From freezing the anatomy of tetropters to op art (Tetropters VIII)

Tetropters have been discussed in this blog several times. Eight posts were devoted to them (one, two, three, three bis that doesn't really count, four, five, six, and seven) and they were mentioned more often. In fact, they first featured in the third post ever, published on April 27, 2008. Attentive readers may note that the 10 year anniversary of this blog is coming up, and I intend to write more posts this year to honour the occasion.

Tetropters are flying animals with a radial symmetry, something that was not common at their time of invention, well before they featured in the blog. They fly with a 'clap and fling' mechanism, also used by various flying animals on Earth: the animal brings its wings together over its back and then separates them again, starting at the top. This apparently creates a lower pressure above the animal, which helps the animal to stay in the air. Earth animals, with their two wings, have one 'clap' in each movement cycle of the wings; Wikipedia has a short section on it. Tetropters have four wings and move them in such a way that there are two 'clap' events in every wing cycle. I was delighted to learn, years after their 'evolution' as Furahan animals, that someone had had the same idea but with the purpose of building an actual flying robot using the double flap and wing scheme. I wrote about that in this post.

At present I am working on the second of what will probably be three two-page spreads on tetropters. The first detailed paintings of a specific animal (or plant or mixomorph) always represents a bit of a crisis, as the characteristic features of a group, its Bauplan, have to settled for good: it has to be frozen. Tetropters had  four wings and I knew their movement pattern, but that left many other decisions to be made. How many eyes should they have and where are these eyes placed? They are presumably related to spidrids, so which features should they share? Should they have eight legs or four? If the mouth is placed at the underside of the animal, how does that reach its food? The list goes on. I have frozen the Bauplan of spidrids, cloakfish and Fishes I to VI in the past, so the process is familiar by now. I confess that I have kept one of the most difficult decisions for last, and that is the suspension system and leg anatomy of large hexapods: I wish to avoid a mere doubling of hind legs or of front legs, which is how most illustrators solved the problem of designing animals with six or eight legs (see my posts on Avatar and thoats).

The first decision regarding tetropters was that it dawned on me that I had not considered the etymology of the word well. The word is derived from the Greek roots for four, 'tetra', and wing, which is either 'pterux' or 'pteron'. In biology just the stem 'pter' is used often. So where did the 'o' come from? I guess I just used 'o' to string the two roots together, or perhaps because of an association with 'helicopter' (a combination of 'helikos', meaning winding, rolling, turning, and 'pter'). As an aside, the Lexilogos websites for Latin and Classical Greek are useful for such things). But as 'tetra' already ends in a vowel, no other sound is necesary to connect the two words, so 'tetropters' are now 'tetrapters'. In the posts I will stick to tetropters or other posts will be difficult to find.

Click to enlarge; copyright Gert van Dijk

Readers will be more interested in what the animals look like. Well, here is a drawing from the famous 'Field Guide to Imparian Tetrapters', showing the male and female forms of the 'Red Baron'. These animals are large, for tetropters that is, predatory tetropters that prey on other tetropters, catching them in flight. They have long wings and are very manoeuvrable. You may note that the outer legs have evolved into grasping limbs, leaving just the inner legs as a landing gear and to walk around on.

To help me get a good idea of tetropter wings in flight I dusted off earlier tetropter animation programs, relying on an unwieldy combination of Matlab, python and Vue Infinite. When I first made these programs I dreamed of producing 5 or 6 minutes high quality films; the one above was made with this idea in mind. Later I realised that these required considerable investments in time, time that might be better used working on The Book directly. So I gave up on nice backgrounds with leaves moving in the wind, etc., and just use animations as a scaffolding for the paintings.

This first animation shows a general undetailed tetropter in 'helicopter mode': the wings are relatively long, and when they move through their 90 degree movement from one clap to the next, they do not move down very much. The 'angle of attack', that is the angle of the plane of the wing compared to the direction of movement, is low. One extreme angle of attack would be a flat plane moving at a right angle to the wind, creating maximum drag but no lift. The other extreme has the plane moving exactly parallel to the wind: no drag, but again no lift. The optimum angle of attack should be one that for a given air speed creates the most lift for the least drag. This is also the flight mode for the Red Baron.

I have played with the structure of the wing, which is transparent with some bright red spots. The structure is much like that of insects, with a thin membrane, taut between 'spars' that give it its shape. There are two main spars to help control the curvature of the wings during flight. I tried to envisage completely unearthly spar structures, but all my attempts ended up looking like insects; let me know if you find a workable unearthly design. Note that the speed of movement shown here is not at all the natural one: for earth insects, wing frequency varies between 4 and 250 Hz, with low frequencies for large butterflies (I might write a short post on tetropter wing beat frequency taking air density and gravity into consideration).

This second example shows a 'rowing' mode of flight. Here, the wings beat down over a large angle and the plane of the wings is at a large angle to the direction of movement, somewhat in the way the blade of an oar is at a right angle to the direction of the stroke.  I should probably have made the body a lot smaller in relation to the wings, so the animal can beat its wings like Earth moths or butterflies: slowly, so the colour pattern can be appreciated. Just think of the animations as showing the animal in extreme slow motion.

Still, I could not resist adapting the animation to show a wing movement at 4 Hz, which is really low for Earth insects.  The colours stand out less.

The third and last movement concerns a mixture of the two flight patterns shown above: not too flat nor too steep, but just right. Again, this should be probably a large animal with a smaller body. There is some reasoning behind the bold colours.

I assumed that animal vision in relevant Furahan animals deals separately with colour contrast and with luminance contrast, just as the human visual system does. Generally, if you wish to see detail, use a large contrast between light and dark (i.e., a big luminance difference). You might think that colour differences are more important, but they are not. Designers know such things; here is a nice NASA image that explains the use of both types of contrast from a design point of view.

Click to enlarge; copyright Gert van Dijk

Colours can do strange things: the visual resolution differs between colours, with blue as a particularly poor colour to use for spatial information. Here is a trick to show that: the original is at the top left. I used that to blur two of the three colours red, green and blue, leaving one colour in its original sharp form. You will see that the clarity of the image really suffers f the green channel is blurred, but that the image does not suffer that much if green is unaffected. What this simple experiment shows is that blue and red, mostly blue, have a poor spatial resolution. Interesting things happen if you put two contrasting colours side by side, and tweak their luminance until they appear equally light or dark. This contrast has a poor spatial resolution, making shapes seem to float or flicker. This is just one of the properties of the visual system that op art relied on. Just type in op art in Google and do an image search.

I used such a design for the wings in this tetropter. Each wing has a bold pattern of two colours. Two wings have their colours placed opposite to the other two. The idea was that the wings in a near-clap position would provide a visual shock. Theoretically the to and fro sides of each wings could have contrasting patterns as well, to provide even more dazzle.

Here it is again, manipulated to result in a 4 Hz cycle. Does it work? Such visual effects rely on the properties of the visual system, and those will differ greatly between different animals. One species op art is another species' drabness. I have often wondered whether the colour patterns of some  Earth animals evolved to create a specific visual effect in a specific visual system. For instance, what effect do the stripes of a zebra have on the visual system of a lion, perhaps at night?

I will equip a Furahan farfalloid tetropter with a similar pattern, in the expectation that its colours will ignite at least one visual system, probably those of potential mates, to make it something like 'Wow!'.

Flying animals or true 'weight lifting'

In response to a question on the Speculative Evolution website I thought it might be useful to write a short post on animal flight, with an eye on other worlds besides Earth. It will turn out that the logic is very similar to that of leg design. That subject, focussing on bone thickness, was discussed in two earlier posts, here and here. There is some math involved, but nothing more complicated than understanding powers and roots.

This post will only deal with the most basic aspect of flight, which is staying aloft. Let's start by considering an animal that is in stable flight. This means that is neither losing nor gaining height, with some kind of propulsion we will ignore (the same reasoning will also hold when the animal stably glides downwards, so lift is a bit less than weight). When the animal says aloft without sinking two forces must be equal in size: gravity pulls the animal down and lift pulls it up. We need to take a closer look at each. First, weight; it is the force induced on a body by gravity:

   weight = gravity constant (g) * body mass (m),

Click to enlarge; copyright Gert van Dijk

The image above shows several views of a general avian of the genus 'Avidisneius'. We will deal with the size of the animal's body; the wing will come later. For that we need to understand that its mass equals the product of its density and its volume. We will not alter density at all but will play with the volume. Volume is determined by its length, height and width. If this sounds as if the animal is shaped like a rectangular brick, that is essentially correct. But a more complex shape than a brick does not alter the principle that its volume depends on the product of its length, width and height; there will just be various constant factors thrown in, that we ignore. All three are measurements of length that we can label as 'L'. Volume thus equals L* L* L, or L to the third power: L^3. Weight was g*m, and we now replace 'm' with L^3:

    weight =  g * density * L^3.

Now we can go on to lift. Textbooks will tell you that it depends on a fairly simply equation:

  lift = rho * area * v^2.

Rho is the density of air.  'Area' is the wing area as seen from above, and 'v^2' is the square of the velocity of the animal with respect to the air. What this tells us is that there are three ways to get more lift. Obviously we cannot change the first, atmospheric density, but the equation tells us that an atmosphere with double the density doubles lift. Halve that density and you get half the lift, which is bad news for animals trying to fly on planets such as Mars. We can and will play with wing area: double the wing area and you obtain double the lift. The real winner here is velocity: doubling velocity gives four times the lift, because the formula contains velocity squared.

Of course, a wing can also be described by length and width and height. We will not use 'L' here but 'W' to specify that we are dealing with the wing. The mass of the wing will be W^3, but its area is proportional to W^2. So the equation for lift becomes:

  lift = rho * W^2 * v^2.

Click to enlarge; copyright Gert van Dijk

The two equations for weight and lift are all we need, for now. Let's take 'Avidisneius' and either double or treble its size as in the image above. To get the new forms, we replace 'L' in the weight equation with '2L' or '3L'. The volume becomes (2L)^3 or (3L)^3, giving us 8*L^3 and 27*L^3. The mass and weight change linearly with volume, so weight will increase by a factor of 8 or 27.

Unfortunately, this 'simple scaling' disturbs the balance between weight and lift. Why? Remember that lift depends on area, and hence on W^2. So if we double or treble W just as we did for L, we get new wing areas of (2W)^2 and (3W)^2, or 4*W^2 and 9*W^2. These wings, even though they are larger, are too small to hold the larger weight up. It's because of that infernal third power for weight versus the square for lift. (As an aside, the usual expression for 'scaling every dimension by the same amount' is 'geometrically similar'.)
              

Click to enlarge; copyright Gert van Dijk

Alas, the wings will have to be made even bigger. The image above shows the results, first for the previous 'simple' scaling and for a 'corrected' scaling attempt. We have seen that doubling body size (L=2) will make weight increase 8 times. What we therefore need is a new wing with 8 times the area. Similarly, if we make the body three times larger (L=3) then the new wing area must be 27 times the original one. We can find out how much we need to change the wing dimension 'W': area was the square of W, so to find the new 'W' we take the square root of 8 and of 27: the numbers are 2.82 and 5.20 respectively. So, if the body dimension (L) is to be 2 times bigger, the wing dimension (W) has to become 2.82 times bigger, and if the body becomes 3 times larger, the wing has to become 5.2 times larger.
 
Is everything solved now? Alas again... The additional increase in wing generates just the right amount of lift to compensate for the larger body. But the wing itself will also become heavier. Remember that volume corresponds to length to the third power? If our new wing dimension W is 2.82 times the original, the new wing mass will be 2.82^3 larger than the original, or 22.43 as much! This is not funny anymore. For the animal that became 3 times larger, the wing dimension W had to become 5.2 instead of 3, meaning the new wing volume is 140 times the original one, even though the body became only 27 times heavier. The 'corrected' scaling definitely falls short...

There is no escape from these cubic effects. Here is another way to look at its devastating effects. Suppose that the mass of the wing was originally about 20% of the mass of the animal. For an original Avidisneius of 500g in total, the wing would have a mass of 100g and all the rest has a mass of 400g. Let's take the animal we made three times larger using the 'corrected' scaling: the new mass of the body will be 27*400g, or 10,800g. The wing mass of 100g becomes 140*100g, or 14,000g. So our original 0.5 kg beastie now weighs 24.8 kg, and a staggering 56% of it is now wing. That's good if you like wing meat, and the animal should be easy to catch: can its heart and lungs even keep up with these massive wings? Mind you, in reality the bones and muscles themselves also need to increase by additional amounts, as their strengths depend on diameters (for examples see the posts mentioned above or my discussion of the giants in Game of Thrones here). 

The lesson is that, if you increase a flying animal's size, the wing dimension must increase more than the body. This actually happens in nature: larger birds have relatively larger wings. But there is no escape from the merciless differences caused by weight depending on cubic effects and lift, while bone and muscle strength depend on cross-sectional areas. At some size, the only weight that the wing can lift is that of the wing itself! But long before that point is reached, the construct will no longer be a viable animal. It is difficult to say where the 'Limit of Flight Plausibility' lies. On Earth now, kori bustards are arguably the largest flying birds. They weigh up to 18 kg, with a 275 cm wingspan. But extinct birds may have weighed 40 kg or even 72 kg. Did pterosaurs really weigh up to 250 kg? There is room for speculation here (which convinces me that Furaha needs some gargantuan beasts in the sky). However, please do not just geometrically enlarge a sparrow and present it as a 250 kg avian: approaching the Limit calls for profound changes in anatomy and flight efficiency.      

Other planets

The lessons for other planets are not very complicated: if you increase gravity by a certain amount, you increase weight by the same amount. If you transplant an Earth-like flying animal to a heavy world, you should make certain that lift increases by the same amount. Enlarging the wings will do that, but, because of the heavier wing, you should trim off a considerable amount of weight wherever you can. You could also make your animal fly faster, but do not think that that solves everything: your animal must be able to fly at low speeds to start and stop. You could evolve the propulsion system in such a way that it provides additional lift.

A high atmospheric density is a luxury, in contrast: if you double density, you can get away with half the wing area, which means that 'W' need only be 0.7 of what it would be on Earth. The soupier the air is, the more you can dream of avians with short stubby wings, resembling flying penguins.
  
But could you have something as large as a 'Game of Thrones' dragon flying around on a planet with an apparently Earth-like gravity and atmosphere? Of course not. Don't be silly. Dragons fly through magic. Birds don't.          
   

Tetropters V: a livelier animation

Regular readers may know that I return to the subject of tetropters from time to time, in a slow and fragmented effort to produce a documentary video showing the little beasties hovering through the air as if they were real, perhaps with an appropriate narrator (as I wrote earlier, David Attenborough would be perfect).

New readers may however respond by saying "What on Earth is a tetropter!?". Part of the answer lies in rephrasing that as "What on Furaha is a tetropter!?" Well, tetropters are small exoskeletal insectoids with a radial base-four Bauplan using a double clap-and-fling wing movement. That is about as short a description as can be given, I think. Those who wish to read more can find the latest instalment ('Tetropters IV') right here, with links to the previous three chapters.

'Tetropters IV' had reached the stage where I could simulate tetropter wing movement, resulting in animations showing a completely immobile body in a completely immobile environment and a fixed camera position. To get there had required a lot of work, but so much more was needed: the animals' bodies should be detailed -and should probably have internal movement as well-; there should be a larger variety of wing shapes; the animal should tilt a bit in the direction of movement, and larger tetropters with slow wing beats should bob up and down in flight, like a butterfly does when flying. And to mimic the effect of a macro lens the scene the depth of field should be narrow, with blurring of nearby and far objects.

 Copyright Gert van Dijk

I used some time in the holiday season to work on the animation, pushing against the limitations of time and capability. The first result of that push stage is shown above, and had the animal moving about freely in three dimensions. To do so I wrote a program in Matlab to define a 3D path in x, y and z-coordinates. The movement is based on the number of frames per second and the numbers of seconds the film should last for. To keep the wings moving there is the number of frames per cycle to consider. I added a little tremor to the vertical component of the movement, so the animal bobs up and down a bit, in phase with its wing beats. All this resulted is a text file with a lot of numbers stipulating where the animal is and at which phase its wings are. The more difficult part was convincing the rendering program 'Vue Infinite' to accept all these numbers and produce a nice image per frame. I had to work on a program in the language 'Python', which I am hardly familiar with, but which can be used to control almost any function in Vue Infinite. I got over that and made the animation above. Not too bad, is it? In an earlier version Evan Black commented that an improved animation might have the effect that the coarser aspects of the design, such as wing attachment, would be less noticeable if the animation would be developed more. I think that that now proves to be true. By the way, the three axes and the balls are there to tell me whether the animal is with regard to local space. I also did not bother to set the wing cycle to match with the movement; in a real scene the wings should beat much more often over the course of such a movement.

Copyright Gert van Dijk

The next stage, shown above, involved 'lens blurring' and body tilting. After various tries and errors Vue Infinite could do lens blurring, but in a very complicated manner: there was a variable that had to entered as a percentage, so I stopped at 100%. The blurring only worked as intended when I set it to 2000%, something I learned after having received help from the Vue Infinite forum at E-on software.
  As for the body tilting, that involved rotations around all three axes. I wrote the program so i could control the rotations by hand, but added an automated feature that differentiated a position path. There should probably be a time delay in that the body should probably start to tilt in a given direction slightly before it starts to move that way, but the lack of such a delay is not noticeable. There could be various way for tetropters to change direction; they could change the aspect ration of specific wings or during specific phases of wing movement, or they could bend their bodies to change their centre of gravity. Regardless, I think the tilt adds a nice touch, rendering the flight a bit like that of a helicopter.

Copyright Gert van Dijk 

The animation above shows where I am now: camera movement. The camera follows the tetropter. As all this is a simulation that could be done perfectly, so every bob up and down would be followed, and the body would stay centred on the image with mathematical perfection. That would look very artificial: a human camera operator would lag behind the movement and would not follow tiny variations. I mimicked that by having the camera follow a smoothed path rather than the actual one, but I do not think the smoothing is good enough yet; it probably needs a delay function as well.

Oh well, there are enough things left for the next stage, such as adding a suitable body. I will probably sculpt one in Sculptrix or build one in Vue itself. The latter option will result in an artificial technical look but has the advantage of colouring the animal with ease. The Sculptrix option will produce a much more biologically looking body, but requires colouring in some other program, another new task to learn (apparently Photoshop can be used to paint 3D objects). So, do not hold your breath, but 'Tetropters VI' will probably be the final documentary, adding all the items mentioned earlier.

Except for the narrator, I am afraid...

Tetropters

Those who visited the land and sea pages on the site may have wondered why there is no page on the biomechanics of flight, seeing there are pages on 'Walking with...' as well as on 'Swimming with...'. So where is the 'Flying with...' page?

The simple answer is that I am not happy with it yet. There is material on ballonts and on tetrapterate (four-winged) large flyers, and some are in fact shown on the 'air' page. But these organisms are fairly like Terran birds. For a true oddity the tetropters should be considered; in their case, 'oddness' does not reside in them having four wings. After all, Furahan tetrapterates and Terran insects also have two pairs of wings, i.e., four wings, so that isn't really extraordinary. No, the ordinariness resides in the description: 'two pairs of wings', and that says it all: the wings are arranged in pairs. Insects and birds (and Furaha tetrapterayes) all have a body scheme with bilateral symmetry, so their limbs are arranged in pairs.

Not so the tetropters. To be honest, the very first sketches I did of them did show bilateral symmetry. The top animal in the following image shows that primordial tetropter. In fact, their wing movement patterns had already been worked out, and showed a pattern that was exactly the same for each wing. So the wings showed a radial pattern, like a four-pointed star, but the body hadn't kept up. Here is one of those early tetropter designs under attack from a larger tetrapterate.

With their radial wing pattern tetropters were excellent hoverers, and control over wing movements should have allowed omnidirectional movements. But the animals still had a front and aft side. If you have a flight system that allows such tremendous manoeuvrability, why limit it with a limited body design? That is where the concept of complete four-sided symmetry came from. The next sketch illustrates the next logical step in the evolution of the tetroper concept. The top animal has bilateral symmetry, but the bottom one represents a conceptual novelty: the body follows the wings! So the entire animal now shows complete quadriradiate symmetry. By the way, I was taught that you should not mix Latin and Greek roots, and that is explains the switches between 'tetra-' (Greek) and 'quadri-'(Latin). Can't be helped.


I know of no such designs on Earth, although there are animals with five-sided symmetry (starfish, sea urchins etc.). The rest of their body scheme hasn't been worked out in much detail. The design problems are like those of octapods, with eight-sided symmetry. Tetropters too have eyes above as well as below their 'equator', and the mouth parts are all on the ground side. But I haven't done a complete drawing or painting of one yet, so the details remain sketchy - for now.

I found that the flight patterns had much in common with the movement patterns of flying and walking organisms. After all, each leg or wing or flipper moves repetitively, and any gait is no more than a cycle in which the limbs move with a specific set of phase offsets. Although there are an infinite number of ways in which you can do so, only a limited number make much sense. And so tetropters have 'walks', 'trots' and 'paces'. The first pattern shown here is a rather silly one.



I can't show the animated gif here; to see it click here.

Imagine the sphere as the animal's body, seen from below and to the side. The four wings are rotated around their axis to provide lift while going clockwise as well as moving back anticlockwise. While such a pattern does provide lift, it would also result in a net rotation effect on the animal as a whole, which is impractical. To offset that, the rotation forces need to be cancelled by having two wings swing in the opposite direction from the other two. A rough animation of just such an effect follows:


I can't show the animated gif here; to see it click here.

You can see the wings going through one another; in reality they would of course not do so. That's why I need to do a better animation, but it is not as easy to do as it sounds. In the smallest tetropters the wings clap against one another and then to move in the other direction. Many Earth insects use this technique, by the way: the clap their wings together behind their backs, and this apparently generates lift when the wings move away from one another again. In insects, there is just one such 'clap' in each cycle. Tetropters have taken the idea one step further, and the clapping occurs twice in a wing cycle, not just once.

Larger tetropters usually avoid clapping the wings together, so they reverse direction without touching. Whether this is aerodynamicaly better or simply avoids damage to the wings remains to be seen. There is so much to be researched...