December 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Decimal digits
by Henry Baker 21 Dec '10

21 Dec '10

Having exactly 5 toes would appear to be unimportant for running/walking, as shown by a large number of animals which don't have 5 toes. However, the human is pretty unique in its upright stance, so perhaps the jury is still out. There is a huge amount of interest currently in the evolution of human feet in the context of "barefoot running", as evidenced by the best-selling book "Born to Run". Based on the work by Daniel Lieberman at Harvard, the view of the human foot as a compromise forced by the descent from trees is being revised, and the newer view is that the present human foot is an exquisite result of upwards of 10 million years of evolution to adapt to long runs & long walks (perhaps averaging >20 miles per day). A side-effect of this research is the conclusion that -- for running on natural surfaces at least -- wearing shoes (particularly shoes with a built-up heel) is actually counterproductive & leads to poor running style & lots of injuries. http://www.barefootrunning.fas.harvard.edu/ Unlike many hoofed (hooved?) animals, humans have very sensitive feet, which enables them to instantaneously react to surface conditions while running. This is particularly important for bipedal locomotion, since humans don't have an extra set of legs to "fall back on" and the alternative to instantaneous reaction may be a face-plant. I've watched movies of horses & mules walking on narrow paths with treacherous footing, and these animals tend to rely on their extra feet to overcome a stumble. Separated toes appear not to be particularly useful for running/walking, and make them more prone to injury & frostbite (a relatively recent problem arising after leaving Africa). A comparison of chimp toes & human toes show the loss of individual grasping ability of human toes. Chimps & gorillas, on the other hand (pun intended), can't run/walk for long distances upright & utilize their longer arms to help in locomotion. However, as walking robots become more & more common, roboticists are learning more & more about the why's and how's of leg, ankle & foot design, and they may yet shed some light on the function of toes. At 06:35 PM 12/20/2010, Bill Gosper wrote: >5. The detailed topological equivalence yet functional inequivalence of > hands and feet needs to be explained. The present design of toes as ridiculous > parodies of fingers, right down to fourteen phalanges and nails, confers no > advantage over a simple flexible tab, like being born with socks on. (Except > for people without hands.)

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[math-fun] Decimal digits
by Bill Gosper 21 Dec '10

21 Dec '10

1. Most limb anomalies are from non-heritable fetal injuries. 2. The 5-digit design is older than the primates, and hard to change without breaking something, so 3. Heritable missing or extra digits are usually parts of syndromes featuring serious problems, e.g. heart, metabolic, or brain defects. 4. Exception: Some ectrodactylies, but then the actual limb defects are severe enough to impact reproductive probabilities. 5. The detailed topological equivalence yet functional inequivalence of hands and feet needs to be explained. The present design of toes as ridiculous parodies of fingers, right down to fourteen phalanges and nails, confers no advantage over a simple flexible tab, like being born with socks on. (Except for people without hands.) 6. The 14-phalanx, opposable thumb design was optimized over a long time by our tree-dwelling ancestors. Toes are just vestigial fingers, catagenetic to the point of diminishing returns. Aside: I've never seen convincing evidence that, even in those rare cases of complete duplication of a metacarpal ray and forearm machinery, extra fingers can move independently. (Except for depressing medical articles admitting that it's sometimes hard to decide which childish finger to amputate.) Is the infant brain plastic enough to build drivers for nonstandard hardware? Especially interesting would be those cases of "mirror hand" (dimelia. Very rare.) --rwg

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Re: [math-fun] Five-fingered hands
by Henry Baker 21 Dec '10

21 Dec '10

Are you telling me that picking slippery fruit may be a life&death issue for homo sapiens? Or that the Old Testament is correct: it takes a human hand to grasp the forbidden fruit? ;-) At 04:10 PM 12/20/2010, Fred lunnon wrote: >Well, I could have got this more wrong if I had tried --- but not by very much! >In fact, it's obviously impossible to grasp a slippery sphere securely at all: >there is no mechanism by which (freedom-3) torques around the centre can >be countered. As Adam observed, the remaining freedom-3 linear forces >are countered by 4 fingers. > >Thinking about it (late in the day), I've come to the conclusion that the 7-finger >minimum applies where the object is slippery but irregular, and actively >attempting to escape by generating wrenches / screws (e.g. via attachment >to an opponent). But this is all supposition, until somebody can lay hands >on a copy of Mishra et al (1987). > >Of course, a five-fingered primate hand also utilises the palm to assist in >grasping: so I wonder, to how many extra fingers might a palm be equivalent? > >Fred Lunnon > >On 12/20/10, Adam P. Goucher <apgoucher(a)gmx.com> wrote: >>> Fred Lunnon wrote: >>> >>> There's little light to be cast by screw theory here. Exactly the same analysis >>> applies as to bicycle spokes --- or rather cartwheel spokes, since fingers act >>> >>> as struts (in compression) rather than ties (in tension). Picking up a slippery >>> sphere requires (at least) 7 fingers. WFL >> >> What? Wouldn't four fingers, positioned at the vertices of a regular >> tetrahedron, suffice to hold a sphere in position? Or three, if you allow >> gravity to assist. >> >> Sincerely, >> >> Adam P. Goucher

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Re: [math-fun] Five-fingered hands
by Henry Baker 21 Dec '10

21 Dec '10

I keep asking my friends in the evo-devo (evolutionary development) community why 5-fingered hands and 5-toed feet in humans are so resistant to variation -- particularly relative to the variation found in many other traits. From time to time, there are a minute fraction of people who are born with extra fingers and/or toes, but this seems far less common than many other not-so-rare traits which are incredibly debilitating. Given the effort required during development to enforce this 5-digit constraint, there must be extremely powerful evolutionary factors involved. Darwin's Theory of Sexual Selection provides one possibility: that having exactly 5 digits/hand/foot is extremely sexy, but this is a cop-out. For some reason, humans tend to think that cartoon figures with only 4 digits (thumb + 3 fingers) are extremely cute. There must be some extremely important tasks for which are impossible with only 4 digits/hand, as well as other extremely important tasks which are impossible with more than 5 digits/hand. Even if the evolutionarily important tasks involved counting, what on earth is so important that 5 or 10 is just the right number? Are the digits used for counting the months during gestation? Perhaps the husband has to be back in 10 months from his latest war? If he comes back too soon or too late, terrible things happen? Perhaps certain important foods only achieve ripeness after 10 days, but become rotten in 11 days? At 09:33 AM 12/19/2010, Fred lunnon wrote: >Jon Selig tells me that the theory is familiar among the robotics community, >having been developed 20+ years ago in the somewhat different context of >the number of fingers required by a mechanical hand --- imagine trying to >grasp securely a light, slippery ball, without it touching the palm. With one >hand --- could be a good game for party forfeits in there! > >The reference he gives is > B. Mishra, J. T. Schwartz and M. Sharir, > "On the existence and synthesis of multifinger positive grips", > Algorithmica, Volume 2, Numbers 4, 541-558 (1987)

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Re: [math-fun] Five-fingered hands
by Henry Baker 20 Dec '10

20 Dec '10

Re: redundancy True, but then why not have 6,7 or 8? There must be a steep marginal cost associated with having 1 more finger, so that evolution calls a halt to digit inflation. According to some theories, every square inch of skin is mapped to a distinct area of the brain, where the fingertips get a much larger portion of the brain than would the back of the arm, for instance. Since brain area is very valuable, additional digits wouldn't be able to get their fair share of this area & would be at even greater risk for damage due to less feeling. I can't come up with a theory that explains why evolution places such a huge value on exactly 5 fingers, not 4 or 6. Evolution is happy to make huge mistakes about all kinds of endocrine & mitochondrial diseases, but not about the # of digits. There is a huge variation in body size & amount of body hair, but not in the # of digits. A conundrum. At 08:17 AM 12/20/2010, Torgerson, Mark D wrote: >Maybe the issue is about redundancy. > >Back when folks were using rocks and sticks to pound stuff, the loss of a digit or two had to be common. Up until even a few hundred years ago, when a finger got broke it had a good chance of being essentially useless thereafter. There was no chance of recovery of the digit if it were severed. > >So maybe the question isn't "Is X digits optimal for the tasks at hand?" But rather "How far off from optimal is X-1 or X-2?" > >Looking at it this way 5 digits makes some sense. 5 may be more than is needed and life can continue on just fine if we suddenly have 4. If we start with 4 and lose 2... Though, I know a guy who was born with just a thumb and a single other digit on each hand. He gets along just fine, but he does admit that having another digit or two would be nice. > >-----Original Message----- >From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker >Sent: Sunday, December 19, 2010 3:51 PM >To: Fred lunnon >Cc: math-fun >Subject: Re: [math-fun] Five-fingered hands > >I keep asking my friends in the evo-devo (evolutionary development) community why 5-fingered hands and 5-toed feet in humans are so resistant to variation -- particularly relative to the variation found in many other traits. From time to time, there are a minute fraction of people who are born with extra fingers and/or toes, but this seems far less common than many other not-so-rare traits which are incredibly debilitating. > >Given the effort required during development to enforce this 5-digit constraint, there must be extremely powerful evolutionary factors involved. > >Darwin's Theory of Sexual Selection provides one possibility: that having exactly 5 digits/hand/foot is extremely sexy, but this is a cop-out. For some reason, humans tend to think that cartoon figures with only 4 digits (thumb + 3 fingers) are extremely cute. > >There must be some extremely important tasks for which are impossible with only 4 digits/hand, as well as other extremely important tasks which are impossible with more than 5 digits/hand. > >Even if the evolutionarily important tasks involved counting, what on earth is so important that 5 or 10 is just the right number? Are the digits used for counting the months during gestation? Perhaps the husband has to be back in 10 months from his latest war? If he comes back too soon or too late, terrible things happen? > >Perhaps certain important foods only achieve ripeness after 10 days, but become rotten in 11 days? > >At 09:33 AM 12/19/2010, Fred lunnon wrote: >>Jon Selig tells me that the theory is familiar among the robotics >>community, having been developed 20+ years ago in the somewhat >>different context of the number of fingers required by a mechanical >>hand --- imagine trying to grasp securely a light, slippery ball, >>without it touching the palm. With one hand --- could be a good game for party forfeits in there! >> >>The reference he gives is >> B. Mishra, J. T. Schwartz and M. Sharir, >> "On the existence and synthesis of multifinger positive grips", >> Algorithmica, Volume 2, Numbers 4, 541-558 (1987)

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Re: [math-fun] q. re Farey-ish fractions with weighted mediants
by James Propp 20 Dec '10

20 Dec '10

Allan Wechsler wrote: >Aaaaaand I screwed up. > >123446556666677777787; I left out a(6)=6. My Mathematica code gives {0,1,2,3,3,5,4,4,5,5,5,5,5,6,6,6,6,6,6,7,6,7,7,7,7,7,7,8,7,7,7,8,8,7,8,8,8,9,8,8,8,9,8,8,8,8,8,9,8,8,9,9,9,10,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,9,9,9,9,10,10,9,9,9,10,10,10,10,10,10,10,9,11,10,10,10,10,10,11,10,10,10,10} as the earliest occurrences of the respective denominators (though mathematically speaking that initial "1" is a glitch and should really be a "0"). I can easily generate more terms if anyone wants me to, or you can do it yourself with my code: FLengthen[L_] := Module[{i, M}, M = Table[0, {2 Length[L]}]; M[[1]] = Numerator[L[[1]]]/(1 + Denominator[L[[1]]]); For[i = 1, i < Length[L], i++, M[[2 i]] = L[[i]]; M[[2 i + 1]] = (Numerator[L[[i]]] + Numerator[L[[i + 1]]])/(Denominator[L[[i]]] + Denominator[L[[i + 1]]])]; M[[2 Length[L]]] = L[[Length[L]]]; Return[M]] F[n_] := F[n] = If[n == 0, {1}, FLengthen[F[n - 1]]] FEarliest[k_] := Module[{i}, For[i = 0, Length[Denom[F[i], k]] == 0, i++]; Return[i]] To reconcile this with Allan's numbers, note that I call {0/0,1/1} the 0th row, {0/1,1/2,1/1} the 1st row, etc. My rationale for this choice is that the nth row has (2^n)+1 elements under my convention. Jim

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Re: [math-fun] Bicycle spokes update
by Henry Baker 20 Dec '10

20 Dec '10

I'd rather just download a PDF file. I hate being tethered to the Internet. Storage is cheap, thanks to Feynmann (lotz of room at the bottom). At 12:06 PM 12/19/2010, Fred lunnon wrote: >I've had 3 reports of success and 0 failures, so I guess it works OK. > >If anybody else wants to try the GoogleDocs route to vanity publishing, >it may save them some hair-tearing to bear in mind the following mantra: >to access all file-management functions, ignore the customary plethora >of disorganised buttons, along with the strangely deluded "help" file, >and instead just right-click for the neat, self-evident drop-down menu. > >WFL > >On 12/19/10, Fred lunnon <fred.lunnon(a)gmail.com> wrote: >> As an experiment, I've uploaded a PDF of "Six Screws for Seven Spokes" >> to GoogleDocs at (no linebreaks; just trips of the tongue, don't it?) >> >> https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B6QR93hq… >> >> I'd appreciate it if somebody would let me know whether this can (as promised) >> be downloaded directly from this link by Joe Public --- and even more if I can >> have some feedback about the content!

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[math-fun] Signed adjectives
by Bill Gosper 20 Dec '10

20 Dec '10

Most(?) human languages have positive, comparative, and superlative, corresponding to 1, 2, and infinity. But we clearly need forms for 0, -1, -2, and -infinity. (Provisionally: inoperative, depreciative, pejorative, and besmirchative.) Also adverbial forms. Tongue-in-cheeklier, --rwg

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[math-fun] GoogleDocs (was Bicycle spokes update)
by N. J. A. Sloane 19 Dec '10

19 Dec '10

Does anyone know how long GoogleDocs will store your document for? Is this, finally, a solution to my Eternal Home Page suggestion (see http://www2.research.att.com/~njas/doc/eternal.html) ? Neil

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[math-fun] Bicycle spokes update
by Fred lunnon 19 Dec '10

19 Dec '10

As an experiment, I've uploaded a PDF of "Six Screws for Seven Spokes" to GoogleDocs at (no linebreaks; just trips of the tongue, don't it?) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B6QR93hq… I'd appreciate it if somebody would let me know whether this can (as promised) be downloaded directly from this link by Joe Public --- and even more if I can have some feedback about the content! Jon Selig tells me that the theory is familiar among the robotics community, having been developed 20+ years ago in the somewhat different context of the number of fingers required by a mechanical hand --- imagine trying to grasp securely a light, slippery ball, without it touching the palm. With one hand --- could be a good game for party forfeits in there! The reference he gives is B. Mishra, J. T. Schwartz and M. Sharir, "On the existence and synthesis of multifinger positive grips", Algorithmica, Volume 2, Numbers 4, 541-558 (1987) Fred Lunnon

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