Evolution and Probability

Probabilistic thinking is pervasive in evolutionary theory. It’s not a bad thing, just something that needs to be acknowledged and appropriately handled.

Denial

Some go so far as to deny it, but in my experience these people are ideologues. These are critics of ID who complain about the lack of any numbers being attached to the probability arguments of ID proponents, and their denial is perhaps rooted in their fear of a tu quoque.

Where are their own probability calculations?

Incredulity

Another reason for their denial could be that they also love to accuse ID proponents of making arguments from incredulity, while being unwilling to face up to the fact that they are guilty of the same thing. Does evolutionary theory depend on arguments from incredulity? Almost certainly.

Take for example the idea that all extant life shares a common ancestor. It is based upon the idea that it is simply too implausible that life should arise more than once and yet share common features such as the genetic code.

We can be very sure there really is a single concestor of all surviving life forms on this planet. The evidence is that all that have ever been examined share (exactly in most cases, almost exactly in the rest) the same genetic code; and the genetic code is too detailed, in arbitrary aspects of its complexity, to have been invented twice.

Dawkins, Richard. The Ancestor’s Tale: A Pilgrimage to the Dawn of Evolution


An argument from incredulity.

Probabilities are Important

The importance of probablity in evolutionary thinking might best be seen in the following text:

If there are versions of the evolution theory that deny slow gradualism, and deny the central role of natural selection, they may be true in particular cases. But they cannot be the whole truth, for they deny the very heart of the evolution theory, which gives it the power to dissolve astronomical improbabilities and explain prodigies of apparent miracle.

Dawkins, Richard. The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design

Independence of Events

While having said all this, I’d like to focus on the idea that evolutionary events are not independent and that this somehow rescues evolutionary theory from being guilty of appealing to vastly improbable outcomes, aka miracles.

Consider a toss of the dice in a game of craps. The odds of double six is 1/36. Sure, we can roll a single die twice, and the odds of a six on each roll is now only 1/6 vastly more likely to occur by chance (not really). 1/6 x 1/6 is still 1/36. The probabilities are multiplicative because the events are independent events. The fact that if you have two dice and you roll the first die until you get a six and then you keep that (by cumulative selection) and then roll the second die until you get the second six and now you have two sixes doesn’t change the probabilities one whit. Doesn’t that demonstrate that cumulative selection is helpless in reducing probabilities?

Well, you might say, you need to roll BOTH dice until only one of them shows a six and then keep that one and then roll the second die. But what in evolution is analogous to that?

Sure, if you roll two dice trying to roll a six you have a better chance of a six showing on one of the two dice than if you roll just one. It’s like rolling one die twice in an attempt to get a six rather than just once. Of course the probability of the second six would still be 1/6. But why aren’t justified in adding a third die after our first six is rolled so that once again we are trying to get a six from two dice and not just one? And doesn’t this again demonstrate that it is not cumulative selection at all that is responsible for the reduction in probability but rather the number of trials we allot each attempt to roll a six?

Closing

The fundamental question is why aren’t evolutionary events independent and thus multiplicative?

The secondary question is what is the true role of cumulative selection in reduction from the miraculous to the mere appearance of the miraculous?

291 Replies to “Evolution and Probability”

  1. Mung Mung
    Ignored
    says:

    Rumraket: It’s at generation 42 now, and it’s an octagon, essentially the best approximation of a slightly off-balance wheel the simulation allows.

    It somehow knows in advance what a wheel should be like? Essentialist then.

  2. Mung Mung
    Ignored
    says:

    Rumraket: And yet we know there are no targets in the BoxCar2D simulation. And yet better and better cars incrementally evolve, where after something like 100 generations we end up with cars that work very well. And we can then go and look at the car chromosome of some car that results from 100 generations of evolution in the simulation, and see that it is some extremely unlikely combination. So how did it evolve without being a target?

    How do you know there are no targets? The target is not a “better and better” car? And yet, rather magically, better and better cars incrementally evolve.

    How do you explain that? Magic?

  3. Mung Mung
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    says:

    dazz: I’m sure you can make up some shit about the weasel and language, but that doesn’t change the fact that it was NOT about coming up with a meaningful sentence, let alone one in english in particular.

    Sad really, that Dawkins chose that phrase. He could have chosen gibberish. And then explained how cumulative selection makes gibberish more probable. That’s a great way to sell Darwinian evolution.

  4. Mung Mung
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    says:

    Alan Fox: Regarding “Weasel”, the program Dawkins came up with nearly 40 years ago is not considered by anyone, including Dawkins, as anything other than a simple demonstration of the difference between cumulative selection against random selection.

    Random selection of what? Do people really think that Creationism is based on random selection of targets?

  5. dazz dazz
    Ignored
    says:

    Mung: Sad really, that Dawkins chose that phrase. He could have chosen gibberish. And then explained how cumulative selection makes gibberish more probable. That’s a great way to sell Darwinian evolution.

    Gibberish like this:

    MLSVRVAAAVVRALPRRAGLVSRNALGSSFIAARNFHASNTHLQKTGTAEMSSILEERIL
    GADTSVDLEETGRVLSIGDGIARVHGLRNVQAEEMVEFSSGLKGMSLNLEPDNVGVVVFG
    NDKLIKEGDIVKRTGAIVDVPVGEELLGRVVDALGNAIDGKGPIGSKTRRRVGLKAPGII
    PRISVREPMQTGIKAVDSLVPIGRGQRELIIGDRQTGKTSIAIDTIINQKRFNDGSDEKK
    KLYCIYVAIGQKRSTVAQLVKRLTDADAMKYTIVVSATASDAAPLQYLAPYSGCSMGEYF
    RDNGKHALIIYDDLSKQAVAYRQMSLLLRRPPGREAYPGDVFYLHSRLLERAAKMNDAFG
    GGSLTALPVIETQAGDVSAYIPTNVISITDGQIFLETELFYKGIRPAINVGLSVSRVGSA
    AQTRAMKQVAGTMKLELAQYREVAAFAQFGSDLDAATQQLLSRGVRLTELLKQGQYSPMA
    IEEQVAVIYAGVRGYLDKLEPSKITKFENAFLSHVVSQHQALLGTIRADGKISEQSDAKL
    KEIVTNFLAGFEA

  6. Mung Mung
    Ignored
    says:

    dazz: Gibberish like this:

    Exactly! So why did Dawkins choose Shakespeare? Rhetoric.

  7. Mung Mung
    Ignored
    says:

    dazz: You could even tweak the weasel algo to work with made up symbols, or even pictures.

    You are correct, but so what.

    If you care to have any chance at all of “finding” the target phrase you’d better use the same symbol set for both. That’s design.

  8. Mung Mung
    Ignored
    says:

    Rumraket: He has completely misunderstood what you and I have been arguing about.

    Probably true. 🙂

    But possibly irrelevant.

  9. Mung Mung
    Ignored
    says:

    Joe Felsenstein: As someone who has worked for years on theoretical population genetics, which is mostly about changes within species, I am very unimpressed by this restriction on what is “evolution”.

    What is “evolution” in theoretical population genetics?

  10. Mung Mung
    Ignored
    says:

    Rumraket: My claim is that the BoxCar2D simulation proves this claim wrong, in that the cars and the chromosomes that encode them, are analogous to proteins and the genes that encode them.

    So there is no taxonomy of proteins, and there is no taxonomy of genes, and there is no taxonomy of cars, and taxonomy and classification only applies to species? And there are no species of proteins, and no species of genes, and no species of cars. And protein “families” are not analogous to the “family” in taxonomy?

    I’m just wondering why the same reasoning that is applies to species does not apply to genes and proteins. Is it because they do not reproduce?

    How about box cars? They don’t reproduce so they don’t really evolve?

  11. colewd
    Ignored
    says:

    Neil Rickert,

    But here we see colewd demanding a mechanistic materialistic account.

    I am saying that you need a model to back up your claim. You are the one who keeps repeating science builds models.

  12. Rumraket Rumraket
    Ignored
    says:

    Mung: It somehow knows in advance what a wheel should be like?

    No.

  13. Rumraket Rumraket
    Ignored
    says:

    Mung: How do you know there are no targets?

    By being able to read. The algorithm is described in detail.

    The target is not a “better and better” car?

    No, there is no target. There is a selection criterion of course, according to which the performance of different phenotypes are evaluated (ability to move to the right). This has the effect of producing better and better cars as generations pass, but no particular car is a target.
    (Even better, if you enable “roulette wheel selection” and disable “elite selection”, cars are not picked deterministically by their performance (tournament selection ensures the “best” car of every generation is always picked for mating, which is rather unrealistic for natural selection), instead roulette wheel selection merely increases the odds of being picked for mating, with the odds being in proportion to car performance. This much more realistically mimics natural selection, in which even the most fit individual can still be unlucky and find themselves in the wrong position at the wrong time).

    This is why I’m claiming the car is like a protein (and the car’s chromosome is like the gene encoding the protein), and not like a whole organism. The car’s can’t reproduce themselves, they are reproduced imperfectly by the algorithm, in the same way protein coding genes are reproduced when cellular organisms reproduce.

    So the selection criterion (move to the right) is here analogous to the function of a protein, and it’s effect on the reproductive success of the host organism.

    I would not claim the BoxCar2D simulation is like life in all crucial aspects relevant to evolution. It isn’t. But it is close enough in the aspects for which I have employed it in my arguments.

    And yet, rather magically, better and better cars incrementally evolve.

    How do you explain that? Magic?

    I’m astonished that I’m being called to even explain it, we can sit here and see it with our own eyes. It’s random mutation and selection.

  14. Rumraket Rumraket
    Ignored
    says:

    Mung: Rumraket: My claim is that the BoxCar2D simulation proves this claim wrong, in that the cars and the chromosomes that encode them, are analogous to proteins and the genes that encode them.

    Mung: So there is no taxonomy of proteins, and there is no taxonomy of genes, and there is no taxonomy of cars, and taxonomy and classification only applies to species?

    Why do you think that follows?

    And there are no species of proteins

    I suppose that’s really just a matter of semantics. You can call different proteins species if you really want, Mung. Have at it.

    The terminology for protein classification is already heavily inspired and influenced by cladistics and taxonomy, and proteins are sorted into families and superfamilies, a database of which you can find here:
    https://pfam.xfam.org/

    If you’re actually interested, here’s something about methods for protein classification:
    III. Classification Of Proteins By Patterns Of Tertiary Structure

    and no species of genes, and no species of cars.

    You can call them species if that floats your boat Mung. I’m totally okay with it.

    And protein “families” are not analogous to the “family” in taxonomy?

    I think they are in many respects.

    But Mung, YOU were the one that said:

    There’s no speciation? So no “On the origin of species of cars“? So really, no evolution?

    The implication being is that, if we don’t call it speciation, or if it doesn’t result in something we (or you?) would recognize as speciation, it wouldn’t qualify as evolution in your view.

    It’s not clear what your argument here is even about. It has that distinct feel of being spawned for no other reason that you felt like being argumentative just for the hell of it. What is your problem?

    I’m just wondering why the same reasoning that is applies to species does not apply to genes and proteins. Is it because they do not reproduce?

    Proteins normally don’t reproduce themselves (though exceptions are known), but they still evolve, because the genes that encode them are reproduced, and they exhibit descent with modification.

    So I would argue that the same reasoning actually does apply to genes and proteins.

    How about box cars? They don’t reproduce so they don’t really evolve?

    Boxcars, and the chromosomes that encode them, are reproduced by the algorithm, and they exhibit descent with modification, just like genes that encode proteins are reproduced by the organisms that host those genes, and so those genes also exhibit descent with modification.

    So I claim they really do evolve.

  15. Rumraket Rumraket
    Ignored
    says:

    Mung: Do people really think that Creationism is based on random selection of targets?

    No, I think his point was that some creationists think that evolution is based on random selection of targets.

  16. Allan Miller
    Ignored
    says:

    Mung: Anyone specific in mind?

    All of ’em … or, anyone advocating the notion, complete with hokey probability calculations, that the space is not navigable without tinkering. Which is all of ’em, round here at least.

  17. phoodoo
    Ignored
    says:

    Mung: Rumraket: And yet we know there are no targets in the BoxCar2D simulation.

    Anyone who read this and believed it, must be part of a simulation.

    Maybe he meant BESIDES the target of making a fast car that can jump?

    Are the cars allowed to morph into fat people that sit around eating bugles and watching Mr. Bean? Maybe that’s better?

    Why oh why does Rumraket write such horseshit? Any simulation can solve that?

  18. OMagain
    Ignored
    says:

    phoodoo: Maybe he meant BESIDES the target of making a fast car that can jump?

    Sounds quite like the target of “staying alive”. Quite a general one that, and many ways to achieve it. Likewise, the target of boxcar can be approached in a very large number of ways.

    phoodoo: Are the cars allowed to morph into fat people that sit around eating bugles and watching Mr. Bean? Maybe that’s better?

    Does smearing shit on everything make you feel better?

  19. phoodoo
    Ignored
    says:

    OMagain: Sounds quite like the target of “staying alive”.

    So there is a target then?

    If the target of evolution staying alive? That’s interesting, I thought the staying alive and reproducing in evolution was just a happenstance, a fluke that continues, I didn’t realize it was an actual target.

  20. OMagain
    Ignored
    says:

    phoodoo: So there is a target then?

    There’s very little point having a conversation with you. You have your “thing”, your “whipped cream” knee jerk response and you are sticking with it.

    phoodoo: If the target of evolution staying alive?

    I could assume you meant “is” not “if” as if makes no sense. But then again, that’s not necessarily an unwarranted assumption with you. So I’ll go with what is most likely and just assume it’s more nonsense.

    phoodoo: That’s interesting, I thought the staying alive and reproducing in evolution was just a happenstance, a fluke that continues, I didn’t realize it was an actual target.

    Is that what you thought? Given that things have been alive now for quite some time it appears your understanding of probability is lacking. Given that fundamental lack there’s probably no point in attempting to point out your misunderstandings. After all, you nurse those misunderstandings, they are what fuels you. You don’t want to lose them. They are what makes you stand out from the pack.

    Water flows downhill because it’s target is to reach the sea.

  21. colewd
    Ignored
    says:

    Rumraket,

    Boxcars, and the chromosomes that encode them, are reproduced by the algorithm, and they exhibit descent with modification, just like genes that encode proteins are reproduced by the organisms that host those genes, and so those genes also exhibit descent with modification.

    In the boxcar case you dealing with vectors that become shapes by connecting the ends of vectors with a line and then attaching wheels. It’s not a sequence with only a 30% chance of forming a structured shape for every 6 amino acids in the sequence.

    The problem for biological evolution is finding function inside a string with almost infinite possible arrangements. This is functional information. Vectors and circles are not.

  22. Rumraket Rumraket
    Ignored
    says:

    phoodoo: Anyone who read this and believed it, must be part of a simulation.

    Maybe he meant BESIDES the target of making a fast car that can jump?

    Are the cars allowed to morph into fat people that sit around eating bugles and watching Mr. Bean? Maybe that’s better?

    The cars aren’t like real organisms, they can’t change beyond the limitations of the algorithm, they’re more like individual proteins which also can’t become anything other than strings of amino acids.

  23. Rumraket Rumraket
    Ignored
    says:

    colewd: It’s not a sequence with only a 30% chance of forming a structured shape for every 6 amino acids in the sequence.

    It’s not clear what this means.

    The problem for biological evolution is finding function inside a string with almost infinite possible arrangements. This is functional information. Vectors and circles are not.

    The string of symbols that make up the car’s genome forms a space of possibilities similar in size to large proteins. The number of ways the triangles can vary in size and shape, combined with the number of ways the circles can be attached, is incredibly large.

    There’s no reason to think this isn’t “functional information” yet simultaneously claim that proteins are.

    There are some limitations to how much the wheels and triangles can change. But of course the same is true for proteins, that can normally only choose from 20 different amino acids, which are “set” in their properties, so the proteins can only change the arrangement of them. The capacity and behavior of the proteins in turn owe to that arrangement of amino acids, in the same way the capacity and behavior of the car in turn owe to the arrangement and properties of the individual wheels and triangles.

  24. OMagain
    Ignored
    says:

    colewd: This is functional information. Vectors and circles are not.

    And yet only a small fraction of the genome-strings produce entities capable of movement.

  25. OMagain
    Ignored
    says:

    Rumraket: The cars aren’t like real organisms, they can’t change beyond the limitations of the algorithm, they’re more like individual proteins which also can’t become anything other than strings of amino acids.

    On another thread:

    phoodoo: Its Alan analogy, did you not get that part?

    Seems somebody has a selective understanding of analogy, when it suits them.

  26. colewd
    Ignored
    says:

    Rumraket,

    The string of symbols that make up the car’s genome forms a space of possibilities similar in size to large proteins. The number of ways the triangles can vary in size and shape, combined with the number of ways the circles can be attached, is incredibly large.

    There’s no reason to think this isn’t “functional information” yet simultaneously claim that proteins are.

    This is clearly false and now I understand why we disconnect sometimes. The combinatorial world leaves the physical world at about 10^80. A number that we could call enormous as it describes the number of atoms in the known universe but in terms of combinatorial mathematics it is relatively tiny.
    Functional information is in the form of strings which lives in the world described by combinatorial mathematical quantities that physical objects don’t. For instance the vector has to have a minimum finite length or it would be impossible to calculate.

    It also has to have a finite number of angles or again you cannot draw the vector within the length of time of our universe if you assign as many possible angles as exist in a 30 unit string with 50 possible symbols.

    Proteins live in the world described by combinatorial mathematics which is a major challenge for evolution that has been defined by Darwin and others. This is why I think Eric’s proof of information non growth may end up being right and can be ultimately reconciled to functional information as functional information and mutual information share a common theme. They both live in the world described by combinatorial mathematics.

  27. Rumraket Rumraket
    Ignored
    says:

    colewd: This is clearly false and now I understand why we disconnect sometimes. The combinatorial world leaves the physical world at about 10^80.

    Wat? That statement makes no sense. The “combinatorial world” leaves the “physical world” at about 10^80. What the hell does that mean? I’m pretty sure that is nonsense.

    A number that we could call enormous as it describes the number of atoms in the known universe but in terms of combinatorial mathematics it is relatively tiny. Functional information is in the form of strings which lives in the world described by combinatorial mathematical quantities that physical objects don’t.

    I have no idea what this word salad is supposed to convey.

    For instance the vector has to have a minimum finite length or it would be impossible to calculate.

    It also has to have a finite number of angles or again you cannot draw the vector within the length of time of our universe if you assign as many possible angles as exist in a 30 unit string with 50 possible symbols.

    The space of possible vectors allowed by the algorithm isn’t infinitely large, no. There is a limitation to how small the incremental changes in the vectors are allowed to take. It is not infinite. Sure.

    If one of the triangles have an angle of 36.02311085 degrees, it cannot change by an arbitrarily small number. There is a limitation to the precision allowed by the program yes, otherwise it would be incalculable.

    But it also isn’t infinitely large for proteins. There are “only” 20 amino acids, and the length of the string string of amino acids. So they are not different in that respect. They are both limited to some finite space of possible configurations. Proteins will fold into particular shapes according to primary and secondary structures, and the environment they find themselves in, they won’t accept any and all possible configurations. They cannot be changed in arbitrarily small increments. The amino acids that determine the properties of the protein do not come in an infinite number of varieties. There are “only” 20 of them.

    Proteins live in the world described by combinatorial mathematics which is a major challenge for evolution that has been defined by Darwin and others.

    Again this makes no sense.

    If you mean to say that the vectors of the cars cannot be changed in arbitrarily small increments drawn from an infinite number of possibilities, then I would agree. But neither can proteins.

    How would you alter the angle between two overlapping betasheets by an increment of 10^-10^-40 of one degree? It’s not like you can put in an amino acid that is 10^-10^-40 picometers smaller than glycine, for example.

    There are many possible arrangements of amino acids, but there aren’t THAT many. The possibilites aren’t INFINITE for proteins of a finite length. And an infinitely long protein can’t be synthesized anyway, as it would collapse into a black hole long before. So your excuse here doesn’t work.

    This is why I think Eric’s proof of information non growth may end up being right and can be ultimately reconciled to functional information as functional information and mutual information share a common theme.

    You are way too impressed by this meaningless information numerology and it’s really a shame you can’t see through it for the mere smoke and mirrors it is.

    They both live in the world described by combinatorial mathematics.

    No, they really don’t. Bill you’re spouting gibberish, and drawing invalid conclusions from irrelevant factoids (and I agree with you it is true the possiblities in the BoxCar2D simulation aren’t infinite, but neither are they infinite for proteins for reasons explained).

  28. colewd
    Ignored
    says:

    Rumraket,

    If you mean to say that the vectors of the cars cannot be changed in arbitrarily small increments drawn from an infinite number of possibilities, then I would agree. But neither can proteins.

    In the case the number of vectors is fixed at 8. The type of vectors is fixed by length and angle. If there are 100 different lengths and 100 different angles that is 10^32 possible combinations of shapes. This is indeed a very large number but minuscule compared to a 400AA protein or a flagellum that is 30000 AAs.

    If like biology you had to build the wheel with extremely small vector shapes (cells) that fit together and were randomly generated the program would never build a wheel within a practical number of tries. The exact same goes for the triangle shapes if they were generated from a large number of smaller shapes.

    At the end of the day this simulation is interesting but dramatically simpler then a simulation of real biological evolutionary model. It also does not demonstrate that natural selection can generate biological functional information.

  29. OMagain
    Ignored
    says:

    colewd: If like biology you had to build the wheel with extremely small vector shapes (cells) that fit together and were randomly generated the program would never build a wheel within a practical number of tries.

    What biological process are you describing there?

  30. colewd
    Ignored
    says:

    OMagain,

    What biological process are you describing there?

    Embryo development; particularly cellular differentiation and body plan development.

  31. Rumraket Rumraket
    Ignored
    says:

    colewd: In the case the number of vectors is fixed at 8. The type of vectors is fixed by length and angle. If there are 100 different lengths and 100 different angles that is 10^32 possible combinations of shapes.

    Unfortunately it isn’t clear from the description of the algorithm what the actual limit of precision is but it’s at least 10^-4 (which would increase the space of combinations to at least 10^64). I don’t know if the author is using the listed email any more but I wrote him asking about the precision of the simulation, let’s see if he gets back.

    If like biology you had to build the wheel with extremely small vector shapes (cells) that fit together and were randomly generated the program would never build a wheel within a practical number of tries.

    Ironically it is the limitation to only 8 triangles that prevent the program from evolving a truly round wheel. But even with a population size of 20, it evolves an octagon in about 25 generations.

  32. phoodoo
    Ignored
    says:

    OMagain: On another thread:

    Seems somebody has a selective understanding of analogy, when it suits them.

    No sorry Omagain, this makes no sense at all. By pointing out to Rumraket that it was Alan’s analogy that life is not so improbable if some “slots” are more likely than others, I am in no way whatsoever abdicating my future opportunities to point out the flaws in other analogies.

    Why you would make such a nonsensical connection is beyond me, and you as well I am sure.

  33. Mung Mung
    Ignored
    says:

    Rumraket: Ironically it is the limitation to only 8 triangles that prevent the program from evolving a truly round wheel. But even with a population size of 20, it evolves an octagon in about 25 generations.

    I don’t understand why you think that is in any way remarkable.

  34. Rumraket Rumraket
    Ignored
    says:

    Mung: I don’t understand why you think that is in any way remarkable.

    I believe you.

    Just to make sure I understand, you take it as an obvious given that if there’s selection for movement to the right, wheels should necessarily follow?

  35. Mung Mung
    Ignored
    says:

    Rumraket: Just to make sure I understand, you take it as an obvious given that if there’s selection for movement to the right, wheels should necessarily follow?

    A triangle is a wheel.

  36. Mung Mung
    Ignored
    says:

    Rumraket: This can be debunked with a simple simulation. In fact the BoxCar2D simulation disproves the claim.

    =======

    The program learns to build a car using a genetic algorithm. It starts with a population of 20 randomly generated shapes with wheels and runs each one to see how far it goes.

    Amazing! Remarkable!

  37. DNA_Jock
    Ignored
    says:

    Mung,
    The wheel-like octagons show up when the parameters are altered to make conventional, circular wheels disallowed.
    Hope this helps you understand.

  38. Rumraket Rumraket
    Ignored
    says:

    Mung: Amazing! Remarkable!

    You missed the post where I described that the closest approximation of a wheel that the algorithm allows, evolves when wheels are disabled?

    Here’s a recording from generation 59-60 BoxCar2D octagonal “wheel”. Unfortunately the algorithm allows a maximum of 8 triangles, so it can’t get any more “round” than this.

  39. newton
    Ignored
    says:

    Mung: A triangle is a wheel.

    Sure


    The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.
    A Reuleaux triangle [ʁœlo] is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question “Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?”

  40. phoodoo
    Ignored
    says:

    Rumraket: Ironically it is the limitation to only 8 triangles that prevent the program from evolving a truly round wheel. But even with a population size of 20, it evolves an octagon in about 25 generations.

    And it never evolves an elephants trunk. More amazing.

    And just imagine, it has no target!

  41. Mung Mung
    Ignored
    says:

    phoodoo: And it never evolves an elephants trunk.

    Despite its usefulness in locomotion.

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