Maths
Evolution Jigsaws
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Plussed.net
Maths
Evolution Jigsaws
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Religious fundamentalists can believe some pretty weird things. When their beliefs run into conflict with reality, then their perception of reality has to be adjusted so that they can maintain their beliefs.
For example, some people who subscribe to the krishna brand of fundamentalism believe that the moon is further away than the sun. This would mean it is impossible for a rocket to travel there in the time that was achieved in 1969. The only way they can maintain this belief is by convincing themselves that the moon landings are hoaxes.
Some people who subscribe to the christian brand of fundamentalism believe that life was created on our planet approximately 6000 years ago. To maintain this belief they have to reject vast amounts of knowledge.
One of the key scientific concepts they reject is evolution. Charles Darwin identified the two key drivers of evolution as being random variation and natural selection. Many anti-evolution propogandists like to ignore natural selection and incorrectly portray evolution as being a purely random process. By ignoring one of the key drivers, they seek to misrepresent the concept of evolution, transforming it into a concept which can be easily discredited.
For example, they argue that humans could not have evolved from far simpler organisms because the probability of a complex organism evolving from much simpler organisms by a purely random process is tiny. They may liken it to taking all the parts required to make a complex piece of machinery such as a watch, placing them in a box, shaking them around for a few minutes, and opening the box to find that the random motion of shaking has caused the pieces to fit themselves together in exactly the right way to produce a working watch. That is, their analogy uses only random variation and ignores natural selection.
This argument has been dished up with many different types of machinery from small items like watches to large items like airplanes. The example I'd like to concentrate on here is jigsaws, since these can be easily simulated on a computer, and we can give a clear demonstration that evolution requires both random variation and natural selection.
For ease of explanation, we'll start small. Take a square picture. Use a pair of scissors to cut through it both vertically and horizontally so that you have 4 smaller squares. You can easily reassemble them to reform the original picture. Place the solved puzzle on a larger piece of paper and draw a square around each of the 4 pieces. Now when you remove the pieces you have a 2 by 2 grid of squares.
Take the 4 pieces of your jigsaw and place them in line. For simplicity, we aren't going to allow any of the pieces to rotate, so in all the following discussion, take it as read that each of the pieces is at all times orientated correctly. We can generate more than enough complexity to demonstrate our argument without allowing for rotation.
A christian fundamentalist trying to discredit evolution might incorrectly argue that evolution operates as follows.
“Take the first piece of your jigsaw and place it at random on one of the four squares in your 2 by 2 grid. For example, you could number the four squares of the grid as 1 to 4, then roll a die until you get a number in the range 1 to 4 and place the jigsaw piece on the appropriately numbered square. Now take the second piece of the jigsaw and place it at random on one of the 3 empty squares in your grid. For example, roll the die until you get the number of one of the empty squares and place the piece on that square. Continue, placing the 3rd piece of the jigsaw at random on one of the 2 still empty squares of your grid. Then place the last piece on the only remaining empty square.
Now ask yourself whether you have produced the original picture. If you didn't, then the first trial of the experiment has failed. Take the 4 pieces off the grid and start over again. Keep doing the experiment over and over again until you manage to reproduce the original image. Even with only 4 pieces in the jigsaw, you'll probably find it takes many trials to solve the puzzle. Once we increase the number of pieces the chance of correctly solving the jigsaw by random placement is miniscule, so something as complex as a human couldn't have evolved.”
There are many factors that make this is an imperfect analogy. The factor I want to concentrate on here is that it incorrectly portrays evolution as being a totally random process and ignores the influence of natural selection. Natural selection ensures that useful variations are more likely to propogate into the next generation. So for our jigsaw model a better analogy is the following.
Place the 4 pieces of the jigsaw in a line. Then, for the first trial of the experiment, place them at random onto the 2 by 2 grid, one piece per square, using exactly the same process as described above. Now examine the result to see if you have succeeded in solving the jigsaw. If you have, you've finished. If you haven't, ask whether any individual pieces are in the correct position. Natural selection preserves useful variations, so here the useful analogy is that we will leave any correctly placed pieces on the grid. Only remove the pieces that were incorrectly placed. Now, for the 2nd trial, take the pieces that were placed incorrectly on the first trial and place them at random on the empty squares of the grid. Again, examine the result, remove any pieces still incorrectly placed. Repeat until the jigsaw is solved.
Note that this model is a better representation of evolution. The original experiment used only random placement, so it only contains one of the two drivers of evolution, random variation. The improved experiment combines random placement with the ability to leave any correctly placed pieces in their position, so it contains both required drivers of evolution, random variation and natural selection.
To get a feel for the numbers, it useful to run both versions of the experiment side by side. This is tedious by hand, so instead run the simulation. (Shockwave required. File size approx 140KB.)
The image used is a cropped version of a picture of the young Charles Darwin painted by George Richmond.
OK. You probably guessed that the second model, involving natural selection, would usually get to the solution faster that the first model, but perhaps the magnitude of the difference was a little surprising. The random placement model can finish before the model incorporating natural selection, but it happens pretty rarely.
For the random placement model, a little mathematics can show that on average the number of trials to complete the experiment is4 × 3 × 2 × 1 = 24
(In general, mathematicians write the product of the first n positive integers as n! and pronounce it “n factorial”. Here we have 4! = 24.)
We're not saying it always finishes in 24 trials. It's a random process so the number of trials varies. I've seen the simulation produce the correct picture on the first trial. I've also seen a run that took over 150 trials. What we're saying is, if you run the simulation a large number of times, and write down the number of trials it takes each time, the average (arithmetic mean for the pedants) of the numbers you write down will be around 24, and the more times you're willing to run the experiment, the closer to 24 the average is likely to be. So, while most readers won't have the mathematics required to prove the average number of trials is 24, by running the simulation several times, you can verify that a number around about 24 does appear plausible.
By contrast, on average the experiment incorporating natural selection takes only 4 trials to finish. Again, we're not saying it always takes 4 trials. I've seen it finish in one trial and I've seen it take as many as 25 trials. But if we run the experiment many times, noting the number of trials each time, the average (arithmetic mean) of those numbers will be about 4.
Note the large difference that natural selection makes. Without natural selction the average length of the experiment is 24 trials. With natural selection the average drops to 4 trials. Natural selection looks pretty powerful, doesn't it? Well, actually we still aren't getting a proper appreciation of its power, because we've only got a 4 piece jigsaw.
Those who have some skill with probability and/or combinatorics can delve a little deeper here. Others may prefer to skip this section.
Consider again the random placement model. When we place the 4 jigsaw pieces on the grid, the 1st piece can be placed in any of the 4 positions. For each such placement, the 2nd piece can be placed in any of the 3 remaining places. For each way of placing the first two pieces, the 3rd piece can be placed in any of the 2 remaining positions, and the final piece can be placed in the sole remaining position. Thus the number of different ways of placing the 4 pieces is
4 × 3 × 2 × 1 = 4! = 24
Our random placement ensures each of these 24 arrangements is equally likely to occur, so the probability of success on the first trial is 1/24.
We are dealing with independent Bernoulli trials, and so the number of trials till the first success is a geometric random variable with parameter 1/24 and hence its mean is 24.
Since the number of jigsaw pieces is only 4, students of probability should be able to thrash out the expected number of trials required for the natural selection model by a brute force recursive approach. As stated above, the answer is 4.
Hmm. 4 pieces. Expected number of trials to complete is 4. That's suspicious. You may already be wondering whether there is a pretty generalisation with a cute proof waiting in the wings. The pretty generalisation is there for the taking, but the cute takes a little work.
Try the same simulation with a 16 piece jigsaw. (Shockwave required. File size approx 140KB.) You might like to set it running in another window so you can come back here and read some more while it's running.
Our 4 by 4 jigsaw has 16 pieces. It turns out the expected number of trials for the experiment with natural selection to complete is 16.
So, how many trials did the random placement part of your simulation take to complete? What do you mean it hasn't finished yet?
The 2 by 2 jigsaw had only 4 pieces. We said the expected number of trials for the random placement experiment to complete was 24, this number being the product of the first 4 positive integers. The 4 by 4 jigsaw has 16 pieces. The expected number of trials for its random placement experiment to complete is
16×15×14×13×...×3×2×1
(The 3 dots are the mathematician's way of writing "I don't have the space and/or time to write the whole thing out, so just follow the obvious pattern.&rdquo) This is product of the first 16 positive integers, written 16! and pronounced 16 factorial. Any decent scientific calculator will tell you that 16! is approximately 2×1013, or about 20 trillion.
To get an idea how big this is we'd have to talk about massive things like North America's GDP. The trial count in the simulation will probably crash before the simulation finishes, because the number of trials is likely to exceed the largest integer the program can store. But if the program keeps running, your computer will probably die before the simulation finishes, but then, so will you. At the speed this simulation runs, the expected time to completion is around a million years. Factorials get very big very quickly, which is perhaps why mathematicians chose the exclamation mark to denote them.
I don't actually have a simulation of this one here. I'm just mentioning it as a significant case for anti-evolution propogandists. I've seen several anti-evolution articles which jubilantly announce something like "The chances of this evolution related thing happening are less than 1 in blah, where blah is a number so big that it is bigger than the number of particles in the universe."
If you want to fraudently produce such a number, the 8 by 8 jigsaw will do it for you. It has 64 pieces so the chance of success on the first trial is 1 in 64!, where 64! is about 1.2×1089, putting it above most estimates of the number of particles in the universe.
That is, in an 8 by 8 jigsaw, the expected number of trials for the random placement experiment to complete is about 1.2×1089. If you add in the natural selection effect, the expected number of trials to complete drops to 64. Perhaps that gives a better indication of the power of random variation and natural selection working together compared to random variation acting alone.
There are many factors that make the random placement jigsaw model of evolution an imperfect analogy.
For example, a jigsaw puzzle only has one solution, where in real life only an exceedingly egocentric person could argue that the evolution of humans is the only solution that matters. If there was some way to hit a rewind button to go back in history a billion years or so, give the world a bit of a shake to randomise things a little, and then hit the fast forward button to let it replay back to the present, then we would almost certainly find humans don't exist in the alternative reality we have created, but evolution will have produced a whole heap of lifeforms just as interesting, (unless somewhere in the process the dominant lifeform managed to fry the planet in a nuclear holocaust.)
Biologists could have a field day listing all the problems in the random placement jigsaw analogy. I'm not a biologist, so I won't. What fascinates me is that we could build a random placement model comparable to those propounded by many anti-evolution propogandists, and simply by making a small addition of natural selection we could reduce the scale of the resulting numbers from “out of this world” to “right before your eyes”.
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Last modified: 25 Dec 2009
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