Tuesday, 13 October 2020

Ballonts IX: Blue Sky Thinking (Part 2)

By Abbydon

The previous article discussed the possibility of using soap bubbles to create small lighter than air organisms. A second possibility is more sophisticated and relies on the wonder material, graphene. It was discovered in 2004 by two scientists at the University of Manchester who were subsequently awarded a Nobel prize for their work.

Graphene is made of a flat sheet of carbon atoms, so it is a little bit like 2D diamond. It has many interesting properties but is most well-known for its high strength (100 times stronger than steel) and low density (less than 0.001 grams per square metre). An amusing illustration of this was provided in the Nobel prize paper. Imagine a one square metre graphene hammock tied between two trees. This could hold a 4 kg cat before breaking yet would only weigh as much as one of the cat’s whiskers.

As impressive as this is, recent work suggests that graphene has another property that is extremely relevant to ballonts. Graphene is also impermeable to gases. With such an amazingly low density it effectively produces a matte-black massless membrane even when a few thousand layers are used.

It would however be extremely challenging for life to produce a graphene balloon with absolutely no defects. A more robust approach is to copy the bubble foam concept and use a mass of small graphene “bubbles” instead. Conveniently, a material extremely similar to this called aerographene has been invented by scientists. It consists of a complex 3D network of graphene and carbon nanotubes where most of the volume is air. For this reason, when measured in a vacuum it has a density slightly lower than helium at 0.16 kg/m3.

Click to enlarge; copyright unknown; source here

While aerographene itself is not airtight it is not inconceivable that a material like it could be produced that contains many small airtight compartments in a similar way to a bubble foam. If these compartments contained a lifting gas, then the entire structure would float. When I mention aerographene below I am actually referring to this possibility and not the real material. The chart below shows that both a 1000-layer graphene membrane and a “solid” aerographene balloon would easily enable small (or large) ballonts.

Click to enlarge; copyright Abbydon


This strongly suggests that an “magic” aerographene like material could be used to enable small ballonts to exist. Since they cannot have a membrane then any aerographene produced would be an external structure grown from the bottom up, like hair, and could not be repaired only replaced. Additional graphene would be produced on the lower surface where the organism was hanging while the top surface would slowly degrade.

It is unfortunately unknown whether life could produce graphene in the first place as it does not appear to be produced naturally by life on Earth. Industrial processes for producing graphene based products typically require temperatures beyond the reach of even the most extreme of extremophiles. However, Shewanella oneidensis bacteria can be used to produce graphene from graphene oxide which is at least a start.

I am not a chemist and this is not the blog to delve deeply into the exact chemical process that life could use to produce graphene at ambient temperatures. There is some justification that it would not be implausible though. Inspired by photosynthesis the direct conversion of carbon dioxide into graphene at high temperatures has been demonstrated. A slightly different approach has even been shown to work at room temperature.

As an example of what is possible, an approximately spherical lump of dark grey aerographene with a 9 cm radius can lift about 3 g. This could support something like a praying mantis that is about 7 cm long though could perhaps be longer and thinner. A cup-like abdomen could contain the graphene and hydrogen producing organs while the four long rear legs partially surround the sphere to maintain a grip. Four independent wings could provide mobility. The long forearms would then provide good reach to gather food. Due to the benefits of aerographene this hanging mantis would be able to float even when small and could grow to a size only limited by other factors. 

Click to enlarge; copyright Gert van Dijk. Note that the graphene 'floating body' is narrower at the tope than the bottom, because the animal grew in-between forming the early top and the more recent bottom graphene.  

The soap bubble and aerographene concepts don’t mean that the small ballont problem is solved as they are really just meant to inspire others to come up with their own ballont concepts. In the absence of any lighter-than-air organisms for comparison on Earth there are still many unanswered questions to be considered and that is all part of the fun of speculative evolution.

For example, why would an organism evolve to be lighter-than-air? Is it always lighter-than-air or is it only temporary? How does it control its movement when floating? How does it protect itself from predators? How does it feed? Can it repair or replace its balloon if punctured? How is the lifting gas generated? How fast does the lifting gas leak from the balloon?

If we were only interested in creature design then this would all be sufficient to inspire a range of organisms based on shimmering soap bubbles or black graphene. Speculative evolution requires more than that though. Demonstrating that physics supports the idea and that there is a plausible way for the organism to implement the idea are both important. To be thorough, it is also important to consider whether the proposed organism could have evolved through a series of plausible steps, rather than just spring into being fully formed.

There are various ways that life on Earth already generates hydrogen, such as through fermentation, so that part is not unusual. The formation of chemical laced water to form longer lived bubbles is also fairly common as fish, frogs, snails and insects are already known to do this. A soap bubble based ballont as shown in the previous article therefore seems reasonably plausible.

On the other hand, the formation of graphene is more challenging to justify in a series of steps and I cannot give a solution to this. Graphene does have the ability to absorb light efficiently at all wavelengths, which is why it is black after all. A plausible evolutionary path could involve algae or plants in low light environments developing graphene for photosynthesis related reasons. Since hydrogen can be produced by algae as part of the nitrogen fixing process perhaps forests of lighter-than-air pitch black plants could feasibly evolve. That is however an idea for another time but perhaps it will eventually appear on my recently created blog. It describes the tidally locked world Khthonia, which orbits twin red dwarfs.

Finally, I am very grateful for the opportunity to share my thoughts on this matter in this blog and I hope that they inspire people to develop their own ideas in this area. Please comment to let everyone know what you think about graphene and its possibilities.

Thursday, 10 September 2020

Ballonts VIII: Blue Sky Thinking (Part 1)

By Abbydon

Over the past few years, I have discovered an interest in astrobiology as it combines my childhood fascination with both astrophysics and biology. Linked to this I also enjoy speculative biology though sadly my artistic talents are embarrassingly poor (For reference, I did not produce any of the pictures shown below). For this reason, I have followed the various articles on lighter-than-air organisms (also known as ballonts) with interest.

Like many other people, I have found it disheartening that designing a plausible small ballont has proven challenging. Large ballonts appear to be viable as they can generate sufficient lift to allow both the balloon and a body to float but smaller ballonts cannot get off the ground.

This is unfortunate as it means infant “large” ballonts would be unable to float until they had grown sufficiently large. Small ballonts are also interesting as they can form only part of an organism’s lifecycle which may allow the body to have reduced functionality. This could be a mechanism for spore dispersal from an otherwise sessile organism as a more sophisticated version of a dandelion clock. Alternatively, it could be for reproduction only like an adult mayfly which can have a life measured in minutes.

As previously discussed on this blog, the problem for small ballonts is the membrane needed to contain the lighter than air lifting gas. A small volume of lifting gas without a membrane would of course float but it wouldn’t take anything with it. Therefore, clearly a membrane of some form is needed to separate the lifting gas from the rest of the atmosphere. Unfortunately, the mass of the membrane is proportional to the surface area of the ballont whereas lift is proportional to the volume. Since all membranes have mass this means that for small ballonts the mass of membrane is typically greater than the lift produced by the lifting gas.

Solving this problem requires reconsidering what could be used as a membrane in an attempt to get closer to the “magic massless membrane” than Mylar. Mylar is used for long lived helium party balloons and has a thickness of 0.1 mm with a density 1.2 times water, which is perfectly adequate for large ballonts but insufficient for small ones.

Two possible ideas sprang to mind which might achieve this, the first of which will be discussed in this article, with the second reserved for a following article. The first possibility to be considered is whether soap bubbles could be used. There are many videos online showing lighter-than-air hydrogen filled bubbles being produced, though they do normally come to fiery end which is not what we want to happen to our poor ballonts.

Soap bubbles may not seem the ideal form for a ballont but a bubble film is typically around a thousandth of a millimetre thick and has a density of approximately water. This is much closer to a massless membrane than Mylar so perhaps it will enable smaller ballonts. A ballont would not literally use soap to produce bubbles but would instead use some alternative organic chemical.

In previous articles a hypothetical Mylar based ballont was shown to float only once its radius was above 30 cm but the chart below shows that a hydrogen filled soap bubble could provide lift at much smaller sizes. For reference,

LIFT = AIR MASS - BUBBLE FILM MASS - HYDROGEN MASS.

Click to enlarge; copyright Abbydon
However, while the Guinness world record for bubble size is just under 100 cubic metres we don’t expect plausible ballonts to use single bubbles that large. The next chart shows smaller hydrogen filled bubbles can still lift a few milligrams if they have a radius larger than 3 mm.

Click to enlarge; copyright Abbydon
Scientists are known to do strange things in the name of Science but perhaps one of the weirdest papers I have read involves weighing 90 leaf cutter ants to determine their load carrying performance. Apparently they ranged from 1.2 to 36.8 mg with an average of 9.3 mg. The extreme low end of this is light enough that it could be lifted by a single bubble with about a 7 mm radius. Perhaps an alien leafcutter ant could generate a bubble to get to the top of a tree, harvest a leaf and use the ballast to descend once again. Such an “antballont” might look like a honeypot ant with a bubble for an abdomen as shown in the sketch below.



Click to enlarge; copyright Gert van Dijk. The squares are 1 mm in size. The panel on the left shows a circle with a 7 mm radius and a small leafcutter ant. the panel on the right shows an evolved ant holding a bubble between its hind legs.  

A single bubble could be viable for tiny ballonts such as the ant but it would be too fragile for large ones. A better solution for larger ballonts would be to produce a foam of many small bubbles. Each of these bubbles would produce lift on its own and a foam mass would produce more. This approach has been used to generate floating helium filled foam letters for advertising purposes!




The chart below shows that such a foam mass produces less lift than a single bubble of the same volume but a 5 cm radius spherical foam mass of individual 0.5 cm radius bubbles could still carry just over a quarter of a gram. This again sounds light but it could certainly carry a bundle of seeds or an insect several times larger than an ant. It is probably even enough to carry a vertebrate such as the 7.7 mm long frog, Paedophryne amanuensis with its legs spread out to stay in contact with the bubbles.

Click to enlarge; copyright Abbydon

This all suggests that the soap bubble idea is valid for producing a somewhat plausible small ballont without too much hand waving. This is possible because of the thin water-based membrane but there is one important disadvantage, a short lifespan. The bubble will pop eventually unlike a solid membrane. This can be managed if the ballont does not need to remain aloft indefinitely, perhaps because it is only part of a lifecycle (e.g. a seed or a mayfly) or perhaps because it only creates the foam to float at night. Alternatively, the organism could regenerate the bubbles constantly while in flight to maintain and even adjust lift.

How long can a bubble last though? About a hundred years ago, in a sealed container it is claimed that Scottish scientist James Dewar managed to get a 19 cm diameter bubble to last for over three years and a 32 cm diameter bubble to last for 108 days. This is not likely under realistic conditions but it shows what is possible.

Bubbles burst when the membrane becomes too thin as the water in the membrane flows to the bottom of the bubble or it simply evaporates. While low temperature and high humidity conditions may enable bubbles based ballonts to last longer, a more feasible approach is to add chemicals to the bubble mixture to make the membrane more resilient. For fun at home the Soap Bubble Wiki has several recipes for this.

It remains an open question as to how long a bubble foam could last but several minutes is possible with these home-made bubble mixes. Some species of frogs and fish make bubble nests that last for days, however, these require maintenance and might be too heavy to float. It is therefore conceivable that a biologically possible option in between these two extremes could produce a bubble foam that would be light enough to float but had a longer duration than normal. This would be ideal for seed, spore or larva distribution where a 5 cm radius foam containing a few hundred bubbles could be generated by the parent before being sent on its way to pastures new. This is shown below with the small Furahan “brochos” larvae suspended in a floating foam to enable longer range travel than would otherwise be possible.

Click to enlarge; copyright Gert van Dijk. This is a sketch for a painting that has already been finished and will appear in The Book.

Click to enlarge; copyright Gert van Dijk. The Book will contain many secondary illustrations; this one will accompany the one above.
This article will be concluded with a discussion of the alternative option to soap bubbles for small ballonts. Please let us know what you think of soap bubble ballonts by commenting below.

Since people seem interested in my thoughts on these topics I thought I should perhaps produce my own blog. Since I am a physicist and no artist this will be quite different to the Planet Furaha blog. I have therefore started Exocosm which will discuss the possibilities of planets around other stars (i.e. exoplanets). Time will time whether I can manage to continually produce worthwhile articles though…

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

This was the first ever guest post on this blog, and I hope you liked it. Abbydon and I decided to split it into two parts; the second one has already been written, but I need to produce an additional illustration before it can be posted. 

If anyone else is interested in writing a guest post, write to me about the idea and do not send a complete text yet!

Sigmund Nastrazzurro 
(nastrazzurro  AT gmail DOT com)
 





Sunday, 30 August 2020

Tabulae Mortuae IV (Archives XIV, Ballonts VII)

I thought I might present another post combining old paintings, well, sketches, with ballonts as their main theme. The reason to do so was because of the comments on earlier posts on ballonts. If you read the last one, you will have noted that the troublesome part of ballont physics, the part that conspires against having small balloons, is the membrane enveloping the lifting gas. For small balloons the membrane is simply too heavy for the lifting power of the little bit of gas it envelops. The best solution to deal with that was to think of vary large ballonts, but they would probably be completely at the mercy of the winds as a result. Sadly, small ones could not make it even of the ground. Still, here are some old sketches.

Click to enlarge; copyright Gert van Dijk

The ballonts as I first envisaged them and have presented them here were 'complete' animals, meaning they carried every organ they needed to live, including digestive organs, membranes, nervous systems, etc. But I had also played with another way of using ballonts, and that was a more limited version: they might only serve to disperse seeds or larvae. In this view, the gas and membrane would be produced by a much large parent organism, itself definitely heavier than air. The sketch above shows one expression of that idea: we are looking at something like a water lily, with most of the organism hidden from view below the water's surface. But there are 'ballontogenetic' organs in there; you can see one larva that has just been released, while another is in the process of getting its balloon filled. The idea was that the parent inflates the sac, and that, once released, the larva passively plugs the vents through which the parent injected gas.
 
Click to enlarge; copyright Gert van Dijk
 The image above shown a very similar idea. Here, most of the parent is hidden under the sand of this marsh, or beach, or wherever this takes place. Although it has a Dali-like character, I thought this design visually less pleasing than the previous one.

Click to enlarge; copyright Gert van Dijk

Now this apparently similar scheme rested on another mechanism to form the membrane. Here, a kind of microbial mat under and on a water surface has the ability to produce a lighter-than-air gas, perhaps methane. The gas would push the surface layer of the mat upwards, forming a bulge. As the volume of gas increased, the bubble lifts the membrane up some more. It might dry out to become lighter, while also shrinking at the bottom, forming a kind of tether. I admit that it was not quite clear to me what kind of membrane there would be to keep the bubble intact. If you look closely, you will see that there is a definite round structure within the outer slime mat. What that was, and how it would work, wasn't exactly thought through: while sketching, shapes just coalesce, and the process is only partly a conscious one. With hindsight, I can now add one word that might perhaps solve this problem: foam. But more about that in a later post.

I really liked the concept of some witches' microbial brew forming slimy mats that would in the end produce a nice ballont, carrying the whole microbe culture to wherever the winds would take it.
  
Click to enlarge; copyright Gert van Dijk

This one is very similar, just a bit more worked out, and more pleasing as far as its potential for a painting was concerned. ( I do not know why I wrote 'Colonia volitans' underneath; that should be 'Colonia volans', the 'flying settlement'.)

Click to enlarge; copyright Gert van Dijk

Actually, I could not resist spending an additional 15 minutes on it, quickly adding some colour to see whether life could be injected into this old sketch. It's not too bad, I think; something like this may work artistically. 

Something new for this blog
While 'artistically pleasing' is certainly important when designing speculative lifeforms, it is not the only thing requiring artistry. There has to biological and physical feasibility too, and those are not  easy either. Please keep in mind that the sketches above unfortunately rested on a complete lack of physical feasibility, and that is why they were banned. The previous posts and comments made it clear that small ballonts really need a 'magical massless membrane'. But 'massless' is already 'magical', at odds with 'physics'.

Still, one commenter, Abby, short for Abbydon, came up with an ingenious idea. He wrote that graphene membranes might allow for very small and nearly massless membranes. Whether creating graphene biologically is plausible or feasible remains to be seen. But I liked the idea that this might make ballonts at least physically feasible, so I invited Abbydon to do something I have never done before, and that is to write a post for this blog. Mind you, this doesn't mean that previous commenters were less smart; it's just that I never thought to ask anyone to contribute directly to the blog. Abbydon accepted, so you may expect the upcoming post 'Ballonts VIII' to carry his signature.

Friday, 31 July 2020

Ballonts VI; back from the brink, or still lead balloons?

Over the years, many readers expressed how sorry they were to learn that ballonts, i.e. lighter-than air lifeforms, were almost impossible to achieve on an Earth-like planet. The only way you can have a ballont on such a world is to shape it similarly to actual balloons on Earth: they have to be very, very large. What I had badly wanted was ballonts as aerial plankton floating through forests, or small ballont 'seedlings' released by their parent organisms, or something tiptoeing over the plains, as in the previous post. Such ballonts were an integral part of the Furahan menagerie, until I sat down to do the math. The sobering results are to be found here, here and here.




With that knowledge in mind, I was quite surprised to see a video showing a very small balloon on YouTube. I copied two stills from it to show you, but advise you to see it on youTube.

The video shows a very small balloon, of a size that would lend itself very well to a ballont! Was I wrong to think that small ballonts, or small balloons for that matter, are impossible? The video's narrator says that the balloon weighs 0.3 grams and needs only "naught point five eight litres of helium to float". The video shows a hand releasing a string, or so we assume, as I do not see the string in question, and the image of the balloon then rises. That sounds like a strong assertion that the balloon actually goes up into the air when released. But the video was made for advertising purposes, and advertising and truth are not the best of friends. Beware: the part where the balloon rises after being released looks like an animation rather than live action, and in the rest of the clip there is no actual video of the balloon floating. 

Luckily, they showed numbers: the video specifies a volume of helium, of 0.058 L. Volumes do not depend on weight or mass, so I suppose they meant the volume of the inside of the inflated balloon, as that seems the only thing that would make sense. We can easily find out how large a sphere should be to take up 0.058 L. Turn that into cubic meters, apply the equation for the volume of a sphere (4/3 x pi x R^3 with R as the radius of the balloon), and you get a radius of 0.024 m, or a diameter of 4.8 cm for the balloon. That looks like the size they showed in the clip. Very well, but how much mass can that volume of helium actually lift?

Well, under 'standard' circumstances, meaning 20 degrees centigrade and one atmosphere, the density of helium is 0.179 kg per cubic meter, and of air 1.2019. From that it is easy to calculate the mass of 0.058 L of helium and of 0.058 L of air. Subtract the two, and that is the mass you can lift. You get 0.059 gram. That is almost nothing! Mind you, for a balloon to work it must first lift the mass of its own envelope. In this case, the envelope has to have a mass LESS than 0.059 gram, or else it cannot float. Good luck with that. I am beginning to think that the balloons in the clip were not kept aloft by a bit of string, but that they were held up by a length of stiff wire.     

Still, I wondered if there was anything to be done about the physics. In my previous models, I had used the characteristics of PET for the membrane of the balloon, because that is a strong material that will not let even gases escape. Unfortunately, PET has a density of some 900 kg per cubic meter, so it is only a bit less dense than water.    
Click to enlarge; copyright Gert van Dijk

Here is my old model again. The x-axis shows the radius of spherical balloon in cm and the y-axis shows mass in grams. A balloons works if there is a mass difference between the displaced air and the gas inside the balloon. If that difference is larger than the mass of the membrane, you get lift. In the graph, the red line shows lift: if the values are negative, the balloon sinks, and if they are positive, it floats. This PET-balloon will not float if the radius is smaller than 25 cm; that is a big balloon. Actually, that is MUCH bigger than the balloons you can buy for parties. The membrane of latex balloons must weigh a lot less than the one in my model.

Click to enlarge; copyright Gert van Dijk


Here is the same model, but now the membrane is magically completely massless. That helps a lot! Mind you, that third power is still being difficult: a balloon with a radius of 10 cm can still only lift 5 g. To lift just one gram, you need a radius of 6 cm. Even without such a weightless membrane you cannot have truly small balloons. This raises the question what the mass of a typical children's balloon is, and how small manufacturers can make them?

I asked balloon manufacturers, and they were friendly enough to reply. It turns out that helium balloons of 9 inches can float, and that 5 inch balloons are the smallest ones to float. But they do so weakly and only for a short while, because the helium leaks out though the latex. Now, I refuse to use illogical mediaeval units of measurement, so I will substitute 12.5 cm for the width of five working men's thumbs held next to another. That is a radius of 6.25 cm.  The manufacturer told me that the weight of such a balloon is roughly 0.7 g, if made out of white latex. The colour affects the weight.         

Well, that makes sense if you compare it to the magical massless membrane: the magical one could lift one gram with a 6 cm radius, and now we find that the actual weight of the membrane of a balloon of that size is 0.7 g. That leaves 0.3 g for lift. That is just enough.

Today's lesson is that physics still conspires against ballonts. A secondary lesson may be that advertisers can waste your time. I knew that already. Sigh.


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PS this is an additional figure I had prepared but not posted. However, it ties in so well with Abby's comment below that I decided to add it.

Click to enlarge; copyright Gert van Dijk
 

Saturday, 18 July 2020

Tabulae Mortuae III (Archives XIII)



Clich to increase; copyright Gert van Dijk
 It is with some trepidation that I show you this particular dead painting. (If you wish to see more, just use the search function to find 'Archives'.) It is one of the oldest in the series of paintings that later became the Furaha universe. The subject is neither thought out well nor painted well. The sky used to be ochre, but I secondarily, after a year or so, decided to paint it over with bluish tints to improve the demarcation of foreground and background. That helped a bit, but not enough. I admit that I have now decreased the contrast of the background digitally to improve the sense of depth in the painting.

At the time of painting I was still enamoured of ballonts (there are many posts on this blog about 'ballonts'). I was exploring what you could do with them. So here is a ballont species with the ability to hover a bit above the ground, tiptoeing from stem to stem if it chose to do so. If endangered, it was supposed to be able to release its hold and shoot upwards like a cork in water. I did not stop to think how such animals would be able to regulate the upwards force of their balloon. Perhaps it could contract its bladder forcefully, decreasing its volume and compressing the gas inside.

Anyway, I remember that the tree (?) in the background started out as a gigantic slug-like creature moving across the landscape with interesting mushroom-shaped organisms on top. When the painting was redone, I transformed it into a sessile lifeform, but the mushrooms stayed. These can obviously tilt or bend in such a way that they deflect the wind. Although I have no idea what these mushroom organs are actually for, I still like the idea of them forming a buffer against the wind.

You will not be surprised to learn that the tomatl did not make it into the modern, more sensible, Furaha universe. Now they do, and the quality of the paintings have improved too, or so I like to think. It was fun to simply paint whichever shapes suggested themselves, without wondering whether they made any sense or not. Progress...         


Saturday, 13 June 2020

'Bitrophy' II (that's animals with photosynthesis)

In the previous post on this subject, I discussed some basic aspects of the feasibility of an animal using photosynthesis as an auxiliary energy source. Mind you, I never said that it would work! It is too early to reach a conclusion yet, and if you expect this post to finally answer that, sorry, no, not yet...

This post will have a look at some other factors that would have influence the efficacy of ‘bitrophy’, and I plan to try to pull it all together in a third post on this subject. So how do you judge whether or not bitrophy is of value for an animal? I thought that the ‘leaf area’ needed to catch light in relation body size might be a suitable measure: a very large leaf would be expensive in terms of material and metabolic costs as well as mechanical unwieldiness.   

So here we go; let’s walk through some factors.

1. Photosynthesis is inefficient
The inefficiency of photosynthesis, as we know it in Earth plants, is well known and was discussed previously in this blog. To summarise: many wavelengths present in visible light are not used in photosynthesis at all and are thus wasted. Photosynthesis has an inbuilt maximum, meaning that above a certain point increasing the level of light will not lead to more carbohydrate production. Finally, the chemical reaction runs almost as easily backwards as it does forwards, and given the fact that there is much less CO2 (0.04%) in the air than O2 (20%), taking out CO2 and adding O2 was an uphill struggle to start with.        

2. Let there be light

How much light falls on a plant on a planet's surface? Well, obvious influences are how bright the star is, and how far away the planet is from that star. If the orbit is circular, the amount of light reaching the planet as a whole will constant during a year, but it will vary considerably if the orbit is highly elliptical. Then there is axial tilt: if the planet's axis is not perpendicular to the plane of that orbit, the planet will have seasons, and the amount of light will fluctuate during a year. The planet will rotate around its axis, and with day and night comes a halving of the amount of light on any point of the surface (except if there is 'tidal locking’: then one half will be perpetually lit and the other dark).
 
Finally, part of the surface will receive rays of light at a glancing angle, while others receive sunlight perpendicular to the ground, delivering much more energy. You can estimate that the average amount of light is only about one quarter of the maximum (I can expand with some fingures later). And that is at the top of the atmosphere. Below that, you have atmospheric scatter, clouds, and the shade of mountains, other plants, of being under water, etc., etc.

In the previous blog I used the local maximum amount of light to calculate how large a 'leaf' an animal would need to power its 'minimal metabolic rate' (MMR). Well, if you wish to account for the average amount of light, you should make that area four times as large! That means doubling the length of its side, if the leaf is square, or its radius, if it is circular. Of course, you can decide that such large leaves are unworkable, and you can limit the animal to the tropics. Or you can have it shut down and 'hibernate' through the night. 

3. Energy for an active lifestyle
The MMR only powers the energy needs of an animal ding nothing except being alive. More activity requires more energy, and Alexander's book mentions that the average energy need is about three times the MMR. So, if you wish to cater for that, you should increase the leaf area three times; that means increasing the radius by a factor 1.7.

4. More active animals

In the previous post, we learned that the MMR depends on body mass through an exponential function. The exponent was close to 0.75 for all animals, so that doesn’t matter. However, a multiplication factor differed greatly between animal types: mammals (and birds) need much more energy than some other animals.

Click to enlarge; copyright Gert van Dijk
The image above shows three schematic animals: a mammal, a -warm- lizard and a crustacean, all with a mass of 1 kg. I did not bother to refine the shapes, but they should be recognisable.

Click to enlarge; copyright Gert van Dijk

Here they are again, but now with leaves of the right size to cater for an MMR under maximum light. Remember that if you wish to take astronomical and activity problems into consideration, the radius of the leaf should be made 3.4 times as large (1.7 times 2). However, even with the not-adapted leaf, it is obvious that the mammal needs a ridiculously large impractical leaf. The crustacean's leaf looks more acceptable.  

5. Body size
MMR depends on the mass of an animal, and we have set leaf area to follow MMR. But mass is itself a function of size: increasing the size of an animal by a factor x will increase its mass by a factor of x to the third power. For instance, doubling the size will increase the mass eightfold. In reality things are more complicated, as you cannot simply increase all dimensions of an animal by the same amount and expect it to work. For instance, legs need to become thicker. This was explained in earlier posts, here, here and here. If you want more, here and here are posts on the same them devoted to the giants of  Game of Thrones.

Is anyone still there? If you are, we can work out how MMR responds to size as opposed to mass. First, mass is length to the power of three, and second, MMR depends on mass to the power of 0.75, so MMR increases with length to the power of 0.75 x 3 = 2.25.

Does this matter? Yes: say we double all aspects of an animal’s size. The radius of its leaf is doubled, and the area of the leaf becomes four times as large. Its mass increases by a factor 8, but its MMR only by a factor 4.76 (that’s two to the power of 2.25). That particular MMR would require an increase of the radius of the leaf of a factor 2.18 (that’s the square root of 4.75). But doubling the size yielded an increase of the radius of a factor 2.0, slightly too little. So, larger animals need extra large leaves, but the additional increase is not dramatically large. The influence of size on relative leaf area is not all that strong, but still, if you want your bitroph to have a relatively small leaf, the animal itself should be small.

Click to enlarge; copyright Gert van Dijk
The image above shows three sizes of mammal, warm lizard and crustacean, if 0.1 kg, 1 kg and 10 kg. You can see that the leaves are relatively larger for the big animals than their smaller cousins, but the differences between the three sizes are not impressive.            

Well, that concludes this post. It seems that the required 'leaf area' needs to be very large, and I doubt that having such a large structure would be worth it under most circumstances. But what about other circumstances? I will think it over and try to find solutions. No doubt, readers will have suggestions too.

Wednesday, 3 June 2020

Tabulae Mortuae II (Archives XII)

While I am preparing figures for the next instalment of the ‘bitroph’ series of posts, I thought I would post another ‘dead painting’, again an example of a plant with leaves as large as the sails on a sailing boat.

Click to enlarge; copyright Gert van Dijk

The painting is basically static: there is not a single animal about to enliven the scene, the horizon is completely flat, and the landscape is not exactly spectacular. There are some puddles on the ground, suggesting recent rain in an otherwise dry environment. Only the somewhat unnatural looking clouds add a bit of drama. All this may sound as if I am reviewing someone else’s work, not my own. That is because the painting is old enough to mean that I no longer have a strong image in mind of what I was aiming for. For painters such an ‘intended image’ can obscure judgement of the actual painting. I used to hold paintings up to a mirror to get a fresh look (I now do that digitally, without an actual mirror).

So it is up to the plants together with the clouds to provide any visual drama. The plants grow from underground roots, forming a regular succession of stems resembling telephone poles. In the scene, two roots met and formed special variants of their normal stem. The two entwine one another, and now form a botanic union.

I guess that we are looking at sexual reproduction. I have no idea what happens next; seeds drifting on the wind? Nuts borne by animals? I do remember that new plants form numerous roots that grow out in all directions.

I still like the idea of these underground roots traversing the landscape with maniacal precision, although such linearity looks unnatural. The scientific name of this species, by the way, was ‘Mania predictabilis’. Perhaps I should have envisaged a landscape where these plants are more numerous, and they all criss-cross the otherwise empty landscape.

Unfortunately, the large ‘unileaves’ won’t work for reasons outlined previously. If I do use the idea again, it will be easier to start a whole new painting than to alter this one. That’s why this is a tabula mortua: this painting is dead; it’s no more; it’s expired; it’s an ex-painting.