relationship.

 

Variant 1 assumes isometric scaling, making  A_SW = 1275 ∙ A_FF

             S_SW² ∙ 1275 ∙ A_FF                  =                 30333 ∙  S_FF² ∙ A_FF

             S_SW²                                                 =                 23.8 ∙  S_FF²

We know that the speed of FF = 20 m/s, so

            S_SW²                                                   =                 23.8 ∙  400  = 9516

            S_SW                               =          97.6 m/s or 351 km/h           

This is obviously a ridiculous speed and cannot work. Perhaps larger wings may do the trick if we abandon isometric scaling.              

 

Variant 2 starts a few steps back.

            S_SW² ∙ A_FF                                      =                 30333 ∙  S_FF² ∙ A_FF

If we assume that the skimwing has the same speed as Flying fish, then  S_SW² = S_FF²

            A_FF                                                      =                 30333 ∙ A_FF

This means that the width and lengtn of the wing have to increase by the square root of 30333, meaning they become 174 times larger instead of the original 35.7 times. This is ridiculous, and the real situation would be worse because the arger windg wopuld weigh a lot more, which he haven’t accounted for yet!

 

Are there ways out?

 

The ground effect seems to reduce drag by 14%. That is not the same as increasing lift by 14% but assume that lift is indeed increased by this amount. 

            S_SW² ∙ 1275 ∙ A_FF                    =                 0.86  ∙ 30333 ∙  S_FF² ∙ A_FF

            S_SW²                                                   =                 20.5 ∙  S_FF²

            S_SW²                                                   =                 20.5 ∙  400  = 8200

            S_SW                               =          90.6 m/s or 326 km/h           

or

            A_FF                                                      =                 26086 ∙ A_FF

The wing's length and width have to become 161 times larger. Ground effect does not save the skimwing, and that holds for an isometric as well as for a non-isometruc approach.  

 

What if we make the skimwing relatively more slender than the FF, by halving the volume of its body? This results in:

            S_SW² ∙ A_SW                                    =                 15167 ∙  S_FF² ∙ A_FF

            S_SW² ∙ 1275 ∙ A_FF                    =                 15167∙  S_FF² ∙ A_FF

             S_SW²                                                 =                 11.9 ∙  S_FF²

We know that the speed of FF = 20 m/s

            S_SW²                                                   =                 11.9 ∙  400  = 4758

            S_SW                               =          69.0 m/s or 248.3 km/h           

If we assume that the skimwing has the same speed as Flying fish, then  S_SW² = S_FF²

            A_FF                                                      =                 15167 ∙ A_FF

This means that the length and width of the wing must become 123 times larger instead of the original 35.7 times. This is still preposterous.