Last year I discussed possible gait patterns for animals that walk on an odd number of legs, including various tripod walkers. Around the same time I was contacted by a reader who asked me to help him visualise a gait design. I will not go into details, but the idea was to have an animal, the tiborou, that was to have two paired legs in front and a third unpaired leg in the midline towards the back of the animal.
At that point I thought that this would be similar to the tripod walks discussed in my post, and that all their gaits could be summed up by defining the phase differences between legs. The legs of a walking human have a phase difference of half a cycle or 180 degrees. The hind legs of a hopping kangaroo move together, so the difference is 0 degrees. Back to a tripod walk: a straightforward pattern would be to have legs at 120 degree intervals (there is an animation of that on my old post). Other 'logical' patterns would be to have two legs move together (0 degrees) while the remaining one has an 180 degree difference.
But that was not at all what the reader, Metalraptor, had in mind. He asked for something I at first found very hard to visualise. All the gaits just mentioned have one thing in common, and that is that the walking cycle lasts equally long for all legs. When you think about it, that is how all Earth animals walk, regardless of how many legs they have. The gait of the tiborou had to depart radically from this idea, by having the hind leg go through two cycles in the time it took a front leg to move through one. Now just try to see visualise that...
In the end I thought I should write a small computer programme to illustrate the gait. The following reasoning helped to define its characteristics. Instead of imagining an animal moving over the ground, imagine it staying in place and having its feet slide beneath it (as if it is walking on a treadmill). Now make a film and look at individual frames: when a leg is on the ground its foot will move a certain distance between successive frames. If two legs are on the ground at the same time, that distance between frames must be the same for all legs. If not, one foot will be slipping over the surface. This must be true for any walking gait, and so holds for the tiborou as well. So: the hind leg moves the same distance over the ground between frames as a front leg, and yet moves through two cycles while the front legs move through one. There is only one way out: short steps.
Unfortunately I did not have a handy routine for a lateral view of a leg walking cycle, so I more or less let the matter lie. But later I found out that gaits with different cycle lengths actually occur on Earth! Well, to be precise, I know of just one. I discussed it in my post on brachiation. which showed monkeys that use their tail to grip branches at twice the frequency of the arms.
That knowledge put me back on track and I wrote a rough program to show what such a gait could look like. It is shown above. Please note that the animation makes no attempt at showing feet and it is otherwise also quite unrefined. The white dots are the points where the end of legs end up during a walking cycle.
Here is the animal once more; this time it moves across the surface. That makes it easier to see that the feet of the front and hind limbs do not move relative to one another when on the ground. You can also see that the steps of the hind leg are short, both in time and distance.
Seeing that gaits in which different legs have different cycle frequencies are not just a fictional invention but also something that occurs on Earth, they probably deserves a name. Perhaps it is time for another Furahan neologism, following centaurism and cernuation: how about a 'harmonic gait'? 'Harmonic', when dealing with frequencies, is often used to describe frequencies that differ from one another by multiplying or dividing them by a whole number. That is the case here, as the hind legs moves at twice the frequency of the other ones.
Oh dear... That thought immediately suggests that there could be 'inharmonic' gaits in which different legs have different cycle frequencies, but not related through whole numbers. They probably make little biological sense, but having thought of it I cannot get rid of the concept. I am trying to visualise an animal walking with such a gait right now, and it feels like the insulation of my brain is overheating...
Sunday, 27 June 2010
Saturday, 19 June 2010
Scaling, or 'size matters, but so does gravity'
How do you show a mouse and a dinosaur in the same picture to illustrate the difference in size? That odd question came up while I was preparing for this post. The theme is how making animals bigger, 'scaling', affects not just their size but their shape as well. Scaling is of interest for speculative biology because gravity plays a role, so how do gravity and size interact to determine the overall shape of an animal?
It is not rare to read statements to the effect that animals on a high gravity world must have thick columnar legs and those on a low gravity world will have spindly legs. In fact, I have written several such statements in this blog. The problem is that these statements are not very precise. After all, thick-legged elephants and spindly-legged spiders share the same gravity, so such statements are at best incomplete. There are several excellent books on scaling in animals (here's one that is easily available), but none deal with the added effect of different gravity. This post will not do that either; gravity will be dealt with later. Before discussing it another matter deserves attention: how will legs look if their only function is to act as pillars to support weight. In reality, they move, and that requires other design characteristics as well. Some knowledge of mathematics is needed.
Click to enlarge; copyright Gert van Dijk
Let's start with a simple thought experiment: a small block sitting on a column, marked A in the picture above. The column has just the right width to support the weight of the block sitting on it without collapsing. The cylinder stands in for a bone in a leg. Its capability to support weight depends on the surface area of its cross section. Suppose its diameter is D: the formula for the cross section contains D^2. The actual diameter in centimeters is not relevant; what is important is that an increase in diameter is accompanied by a larger increase in cross section: doubling the diameter increases the cross section four times, and a triple diameter causes the diameter to increase nine times. You all knew that, I guess.
The block sitting on the leg can be described by the length of any of its edges; let's call that L. The volume of the block is given by L to the third power, here written as L^3. We are more interested in weight than in volume, and weight depends on several things: the mass of the object and the force of gravity (which we will forget about for the moment). The mass of an object depends on the relative density of the material (whether it is light or heavy; we will also forget about that) and of course on the volume of the object. The weight of the block is proportional to its volume, and so to L^3.
Now let's make everything bigger by multiplying every length measure by 2: both L and D become twice as large. That is situation B in the image above. The weight of the block is 8 times larger than it was: it was proportional to volume, L^3, and the new volume is (2L)^3=8L^3. You can check that visually: the old block fits 8 times in the new one. The diameter of the column has doubled too. The original diameter was proportional to D^2, so the new one becomes (2D)^2, or 4D^2. In other words, it has become four times as big. It can therefore carry four times as much weight as the original column. That is nice, but it is not good enough, as the block sitting on it has become 8 times heavier!
The only way to get around this is to redesign the width of the column. By how much does the original diameter D have to be changed to support a block weighing 8 times the original one? The answer is that the column's cross-sectional area must become 8 times larger than it was. That equation is not that hard to solve. If a scaling factor x for the diameter is introduced, so the new diameter becomes xD, the new cross section will become (xD)^2 which is x^2D^2. The x^2 bit says how much larger the cross section has to become. which was 8 times; hence x^2=8. X is the square root of 8, or about 2.83. So the new diameter should not be doubled, but should increase by 2.83, and that is what was done in situation C, shown above.
Remember where the '8' came from that we too the square root of: it was the third power of 2, and 2 was the factor x we increased size by. What we in fact did was take the factor x, raise it to the third power to put the new weight in, and then take the square root to get the new diameter factor. Doing it in one go means raising the factor to the power of 3/2.
Click to enlarge; Copyright Gert van Dijk
So this is why large animals need relatively thicker legs than small ones: cross sections depend on squaring a length measure and weights on raising it to the third power. For my next trick, I illustrated this for a stick animal. I made a 'Disneius caniformis (varietas hortiformis)', i.e., the garden-variety cartoon doggie, shown above. Its body and head consist of spheres, and these correspond to the block above: as for a block, its volume depends on a length measure raised to the third power. The legs and its neck are coloured yellow, and correspond to the column above: the diameter of these yellow body parts will be adjusted to the weight of all the brown parts. D. caniformis above is one meter in length, so it is the size of a large dog. The chair is there to give some idea of scale.
Click to enlarge; copyright Gert van Dijk
Let's do some fast evolution and evolve a cousin that is 10 times smaller, so the size factor is 0.1: D. musformis. It is only 10 cm long and weighs 1000 times less than D. caniformis. The diameter of its legs was altered by a factor 0.1^1.5, or 0.0316. The result is an animal with different characteristics, much more fitting with a small animal. You may wonder why I did not splay its legs sideways or bend them more, but such matters will be kept for another post.
Click to enlarge; Copyright Gert van Dijk
Evolution could have gone the other way, so the size factor becomes 10. The resulting D. giganticus is 10 m long and 1000 times heavier than D. caniformis. This animal only has equals among dinosaurs and a few of the largest mammals ever. Its legs have become truly colossal: their diameter has increased by a factor 31.6! (that is 10^1.5). You may well wonder whether such an animal is still practical: after all: the enlarged legs now make up a much larger portion of its body mass than for D. caniformis, while doing the same thing: allowing the animal to stand.
Click to enlarge; copyright Gert van Dijk
Finally, here are the three species together, and this was the problem I started with: how do you depict a mouse and a dinosaur in one picture? Zooming out enough to make D. giganticus visible meant rendering D. musformis invisible, so instead I zoomed in enough to make D. musformis visible while keeping D. giganticus most impressive feature visible: its legs.
A warning for those who are still reading this: there is quite a bit of evidence that the mechanics explained above do work. For instance, within bovids (cows and their kin) bone diameter is related to bone length along the principle explained above. Here is an internet page which explains some of the same things and shows some factual data (my data are from other sources). Most bovids look rather alike, and that may help explain why the relation fits so well. If you take mammals of all shapes and form, the fit is less good and skeleton mass does not increase as much as it 'should'. There appear to be various reasons for this. The most important one is that legs move, which, as said, poses other demands on their design. So please do not start designing animals with exactly these configurations, as that may be incorrect.
Similar thoughts hold for other organs and tissues. For instance, the force a typical mammalian muscle exerts is proportional to the cross section of its fibres. Do you see the problem? When you double an animal's size the mass of its muscles increases eightfold, but their strength only increases fourfold. Relatively speaking you have made the animal weaker! Very complex, animal scaling; it's a god thing the effects of gravity are less complex...
It is not rare to read statements to the effect that animals on a high gravity world must have thick columnar legs and those on a low gravity world will have spindly legs. In fact, I have written several such statements in this blog. The problem is that these statements are not very precise. After all, thick-legged elephants and spindly-legged spiders share the same gravity, so such statements are at best incomplete. There are several excellent books on scaling in animals (here's one that is easily available), but none deal with the added effect of different gravity. This post will not do that either; gravity will be dealt with later. Before discussing it another matter deserves attention: how will legs look if their only function is to act as pillars to support weight. In reality, they move, and that requires other design characteristics as well. Some knowledge of mathematics is needed.
Click to enlarge; copyright Gert van DijkLet's start with a simple thought experiment: a small block sitting on a column, marked A in the picture above. The column has just the right width to support the weight of the block sitting on it without collapsing. The cylinder stands in for a bone in a leg. Its capability to support weight depends on the surface area of its cross section. Suppose its diameter is D: the formula for the cross section contains D^2. The actual diameter in centimeters is not relevant; what is important is that an increase in diameter is accompanied by a larger increase in cross section: doubling the diameter increases the cross section four times, and a triple diameter causes the diameter to increase nine times. You all knew that, I guess.
The block sitting on the leg can be described by the length of any of its edges; let's call that L. The volume of the block is given by L to the third power, here written as L^3. We are more interested in weight than in volume, and weight depends on several things: the mass of the object and the force of gravity (which we will forget about for the moment). The mass of an object depends on the relative density of the material (whether it is light or heavy; we will also forget about that) and of course on the volume of the object. The weight of the block is proportional to its volume, and so to L^3.
Now let's make everything bigger by multiplying every length measure by 2: both L and D become twice as large. That is situation B in the image above. The weight of the block is 8 times larger than it was: it was proportional to volume, L^3, and the new volume is (2L)^3=8L^3. You can check that visually: the old block fits 8 times in the new one. The diameter of the column has doubled too. The original diameter was proportional to D^2, so the new one becomes (2D)^2, or 4D^2. In other words, it has become four times as big. It can therefore carry four times as much weight as the original column. That is nice, but it is not good enough, as the block sitting on it has become 8 times heavier!
The only way to get around this is to redesign the width of the column. By how much does the original diameter D have to be changed to support a block weighing 8 times the original one? The answer is that the column's cross-sectional area must become 8 times larger than it was. That equation is not that hard to solve. If a scaling factor x for the diameter is introduced, so the new diameter becomes xD, the new cross section will become (xD)^2 which is x^2D^2. The x^2 bit says how much larger the cross section has to become. which was 8 times; hence x^2=8. X is the square root of 8, or about 2.83. So the new diameter should not be doubled, but should increase by 2.83, and that is what was done in situation C, shown above.
Remember where the '8' came from that we too the square root of: it was the third power of 2, and 2 was the factor x we increased size by. What we in fact did was take the factor x, raise it to the third power to put the new weight in, and then take the square root to get the new diameter factor. Doing it in one go means raising the factor to the power of 3/2.
Click to enlarge; Copyright Gert van DijkSo this is why large animals need relatively thicker legs than small ones: cross sections depend on squaring a length measure and weights on raising it to the third power. For my next trick, I illustrated this for a stick animal. I made a 'Disneius caniformis (varietas hortiformis)', i.e., the garden-variety cartoon doggie, shown above. Its body and head consist of spheres, and these correspond to the block above: as for a block, its volume depends on a length measure raised to the third power. The legs and its neck are coloured yellow, and correspond to the column above: the diameter of these yellow body parts will be adjusted to the weight of all the brown parts. D. caniformis above is one meter in length, so it is the size of a large dog. The chair is there to give some idea of scale.
Click to enlarge; copyright Gert van DijkLet's do some fast evolution and evolve a cousin that is 10 times smaller, so the size factor is 0.1: D. musformis. It is only 10 cm long and weighs 1000 times less than D. caniformis. The diameter of its legs was altered by a factor 0.1^1.5, or 0.0316. The result is an animal with different characteristics, much more fitting with a small animal. You may wonder why I did not splay its legs sideways or bend them more, but such matters will be kept for another post.
Click to enlarge; Copyright Gert van DijkEvolution could have gone the other way, so the size factor becomes 10. The resulting D. giganticus is 10 m long and 1000 times heavier than D. caniformis. This animal only has equals among dinosaurs and a few of the largest mammals ever. Its legs have become truly colossal: their diameter has increased by a factor 31.6! (that is 10^1.5). You may well wonder whether such an animal is still practical: after all: the enlarged legs now make up a much larger portion of its body mass than for D. caniformis, while doing the same thing: allowing the animal to stand.
Click to enlarge; copyright Gert van DijkA warning for those who are still reading this: there is quite a bit of evidence that the mechanics explained above do work. For instance, within bovids (cows and their kin) bone diameter is related to bone length along the principle explained above. Here is an internet page which explains some of the same things and shows some factual data (my data are from other sources). Most bovids look rather alike, and that may help explain why the relation fits so well. If you take mammals of all shapes and form, the fit is less good and skeleton mass does not increase as much as it 'should'. There appear to be various reasons for this. The most important one is that legs move, which, as said, poses other demands on their design. So please do not start designing animals with exactly these configurations, as that may be incorrect.
Similar thoughts hold for other organs and tissues. For instance, the force a typical mammalian muscle exerts is proportional to the cross section of its fibres. Do you see the problem? When you double an animal's size the mass of its muscles increases eightfold, but their strength only increases fourfold. Relatively speaking you have made the animal weaker! Very complex, animal scaling; it's a god thing the effects of gravity are less complex...
Sunday, 6 June 2010
The Diary of Inhuman Species III
Over a year ago I last paid attention to The Diary Of Inhuman Species. I thought that the sheer good natured fun of the creatures there might offset the somewhat depressing tone of the previous post. That tone was due to the inept designs shown by the Discovery Cahannel in their series "Into the universe".
Stan, the author of The Diary, has been very active in the meantime. He has followed his book by publishing stickers of a few of his creatures, and then asked people to send him photographs of any scene whatsoever, but one of his stickers had to be present in the scene. Together with some friends he has produced a very nice film showing giant spider creatures. Their movements are particularly fluid and quite convincing, except of course for the fact that no creatures of such size can walk with such relatively feeble legs. But in Stan's case I do not mind such obvious errors, whereas I do when the Discovery Channel makes mistakes. The reason is that the Discovery Channel pretends to be serious and therefore deserves a critical response. The Diary has no such pretensions at all. Having said that, what response does the Diary deserve? Have a look at a few of his recent creatures and form your own opinion. Here they are:
Click to enlarge; copyright Stan / Grograou
Click to enlarge; copyright Stan / Grograou
Click to enlarge; copyright Stan / Grograou
For me, they show that the artist had great fun in making them, and I find their exuberance contagious. I also like the changes in style.
Click to enlarge; copyright Stan / Grograou
Recently he posted a 'paper toy', which is a model of one of his creatures that can be downloaded, printed, cut out and glued together. The original is to be found here. I immediately started thinking whether I should do one too...
Stan, the author of The Diary, has been very active in the meantime. He has followed his book by publishing stickers of a few of his creatures, and then asked people to send him photographs of any scene whatsoever, but one of his stickers had to be present in the scene. Together with some friends he has produced a very nice film showing giant spider creatures. Their movements are particularly fluid and quite convincing, except of course for the fact that no creatures of such size can walk with such relatively feeble legs. But in Stan's case I do not mind such obvious errors, whereas I do when the Discovery Channel makes mistakes. The reason is that the Discovery Channel pretends to be serious and therefore deserves a critical response. The Diary has no such pretensions at all. Having said that, what response does the Diary deserve? Have a look at a few of his recent creatures and form your own opinion. Here they are:
Click to enlarge; copyright Stan / Grograou
Click to enlarge; copyright Stan / Grograou
Click to enlarge; copyright Stan / GrograouFor me, they show that the artist had great fun in making them, and I find their exuberance contagious. I also like the changes in style.
Click to enlarge; copyright Stan / GrograouRecently he posted a 'paper toy', which is a model of one of his creatures that can be downloaded, printed, cut out and glued together. The original is to be found here. I immediately started thinking whether I should do one too...
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