Furahan biology and one of its allied matters: origami. Yes, origami!

My series of rusp posts have created something I never expected: an origami rusp. For more on rusps, simply start at the previous post in this blog. Regular commenter Petr (Petr Stuchlý) has sculpted four models of Furahan animals in what for me is a novel medium: origami. He posted photos of one of them, the brontorusp, on DeviantArt under the title of Xenorigami, which I rather like. Petr was kind enough to send me all four models in a neat package, which I appreciate very much.

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Let me show you the models. Here are all four of them along with a prototypical matchbox  for size. Let me close in on all of them.

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 First, of course, the origami model of the brontorusp (Brontocrambis brucus). I have never published a complete image of the brontorusp, so Petr could not know that the front and hind heads are not identical. If you take a look at the image of the origami brontorusp on his DeviantArt page, you will see that the front and hind heads are identical. Once we got to talking about that he has made some changes to the model, and now the hind head no longer has a rostrum (snout). Before you ask about two-headedness (the Janus effect), here is an explanation. The basic ancestral rusp design involves developed segments with eyes on each one. When the lineage increased in complexity, 'cephalisation' set in, so nerve clusters, eyes, whips etc., concentrated in one spot leaving the animal with a head... Actually, no!. That may be what happens usually, but in rusp evolution there was not just an increase in complexity, but cephalisation also involved a loss of some functions, such as visions in middle segments. That, probably coupled with a need for defence on both ends of the rusp, stopped the loss of vision there, so there are eyes there, with a fully functional whip capable of giving any predator a good wallop. There is also a circular secondary brain, the neurannulinus, communicating -we assume!- with the neurannulus in front. There is no hind mouth though, so no rostrum there, and rusps definitely have a front and a hind part.

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Let's look at the details. I think that Petr caught the shape of the brontorusp's head very well. I have only a very vague idea of how you would go about to create such an intricate design from a square piece of paper of 50x50 cm. From his own comments ("hell on earth to fold!!! XD") it was not exactly the easiest thing to do.

 

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The many folds at the rusp's bottom are interesting. Petr explained a particular featre as the result of his folding technique, but regardless of the cause the design parallels my rusp design in a surprising way.  Rusps, as large animals, have vertically-orientated legs that swing forwards and backwards under the body, and not sideways, so there is a risk of them kicking one another. That problem was sidestepped (sorry for that one) by having the legs alternately offset to the centre or the side of the animal. The origami rusp has the same feature!

What you see here is the 'crease pattern' of the rusp laid out on a piece of paper. The 'CP' has lines indicating valleys and ridges, and helps the designer shape a flat piece of paper into a remarkably solid and three-dimensional sculpture.

Click to enlarge; bottom : copyright Gert van Dijk

Petr has also made a hexapod neocarnicore, with its typical 'raptorial appendages'. The model captures the form well, which must be difficult as it is a very small model: he wanted to keep the models within a certain scale range. They are not fully in the same range, but I do not think I ever published enough data on exact rusp size for him or anyone else to judge the size accurately. Above is a new scale diagram for The Book, so you can see how large brontorusps are.

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A marshwallow! It is complete with three horns, and even seems to have the continually irritated expression that humans project on the animal's cranial features...

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 And finally, two sides of the marblebill, here suspended from a Japanese eating stick, to stay  in style. Origami paper can have one colour on one side and another on the other, and here that principle was obviously expanded by having a two-layered sheet of paper with metal foil on one side. Somehow Petr managed to have the dark paper end up on the dorsal side of the upper limbs and the metal side on the palmar side, doing justice to the pattern of the marblebill.

It is obvious that Petr is rather good at origami. If you wish to learn more about his art visit his Flickr page,  DeviantArt page or a page at an English origami site. If you like palaeontology, and the chances of that are high with a blog like this one, you should visit his page on origami versions of extinct animals.They're very good.

Influence of the rostrum linkage system on forage volume in Brontorusps (Brontocrambis brucus)

A Christmas Special!

Ahead of the normal schedule, and with dinosaurs, rusps and biomechanics!

Click to enlarge; copyright Gert van Dijk

The title of this post sounds like that of a proper scientific paper, doesn't it? Something out of the 'Journal of Astrobiological Biomechanics', I guess. It's time to look at rusps again. My big rusp painting is finished, and as it is meant as a double-page spread, it is large: 7200 by 2700 pixels. A spoiler is shown above showing a fragment of a rusp in the background of the painting. The fragment has been halved in size and its area represents just 2% of that of the entire painting. The painting is based on earlier sketches. For more on rusps, either visit the main Furaha site or look at these posts: sketches, anatomy, predation, concept paintings, etc.  

The evolution of new Furahan animals gets more complicated with time. In the beginning I just sketched a pleasing shape and started painting right away. Now, I worry more whether the animal makes evolutionary, mechanical and ecological sense. Well, up to a point; this is science fiction and supposed to be fun, after all. 

Here are some of the steps in rusp 'ontology': they started with some quick sketches, and then the slow evolution began: successive legs were offset medially and laterally to avoid legs bumping into one another, followed by an arrangement for their skeleton. Their fore and aft whips are long and held horizontally rather like the tails and necks of sauropods, and hence have a similar system of internal trusses as compressive elements at the bottom and ligaments at the top to withstand tensile stress. The whip is held up passively by these forces, so avoiding the high cost of doing that with muscle force only. The last stage involved refining the head of the rusp, and in particular its snout, or 'rostrum'. In an earlier post this rusp species was called Mammoth Rusp / Megacrambis, but now it is the Brontorusp / Brontocrambis; yes, that means 'Thunder Caterpillar'!  The Mammoth Rusp still had some intricate limbs functioning as additional feeding aids under its snout. I was not too certain of that arrangement, and my doubts were confirmed by comments on that post. So the Brontorusp no longer has these additional mouth parts. The thing is, now we have a massive animal with a large head. How does it feed itself?

The mouth of the rusp is in its head, which seems obvious but in speculative biology not many things are obvious. Also note that rusps are large herbivores: they need a lot of food and spend much of their time eating. Moving about is costly, so it would be best if they moved the least possible amount to get their food, which does not sound as if there is much room to save energy. Let's tackle that by considering the problem of getting an animal's mouth on vegetation; there appear to be four solutions to do so; rusps use the fourth, but we'll come to that. The first solution, always necessary as vegetation will not come to you, involves walking to the food source.

Click to enlarge; copyright Klein et al; Biology of the sauropod dinosaurs. Indiana University Press 2011

But once an animal arrives at its 'foraging station' a nice way to save energy is to keep most of the body motionless and to have a long neck allowing the head and mouth to move about independently of the gut. For very large animals, needing to feed all day, it pays to divide their anatomy in mouth and guts; the rest is just 'other bits'. Sauropod dinosaurs used that method, and the image above is from a study on how far sauropod mouths could reach, depending on neck length and leg length. The idea is that the neck can move in a horizontal plane 90 degrees to the right and the left, and in a vertical plane straight up and down. If the animal is lying on the ground the volume of space that it can reach is one quarter of a sphere. If the base of the neck is higher up, when the animal is standing, the volume increases. The authors assume that the bottom part of the volume then is cylindrical whereas I would assume that to be spherical as well, but never mind.

Click to enlarge; copyright Gert van Dijk

Swans and geese have very flexible necks and can probably reach every point within that envelope, but if an animal has a neck less flexible than a swan's, only part of the volume is accessible to the mouth. If this is the first time you realised that geese and sauropods might have long necks for a similar reason, good!

The image above shows an adapted 'forage volume' for a sauropod: the outer red sphere is the outer limit of where it can reach, and the inner blue sphere represents the inner limit, assuming that the neck is too stiff for the animal to reach a point closer to its body. The human ('Marlene') is just there to keep the sauropod in its proper place. 

The third solution to get the mouth near food is to use an appendage to shovel food towards the mouth. The best example I can think of is the elephant's trunk, which greatly increases the elephant's reach. The erstwhile rusp mouth limbs were short and not at all good as harvester limbs, and I did not wish to elongate them tenfold; they are gone. I also did not wish to turn the whip into a grasping organ. Rusp whips are not built for that, although in a pickle they can probably be used to knock a branch off a tree. Instead, rusps use a fourth system which is really just a combination of the last two: they carry their mouths towards the food without moving the rest of the head. The 'mouth extender' is extensible and based on a mechanical linkage system. In itself this is certainly not a new idea: Earth fish have such systems in abundance.

Click to enlarge; copyright Gert van Dijk

This image shows a schematic view of the rusp rostrum. Start with the red shape in the foreground: it consists of two V-shapes starting from a vertical axis. All places where elements meet are in fact joints. The pink axis shows that the whole ensemble can rotate, but it can do other things as well: if the two Vs rotate towards one another, the whole shape will become longer and narrower. At its right end, the shape ends in two points on a horizontal line. Now copy the shape, rotate it by 90 degrees, and you get the blue shape in the foreground. The two points where the red shape ends act as connection points for the blue shape. Once connected, some movements from the red shape are connected to the blue one, but not all, and that makes the rusp rostrum quite versatile. In the back you see how the rostrum is formed by stringing red and blue shapes together. In reality the trusses are not formed by straight bones, but by curved ones, so the section of the rostrum is circular rather than rhombic. The cylinder on the right attempts to show the outlines of the bones on a cylinder.

Click to enlarge; copyright Gert van Dijk

And this image shows an as yet unmentioned aspect of movement: if the two starting points are brought closer together, this changes the section of the rostrum as well as its length. The right one is extended, the middle one shortened, and the right one is in neutral position. I expect rusp rostra (yes, that's the plural) to be able to double in length.

Click to enlarge; copyright Gert van Dijk

But we need more flexibility, and that is achieved by rotating the shapes and using the angle between the Vs for additional control. The stylised skeleton in the back shows what can be achieved. So there we are: an extensible and steerable system to get rusp mouths where they would otherwise not reach.

Click to enlarge; copyright Gert van Dijk

Here are two views of an adapted Sculptris model of a rusp head. I take it you will recognise the system of trusses under its hide.

Click to enlarge; copyright Gert van Dijk

And finally, a schematic rusp foraging volume, rather like that of the sauropod (the whip of this model is truncated). Note that the rusp can access a larger portion of the outer foraging volume than the sauropod. The volume itself is smaller though, as rusps are smaller than sauropods, and their rostra extend their reach, bot nearly as much as the sauropod's neck does. Marlene is standing in the forage volume, something I would definitely NOT recommend! In practice, rusps are ground feeders, not bothering about high branches. Have I told you about the ecology of the spotted plains where they live, where post of forests alternate with plains and how rusp feeding habits are to blame for that? No? Oh well, that is another story.  

More future evolution in Japan

Sometimes I like to revisit sites to see whether there is anything new. In this post I will show a few interesting species that came up in this way. The site in question was visited in 2010, and shows the work of the Japanese author and illustrator Satoshi Kawasaki. He specialises in palaeontological illustrations but does not shy away from extending the time line of his work well into the future, up to 200 million years from now, in fact. In palaeontological papers and books you sometimes read 'mya' as an abbreviation for 'million years ago'. As the world of speculative biology is less hampered by ugly facts, perhaps it could profit from having a similar term for 'million years from now': myfn, or perhaps 'million years on': myo.

Click to enlarge; Copyright Satoshi Kawasaki

As I wrote before, Mr. Kawasaki has the sense of humour that allows him not to take his creatures equally seriously, something I like very much (I find mere monsters boring). Some of the animals on the pages showing life 100 and 200 myo are apparently drawn by other artists than himself, so het lets others play along, another nice trait. I would have like to exchange emails, but previous attempts to contact him failed. Let's have a look at some of the creatures.

Click to enlarge; Copyright Satoshi Kawasaki

In Google's translation this one is called 'Nereusu'. By omitting some of the Japanese characters I found out that Nereusu is simply a transliteration of the Japanese characters, so I could not translate it.  I therefore suppose the name refers to Nereus, the mythical being from classical Greece Nereus, who was after all as sort of sea god. Somewhat ironically, there is of course another Nereus in speculative biology...

Anyway, the animal is obviously a large marine predatory bird descendant (probably descended from penguin stock). Students of speculative biology will note that such creatures are very abundant in fictional future seas, as they apparently tend to evolve in the minds of many creators. I do not really mind if such a concept is not completely original; after all, all of science fiction is full of common ideas. While I applaud originality, there is also pleasure in seeing a job well done. Mr. Kawasaki is a very adroit illustrator, and this is an excellent 'future orca-like penguin-descendant marine predatory beast'.

Click to enlarge; Copyright Satoshi Kawasaki

Have a look at this drawing, and you will probably guess what it is about without having to read the text. It can only be a social crab modelled on the pattern of ants, bees and similar colony dwellers. There is one giant 'mother' laying lots of eggs, here very neatly held in a redeveloped abdomen. The ones in the front must be soldiers, and the little ones in the middle must be workers. I cannot see whether or not they have pincers, but assume they do; otherwise, what will workers work with?

Click to enlarge; Copyright Satoshi Kawasaki

Sometimes Mr Kawasaki works on a theme; in my previous post I showed terrestrial cephalopods (I know, I know...), and this time I will focus on a group of his animals that do not seem to enjoy the common attention of future evolutionists: starfish! There is only one on the 100 myo page, shown above. It is not drawn by Kawasaki but by someone else. It may also be the most original of all the future Asteroidea ('starfish'). You cannot beat Google Translate for creating a sense of wonder, particularly where one was not intended: "One of the arm portion becomes large, the remaining portion forms a head lump pseudo part." I guess we would have guessed that anyway: four of the five original arms have shrunken and are now appendages around what is now a proper head. As a result, the animal is now bilaterally symmetrical. I do not quite see how evolution would set off in this particular direction, but like the result. I do not think I have seen anyone else designing this before, either.

Click to enlarge; Copyright Satoshi Kawasaki

The world of 200 myo has more future Asteroidea.The one above is a pseudoplant, a Parasasuteru. It lives in Australian swamps and -I think!- envelops animals moving in its shade, only to digest them at leisure.

Click to enlarge; Copyright Satoshi Kawasaki

And finally, one I rather like: the 'Di pedal stell' ; could that be a 'bipedal star', I wonder? If you count the number of limbs, you will find six rather then five, but the texts suggests that one of the original arms has split to form two legs: "Part of the two-that looks like a foot is what arm once was transformed." Probably. Have a look at Mr Kawaski's site for other interesting creatures, or, if you like palaeontological illustrations -who doesn't?- visit his pages of the past world.


And now something somewhat different
I have been looking for other projects of speculative biology, but have not found any new ones. I searched in various languages, albeit my skills are limited to Germanic and Romance ones. If readers know of projects that deserve attention, let me know, particularly ones I am likely to miss, such as ones in Slavic or non-European languages.

Finally, I have begun considering ending this blog. It is in its sixth year and I feel that some of the freshness has gone. The number of readers has not diminished, by the way: it is stable and in fact grows slowly. I find it a bit more difficult to come up with new subjects, and after more than five years the blog has perhaps become a fixture in the little world of speculative biology, not something that attracts much attention. Perhaps blogs are a bit like television series; at some point you stop caring about the characters, and that may be the time to consider a final episode. 

Red leaves, swaying in an alien breeze...

Readers who have followed my series of posts on alien plants and photosynthesis (here, here and here) will know that I have no objection against plants on other world not being green, so that is why there is  'red' in the title of this post. But this post will not be about photosynthesis, as I think that theme has been dealt with sufficiently. The next theme on plants will probably be about biomechanics, but I have not started that one yet.

This post is about portraying alien plants. Obviously it is possible to do a painting, and that is fine, but it is also a lot of work. Can't computers do part of the work? There are not that many software choices available to populate a landscape with alien plants. The one I have been using over the years is Vue by E-on software. Vue is difficult to handle, in part because there are many options that are not all well-explained in the manual, but also because the software can be very unforgiving depending the hardware you are using. In other words, it may crash. It is the kind of programme that you can easily develop a love/hate relationship with.

It has an ecosystem feature, in which you choose plants or objects, adjust their rations and relative sizes, and when you then press 'populate' the programme does just that. It can even take matters such as height or slope of a landscape into account, placing some species there and others not. The problem in designing alien forests was that Vue's innate plant designer was inadequate: it let you design variations of Earth plants, but made it impossible to design something more interesting from scratch. For that I used XFrog 3.5, a programme that allows the user to come up with intricate new shapes. The XFrog plants could be imported into Vue, and did allow worlds to be populated with alien plants. Some examples of my earlier efforts are here for Epona and here and here for Furahan swamps.

However, there was one disadvantage: Vue's own plants could sway in an imaginary wind, but the imported XFrog plants were static objects. For static images that is obviously not a problem, but for animations a forest in which no leaf moves is just odd. I have stopped doing Vue animations for that reason.

Recently, E-on introduced a new programme: The Plant Factory (TPF), which does let the user design plants from scratch, with the promise of having the result sway the wind. That was attractive, so I decided to try it, even though the user forum made it clear that this is a typical E-on product: it can do amazing things but often in a roundabout or unexpected manner, and sometimes it simply does not deliver. TPF has no manual whatsoever, so anyone wishing to use it should treat it as a voyage of exploration rather than as a productivity tool. There is a 'personal learning' version, so everyone can test it without spending (rather a lot of) money on it.

Click to enlarge; copyright Gert van Dijk

 I first tried whether I could make it design oddly shaped plants, and here is one of first attempts. I tried to obtain a results resembling an earlier XFrog design, and that went reasonably well, as you can see above. As you can see, this tree has its major branches growing from a central trunk as do Earth trees. Its branches curve through the air to reach the ground, where they may take root, providing water and nourishment or simply offer structural support. The proportions are not right yet, it is a start.

Click to enlarge; copyright Gert van Dijk

The image above shows a hillside populated with two species of simple plants, home made in TPF. The scene was intended to experiment with wind animation. The first result was disappointing in that there was hardly any movement. There are lots of sliders controlling wind, which I had left at their original settings. Apparently those are meant for an unnaturally calm day. Very well, let's turn the wind setting up to 100%. That did not do too much either. I remembered an earlier surprise in Vue, dealing with lens blurring; there too a setting of 100% was almost negligible; someone at a forum told me to not to treat 100% as a limit, and so here too I set wind animation to 500%, and now at least the leaves move. Apparently this is more or less a dimensionless unit; just one of those odd Vue quirks.

  

And here is the result; better, I think! It's not a storm yet, but at least there is movement! The quality of videos on blogger is not very good, so it can look a lot better. Meanwhile, there is still a very large number of options to discover or, given the lack of a manual, to blunder into, so do not expect a to see a marblebill brachiating through a Furahan forest. Not quite yet, anyway.

'Zigzag / zagzig' and 'zagzig / zagzig' robots (Walking Machines V)

I have not discussed the theme of walking machines for a long time: the last post on that subject was published back in 2010, but dealt with 'radial robots', meant as toys. As you will know the word robot was derived from 'worker', so it is fitting to go back to posts on robots meant to do proper work (here).

The lack of interest was not because no progress is being made; quite the contrary. There is so much improvement that robots are slowly taking on aspects of animals, and the reason for that is en ever-increasing subtlety of control of the movement. The word 'cybernetics' has its origin in a Greek word for 'steersman', and 'steering' all aspects of movement is a key concept in walking, regardless of whether you are talking about an animal or a machine. The other major theme is mechanics, of course. For some background on leg design, see here and here.

You are probably familiar with 'Big Dog', a walking robot made by Boston Dynamics. That company has developed a range of robots meant to aid the military. The video above shows the level of control their robots have these days: clearly, this thing, the 'legged squad support system' (LS3) can hold its won on difficult terrain and follows its human master on its own. There are more YouTube videos (and with better quality) that are found on YouTube after a search for 'Boston Dynamics'.

Here is another one: a 'cheetah' running very fast on a treadmill. It is tethered and the power source is external, but is still an amazing sight. It is interesting to see how the engineers handled the problem of elongating stride length. Running mammals generally have legs with  three major segments; the foot can be seen as a fourth, minor segment. Cheetahs obtain an additional lengthening of their strides by flexing and extending their bodies as well during the stride. Now compare that to the cheetah robot: it does have a flexing body, but the legs have only two segments and the foot appears to be something like a rubber ball only.

Their latest attempt is called 'WildCat', which is apparently based on the cheetah design. This time both the front legs and the hind legs seem to be linked to the main body by a flexible joint, instead of the legs being fixed to the body directly as was the case for LS3. As a result, the setup is beginning to resemble the setup of shoulder and pelvic girdles common to vertebrates.

The legs still consist of two segments only. Seen from a level of control, having only two segments makes it much easier to find out where the foot should be, as only two angles have to be controlled, and each foot position can be reached with only one combination of angles. If you add another segment, each foot position can be reached in an infinite combination of joint angles. I wonder whether the lack of a foot is due to similar considerations: adding another segment, even a short one, probably adds a considerable computing overhead. 

Another interesting aspects is how the engineers chose the directions where the knees and elbows point to. In an earlier post I discussed whether legs should start with a segment pointing forwards (a 'zig') or backwards (a 'zag'). The next segment than points the other way. The upper arm (humerus) of mammal front legs points backwards and the forearm forwards, so the mammal front leg is a 'zagzig'. Hind legs, in the same jargon, are 'zigzags'. The formula for the entire mammal is a 'zagzig / zigzag'. Are you still there? (Mind you, this is just a simplification paving the way to look at robots; if you include the scapula, -a zig!- and label all three segments, mammals are 'zigzagzig /zigzagzig' animals.)

Now have a look at the LS3 again. Its mid leg joints point away from the body, just the opposite of the mammalian leg bone pattern. The LS3 is a 'zigzag / zagzig' walker. WildCat, in contrast, is a 'zagzig / zagzig' walker. I have no idea why the engineers  chose the designs they did, but the results strengthens my feeling that there is no basic overwhelming advantage inherent in the current mammal pattern. During evolution sideways-pointing legs were turned to have the plane of the leg parallel to that of the body, and in this turn front legs turned backwards and hind legs forwards. Evolution might well have resulted in a different pattern, that of LS3. At least it prevents knocking elbows into knees! Those who wish to add more 'alienosity' to their animals might consider departing from the Earth vertebrate pattern. Have a look at LS3, WildCat, or, of course, at an older post in this blog to see what might be done.

Finally, a word on gaits. Walking consists of a repeated cycle of leg movements, and a gait is nothing more than the phase differences between the various legs. The basic gait of LS3 is a trot, in which front left and right hind legs move together as one pair, and the other two legs from the other pair, moving exactly half a cycle out of phase. If this is confusing go the Furaha 'walking with...' page, where the major gaits are explained. The engineers of Boston Dynamics have managed to proceed beyond the trot, so the thing can bound and gallop as well. I am very surprised though that it seems to use a trot when it is walking very slowly. You would expect a 'walk', which in this context also is a defined gait. The various gaits used by animals have important energy consequences, and a trot is more expensive than a gait. I wonder how much further 'evolution' will take these robots. More segments? More efficient gaits? More legs, even?  

Layers of leaves (Alien plants V)

The last post on alien plants went into some fairly technical details about photosynthesis, and took a look at where the process could be adapted to make it more alien. Today's post has a closer look at just one aspect: leaves.

Photosynthesis obviously depends on catching light and is therefore a process that takes place on the surface of a plant. How much of a surface is needed will depend on many things, such as how much energy is needed. As related before, C3 photosynthesis can only make use of up to 25% of the light falling on them (well, at noon in the tropics, that is). Photosynthesis becomes saturated, doing nothing with that extra light.

For now, let's assume the presence of leaves on a planet of choice as very similar to Earth's broad leaves; flat structures of, say, 10 cm across. They need light, and so face the sun. But not every leaf of every plant will receive full sunlight: the sun moves across the sky (well, as far as the plant is concerned it does), other plants may be in the way, and even its own leaves, placed higher up, will take light away from lower leaves. A typical Earth leaf transmits only 5% of the light striking it. The next leaf down in turn absorbs 95% of the -little- light striking it, leaving only 0.05 times 0.05 of the sun light, or 0.25% of the light striking it.* On Earth the critical level for photosynthesis to be of any use is at about 1% of full sunlight.

It is therefore reasonable to assume that plants would have only one or perhaps two layers of leaves, right? Additional leaves would not contribute anything, and yet trees typically have many more layers of trees. The answer to this riddle is found in the efficacy of photosynthesis and a fact that you might not have considered interesting in this respect: the size of the sun.

Click to enlarge; source: http://en.wikipedia.org/wiki/Umbra

The image above shows the umbra and penumbra as commonly illustrated in astronomy books. The sun is not a point source of light but a sphere much larger than the Earth. Rays of light depart in all directions from any point on its surface, to the effect that there is a conical volume of space behind the Earth where the rays cannot reach, or, in other words, from where an observer can see no part of the apparent disk of the sun. That conical volume of space is the 'umbra' , simply meaning shadow in Latin. Around it there is an area from which an observer can see part of the sun's disk, so that area receives some direct sunlight, but not full sunlight: the 'penumbra' (nearly shadow). Everywhere else receives full sunlight. The length of the umbra cone depends on the diameters of the Sun and the Earth and the distance between them, as a few minutes experimenting with some sketches will show you.

The same applies for objects closer by. All you have to do is to look at the shadow of your hand as you raise it from the ground on a sunny day. Leaves also cast an umbra, an area without direct sunlight, where it would be best not to place another leaf. The length of the umbra can be calculated as explained above, and for Earth the calculations that the umbra is about 108 times the width of a leaf. For a 10 cm leaf that would boil down to 1080 cm, or 10 meters. As we will see the distance is shorter in practice. Leaves may receive enough light in the penumbra to work well. Remember that on Earth photosynthesis is already saturated at 25% of full sunlight, so photosynthesis can work at full capacity even with a fair amount of shade.

I wrote a Matlab program to have a look at how the umbra and penumbra could look for some artificial leaves. The distance between the Earth and the sun is 149,597,870,700 meters and the diameter of the sun is 1,392,684,000 m., both according to Wikipedia. A leaf takes up half the area of a 10 by 1-0 cm square area. In the program, this meant that I could paint half the pixels in a square area black denoting the leaf. All the program does is to cast ray from a raster of points on the sun's disk to all points on the leaf area, and to see which rays are intercepted and which are not. I did that for three distances behind the leaf: 0.5, 1 and 5 meters.

Click to enlarge; copyright Gert van Dijk

And here is the result of a simple roughly circular leaf. Half a meter away our leaf casts a recognizable shadow. I calculated how large the area is that receives less than 25% of full sunlight. That value of 25% is randomly chosen but helps to indicate deep shadow. Depending on the efficacy of photosynthesis, the value could indicate the lower limit of light for photosynthesis to work if it is particularly inefficient, or perhaps the point at which its efficacy becomes impaired. At half a meter an area of 85% of the original leaf receives less than 25% of full light, while at one meter the area decreases to 69%; at 5 meters it is 0%.

Click to enlarge; copyright Gert van Dijk

Let's try with a differently shaped leaf. After all, the umbra depends on the width of the leaf, so a leaf with a thinner shape should do better. This cross-shaped leaf was somewhat disappointing, as its values for 25% full light were only slightly better than for the circular leaf. For 0.5 meter the value was 81% of leaf area, for 1 meter it was 65% and at 5 meters it is 0%. Clearly, some more shape experimentation is needed.

Click to enlarge; copyright Gert van Dijk

This 'clover' has more space between its petals. Does it work better? Yes it does: 0.5 m results in 62%, 1 m in 31%, and 5 meters as usual results in 0%.

Click to enlarge; copyright Gert van Dijk

Finally, here is the ultimate feathery leaf, designed to have thin strands, while its area is still the same as that of the others. Here are the values: for 0,5 meter, only 11% of the leaf area receives less than 25% of full light, and at 1 and 5 meters the value is 0%.

Click to enlarge; source here

So, what does all this mean for the design of trees on other worlds? Firstly, like on Earth, you can have multiple layers of leaves and still have enough light trickling down for lower leaves to be useful. The shape of leaves is also important. Apparently some Earth trees use this effect: the outer or upper leaves of olive trees are thinner than the leaves lower down, which makes sense in view of the experiments above.

An interesting consequence is that the distance between leaf layers would depend on the apparent diameter of the sun's diameter as seen from a planetary surface. Doubling the diameter would half the length of the umbra, so leaves could be closer together and still receive an adequate amount of light. 

Should your alien trees have a few layers of leaves or multiple ones? Theoretical considerations on Earth suggest that fewer layers work better when the amount of light is low to start with: the absolute level decays very quickly with the number of layers. For alien worlds, 'low light' can probably be rephrased as a low capacity to make use of available light. That could be low light with good photosynthesis or good light with poor photosynthesis. The effect of changing the saturation point is more difficult to predict. On Earth, where photosynthesis saturates at only 20-25% of full light, shadows may still leave enough light. But if photosynthesis could use up to 75% of all light, the top layers might generate all the energy needed, so more layers would be superfluous. Then again, the plant might well have evolved to use all that energy, so perhaps lower layers would still be useful.

No doubt, additional demands, such as transport of metabolites and structural stiffness will complicate the picture. Nevertheless, on Furaha various plants carry their leaves in umbrella-like shapes, in which two or three layers of leaves form a nearly completely closed canopy, closing off the sky to potential competitors that might grow up beneath them.                               

* This example is taken from the excellent book 'The life of a leaf' by Steven Vogel

Salsa Invertebraxa : Free pdf available!

In July 2012 I wrote a post on a book I liked very much: Salsa Invertebraxa by Mozchops. The book is about insects on a quest. Yes, that sounds odd, and it is an odd book, as well as a brilliant one! For a closer look got the publishers website: http://www.pecksniffpress.com

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The good news is that it is now available for free as a pdf file. The resolution is fairly low, but it is free. There may be a digital version for Kindle and other devices in the future. I will keep you informed when there is news of that.  Here is a direct link to the pdf file.