January 2015 - mathfun@mailman.xmission.com

[math-fun] USA women & men, height and weight, recommended & actual, plots
by Warren D Smith 25 Jan '15

25 Jan '15

> So what do you reckon is the best-fit exponent e in the equation wt = c*ht^e , based on these data? WFL --no idea. And depends on your goal, are you trying to fit the population, or trying to fit the min-mortality weights (which are quite different)? I think it is much more clear what e should be -- namely e = 2.5 to 2.8 -- if we are fitting the growth curves for children ages 2-19. I do not understand why that e happens, but it does. For adults, damned if I know what is going on. Anyhow here are the formulas: children both genders, ages 2-19, fitting to median weight in population for each height: WeightInPounds = 0.002384649 * MedianHeightInInches^2.593828 (Korevaar) Men ages 25-59, my best fit to mortality-minning weights: WeightInPounds = 99.9 + 8.63*10^(-7) * HeightInches^4.23 Women ages 25-59, my best fit to mortality-minning weights: WeightInPounds = 50.8 + 0.00459 * HeightInches^2.347

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[math-fun] USA women & men, height and weight, recommended & actual, plots
by Warren D Smith 25 Jan '15

25 Jan '15

The following pictures https://dl.dropboxusercontent.com/u/3507527/WomenHgtWgt6.png https://dl.dropboxusercontent.com/u/3507527/MenHgtWgt2.png were produced by me (Warren D. Smith) 2015, with help from Prof. Daniel Kaplan. One shows women, the other men, age range 25-59. Explanation of each picture: The "x" in both pictures shows the mean height and mean weight for the entire US adult population (all ages>19, during years 2007-2010) according to the CDC. The two thick black curves show the weight range, according to Metropolitan Life Insurance Co mortality data (1943, revised 1983, and I have removed their asinine adjustments for clothing weight and shoe height) you want to be in to minimize your mortality. Note, for very tall or short women/men, MetLife did not have enough data, which is why these curves stop. The thin black curve shows a mathematical fit by me, to the geometric mean of the lower & upper MetLife curves; it can be used in an effort to try to figure out the "ideal weights" for especially tall or short people. The tall thin boxes show the optimum height to be (minimizing mortality) which according to USA height distribution data at different ages examined by me, is 5'5"-5'6" for men, and 5'2"-5'3" for women. Finally, what about the bluish/purplish cloud? For every women age 25-59 (or man) examined by the NHANES 1999-2004 national survey measuring some 31000 people, their (height,weight) is plotted using a little circle. Each picture presumably contains about 10000 such little circles in all. People who were off image (mainly, off the top, since too obese) are omitted. This gives you some idea of the distribution of heights & weights in the USA age 25-59 population. As you can see, the US population is mostly quite overweight, and only something like 10% of the population lies within the MetLife healthiest-weight-range between the two thick black curves. Steven Halls MD has disputed the MetLife weight recommendations, feeling they are "unrealistic" especially for tall & short people. But I dispute his disputation, because from the scatterplot it seems the MetLife range (and its extrapolation to taller & shorter) is attainable. I have also collected published height & weight data about especially tall and excellent athletes up to 7'4" (who presumably are atypically healthy) and found they straddle the thin black curves. This also suggests my thin black curve is not "unrealistic" as a health goal, even for very tall people. For very short people, I have not been able to get much data, so perhaps my MetLife-extrapolating fit curves might be unrealistic for them, I do not know. I found a few very short famous athletes (Bill Shoemaker & Julie Krone, two of the best jockeys ever) who both were lighter than my curve. --Warren D. Smith http://rangevoting.org

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[math-fun] Flakey flash memory cards
by Henry Baker 23 Jan '15

23 Jan '15

(This question is also relevant to certain robots on Mars.) I bought a number of questionable flash memory cards (formatted as FAT32), and I wasn't disappointed with their questionableness -- they're failing, so I've been trying to figure out their failure modes w/o having access to their internal management software. They are showing bizarre failure modes. The files don't seem to be failing in a single block, but the failures seem to be more global. For example, a folder will cease being a folder, and look like a regular file, while the files in the folder can be recovered by file checking software. I see almost no failures (= modifications) on files less than 100Mbytes, but nearly 100% failures on files >200Mbytes. Has anyone else here studied flash memory card (uSD card) failures? Is there any software on the internet that can pinpoint failures so that I can simply remove these blocks from consideration? (I realize that it's cheaper to buy new uSD cards, but I thought it would be fun to understand the failures.)

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[math-fun] Human body shape, size, mass
by Warren D Smith 23 Jan '15

23 Jan '15

Tall people are *not* just scaled up versions of short people. Hence, human body weight does *not* scale proportionally to height^3. The so-called BMI (body mass index) suggested by health and fitness scumballs, would indicate that the mass of "healthy weight" adults should scale proportionally to height^2. For children of both genders, median heights are related to median weights at all ages 2-19 by the following empirical law found by Nick Korevaar http://www.math.utah.edu/~korevaar/ACCESS2003/bmi.pdf for median height and median weight USA boys and girls ages 2-19: MedianWeightInPounds = 0.002384649 * MedianHeightInInches^2.593828 fits extremely well. So for example I'm 68.5 inches = 174 cm so if I were age 19 and median among both genders, I "should" be weight 137.7 pounds = 0.002384649 * 68.5^2.593828 = 62.5 kg. Unfortunately I'm considerably older than 19 and currently weigh 155 lb = 70.3 kg. In metric, Korevaar's formula is MedianWeightInKg = 14.848467 * MedianHeightInMeters^2.593828. Other fitters using other data sets for people below age 20 in different countries have found p's throughout the interval [2.59, 2.93]. Why the exponent p=2.59383? Korevaar's formula also would predict (at least naively) that waist circumference should be proportional to Height^q where q=(p-1)/2=0.79691. And empirical fits to data for people below age 20 indeed have found q's throughout the interval 0.68<q<0.82. FAILURE #1: If a human were shaped like a height=H, radius=R cylinder of constant material, and wished to just-survive "thin column Euler buckling" under a load proportional to her weight and hence to H*R^2, then H*R^2 = const * R^4/H^2 hence R is proportional to H^1.5, hence q=1.5. FAILURE #2: Same, but human sprints at stride duration proportional to height^(1/2) [ideal pendulum] and hence ground speed also proportional to height^(1/2) if constant leg angle. This agrees with the fact the fastest sprinters are tall. Usain Bolt is 6'5". Each stride the force exerted scales like weight*height/height^(1/2) = H^(3/2)*R^2, not weight = H*R^2. Hence redoing the Euler buckling argument we find H^(3/2)*R^2 = const * R^4/H^2 so H^(7/2) propto R^2 hence q=7/4=1.75. Well... that only made it worse. I think your body is in no danger of Euler buckling so those two models were irrelevant. We need truly limiting factors for human bodies. FAILURE #3: You generate heat presumably proportional to your mass, which must be dumped thru your surface. Hence you want your surface area which behaves like H*R to scale like your mass H*R^2, hence R=constant, hence q=0. FAILURE #4: Actually, your body heat does not scale proportionally to your mass, if you believe animals have smaller metabolic rates. In fact animal's heat production scales like mass^(3/4) under a fractal-blood-vessel-network model, see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2936637, which agrees with empirical data on animals. So redoing that we find H*R scales like (H*R^2)^(3/4) hence H^(1/4) = R^(1/2) hence R=H^0.5 hence q=0.5. That is better but still too low, outside the observed interval 0.68<q<0.82 for humans. This q agrees with the "body mass index" kludge though. FAILURE #5: To prevent your knees and hips breaking, you need their cross sectional area R^2 to be proportional to your weight H*R^2 (stationary standing) or to H^(3/2)*R^2 (sprinting model). Both would imply H=constant. Wrong. FAILURE #6: To prevent your pelvis breaking, modeling pelvis as "lintel" type beam of length R, thickness also proportional to R (i.e. pelvis assumed to have constant shape and constant material properties), you need your moment-load = weight*R to scale like R^4 (stationary stand) implying q=1/2 (wrong, same as in fail#4 and for BMI) -- or instead... SUCCESS ON TRY #7: ...with the sprinter load model, H^(3/2)*R^2 = R^4 for q=3/4=0.75. Hey, that finally agrees superbly with the empirical data for humans under age 20, which says 0.68<q<0.82. And this q=3/4 would predict p=2.5. OK, so we finally were able to predict the correct power law q for humans in the growing-age range under a model where the limiting factor is the risk of your pelvis breaking when you sprint. This also explains why women are shorter than men. For birth, women need to have wider pelvises. Given that is so, they are forced under our model to be shorter than men. Now once you are an adult, you stop growing and your pelvis and height stay invariant. But your weight and waist circumference need not stay fixed. The "optimum" weight is the one at which (among people your height & gender) mortality rate is minimized. Optimum weights were computed from mortality statistics by Metropolitan Life Insurance co in 1943 with revision 1983. See http://www.ncbi.nlm.nih.gov/pubmed/6707396 http://www.ncbi.nlm.nih.gov/pubmed/8332284 http://www.ncbi.nlm.nih.gov/pubmed/6362514 Here are the metlife tables, where you are warned they are expected to be noisy at especially tall or short heights since less data there. MORTALITY-MINIMIZING WEIGHTS (lb) FOR WOMEN AGES 25-59, HEIGHT SMALL, MEDIUM, LARGE FRAMES 4'9" 102-111 109-121 118-131 4'10" 103-113 111-123 120-134 4'11" 104-115 113-126 122-137 5'0" 106-118 115-129 125-140 5'1" 108-121 118-132 128-143 5'2" 111-124 121-135 131-147 5'3" 114-127 124-138 134-151 5'4" 117-130 127-141 137-155 5'5" 120-133 130-144 140-159 5'6" 123-136 133-147 143-163 5'7" 126-139 136-150 146-167 5'8" 129-142 139-153 149-170 5'9" 132-145 142-156 152-173 5'10" 135-148 145-159 155-176 5'11" 138-151 148-162 158-179 All weights overstated to include 3lb of clothing. MORTALITY-MINIMIZING WEIGHTS (lb) FOR MEN AGES 25-59, HEIGHT SMALL, MEDIUM, LARGE FRAMES 5'1" 128-134 131-141 138-150 5'2" 130-136 133-143 140-153 5'3" 132-138 135-145 142-156 5'4" 134-140 137-148 144-160 5'5" 136-142 139-151 146-164 5'6" 138-145 142-154 149-168 5'7" 140-148 145-157 152-172 5'8" 142-151 148-160 155-176 5'9" 144-154 151-163 158-180 5'10" 146-157 154-166 161-184 5'11" 149-160 157-170 164-188 6'0" 152-164 160-174 168-192 6'1" 155-168 164-178 172-197 6'2" 158-172 167-182 176-202 6'3" 162-176 171-187 181-207 All weights overstated to include 3lb of clothing. Oh wonderful, it seems I am in the min-mortality weight range for my height (assuming medium frame). Metlife says you can assess your frame size by elbow measurement: "Bend forearm upward at a 90 degree angle. Keep fingers straight and turn the inside of your wrist toward your body. Place thumb and index finger of other hand on the two prominent bones on either side of the elbow. Measure space between your fingers on a ruler.(A physician would use a caliper.) Compare with tables below listing elbow measurements in inches for medium-framed men and women. Measurements lower than those listed indicate small frame. Higher measurements indicate large frame." ELBOW MEASUREMENTS FOR MEDIUM FRAME MenHeight ElbowBreadth WomenHeight ElbowBreadth 5'1"-5'2" 2.5-2.875 4'9"-4'10" 2.25-2.5 5'3"-5'6" 2.625-2.825 4'11"-5'2" 2.25-2.5 5'7"-5'10" 2.75-3.0 5'3"-5'6" 2.375-2.625 5'11"-6'2" 2.75-3.125 5'7"-5'10" 2.275-2.625 6'3" 2.825-3.25 5'11" 2.5-2.75 I get 2.8 inches so I'm medium frame. Fitting the medium-frame men optimum weights (now with the 3lb overstatement removed) to a function of height, I find by log-linear-fit of the data (61,log(133)), (63,log(137)), (64,log(139.5)), (65,log(142)), (67,log(148)), (68,log(151)), (70,log(157)), (72,log(164)), (73,log(168)), (75,log(175.5)) that WeightLb = 0.4881 * HeightInch^1.3605 by which formula my ideal weight is allegedly 153.5 lb. Wow, I'm only 1.5lb off! A good linear fit, but it seems slightly worse than the above power law, is WeightLb = 2.93*HeightInch - 45.6. Fitting the medium frame women optimum weights (now with the 3lb overstatement removed) to a function of height, I find a linear fit works very well, WeightLb = 2.93*HeightInch - 55.9 which conveniently has the exact same slope as for the men. Plots: https://dl.dropboxusercontent.com/u/3507527/IdealHumanWeights.png You can't really deduce a good power law from the metife adult data because it spans only a small domain of heights, unlike the ages 2-19 data, so I don't see how to deduce the right exponent without having a huge error bar. Now, you can't control your height (well... certain people apparently are feeding their children HGH to make them taller... which as we'll see seems to be a really stupid idea) but nevertheless we can be envious of those whose heights are optimal. The "optimum" height for your gender is the one that produces the longest lifespan. As far as I can tell from these tables http://rangevoting.org/HeightWeightUSA2008.pdf the optimal heights are about MEN: 5'5" to 5'6" (about 4 inches below median among USA men ages 20-30) WOMEN: 5'2" to 5'3" (about 2 inches below median for 20s-ages USA women).

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[math-fun] The Paradoxes That Threaten To Tear Modern Cosmology Apart
by Warren D Smith 22 Jan '15

22 Jan '15

The claim by Baker's source "What's curious about this expansion is that space, and the vacuum associated with it, must somehow be created in this process...What's more, there is an energy associated with any given volume of the universe. If that volume increases, the inescapable conclusion is that this energy must increase as well. And yet physicists generally think that energy creation is forbidden..." is garbage. In Einstein GR(with Lambda) no energy is ever created or destroyed, it is locally conserved everywhere. There either is no such thing as a total energy of the universe or if is one then it is conserved. No paradox, Einstein theory said so already in 1915 or so. Re the idea mentioned by Meeker that we can resolve the huge discrepancy between the cosmical constant Lambda and the Planck energy density by using "holographic surface principle" somehow... it sounds dubious too. Imagine the universe is finite. As it expands, its total energy stays fixed even though its volume and "surface" expand. This contradicts this "resolution." If we ask not about its total energy but merely about its "cosmical constant" component of the energy, then that scales like the volume, not surface. Hmm. Fortunately in de Sitter nonEuclidean geometry volume and surface can scale essentially the same, at least if you interpret right... so still surviving... this resolution might lead to quantum field theory yielding nonconstant "mass of electron" & suchlike which depend logarithmically on size of universe... which might be compatible with observation... so perhaps there is some hope for this crazy suggested resolution. What are you going to do if the universe is spatially infinite?

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[math-fun] The Paradoxes That Threaten To Tear Modern Cosmology Apart
by Henry Baker 22 Jan '15

22 Jan '15

FYI -- https://medium.com/the-physics-arxiv-blog/the-paradoxes-that-threaten-to-te… The Paradoxes That Threaten To Tear Modern Cosmology Apart Some simple observations about the universe seem to contradict basic physics. Solving these paradoxes could change the way we think about the cosmos Revolutions in science often come from the study of seemingly unresolvable paradoxes. An intense focus on these paradoxes, and their eventual resolution, is a process that has leads to many important breakthroughs. So an interesting exercise is to list the paradoxes associated with current ideas in science. ItÂs just possible that these paradoxes will lead to the next generation of ideas about the universe. Today, Yurij Baryshev at St Petersburg State University in Russia does just this with modern cosmology. The result is a list of paradoxes associated with well-established ideas and observations about the structure and origin of the universe. Perhaps the most dramatic, and potentially most important, of these paradoxes comes from the idea that the universe is expanding, one of the great successes of modern cosmology. It is based on a number of different observations. The first is that other galaxies are all moving away from us. The evidence for this is that light from these galaxies is red-shifted. And the greater the distance, the bigger this red-shift. Astrophysicists interpret this as evidence that more distant galaxies are travelling away from us more quickly. Indeed, the most recent evidence is that the expansion is accelerating. WhatÂs curious about this expansion is that space, and the vacuum associated with it, must somehow be created in this process. And yet how this can occur is not at all clear. The creation of space is a new cosmological phenomenon, which has not been tested yet in physical laboratory,Â says Baryshev. WhatÂs more, there is an energy associated with any given volume of the universe. If that volume increases, the inescapable conclusion is that this energy must increase as well. And yet physicists generally think that energy creation is forbidden. Baryshev quotes the British cosmologist, Ted Harrison, on this topic: ÂThe conclusion, whether we like it or not, is obvious: energy in the universe is not conserved,Â says Harrison. This is a problem that cosmologists are well aware of. And yet ask them about it and they shuffle their feet and stare at the ground. Clearly, any theorist who can solve this paradox will have a bright future in cosmology. The nature of the energy associated with the vacuum is another puzzle. This is variously called the zero point energy or the energy of the Planck vacuum and quantum physicists have spent some time attempting to calculate it. These calculations suggest that the energy density of the vacuum is huge, of the order of 10^94 g/cm^3. This energy, being equivalent to mass, ought to have a gravitational effect on the universe. Cosmologists have looked for this gravitational effect and calculated its value from their observations (they call it the cosmological constant). These calculations suggest that the energy density of the vacuum is about 10^-29 g/cm3. Those numbers are difficult to reconcile. Indeed, they differ by 120 orders of magnitude. How and why this discrepancy arises is not known and is the cause of much bemused embarrassment among cosmologists. Then there is the cosmological red-shift itself, which is another mystery. Physicists often talk about the red-shift as a kind of Doppler effect, like the change in frequency of a police siren as it passes by. The Doppler effect arises from the relative movement of different objects. But the cosmological red-shift is different because galaxies are stationary in space. Instead, it is space itself that cosmologists think is expanding. The mathematics that describes these effects is correspondingly different as well, not least because any relative velocity must always be less than the speed of light in conventional physics. And yet the velocity of expanding space can take any value. Interestingly, the nature of the cosmological red-shift leads to the possibility of observational tests in the next few years. One interesting idea is that the red-shifts of distant objects must increase as they get further away. For a distant quasar, this change may be as much as one centimetre per second per year, something that may be observable with the next generation of extremely large telescopes. One final paradox is also worth mentioning. This comes from one of the fundamental assumptions behind EinsteinÂs theory of general relativityÂ-that if you look at the universe on a large enough scale, it must be the same in all directions. It seems clear that this assumption of homogeneity does not hold on the local scale. Our galaxy is part of a cluster known as the Local Group which is itself part of a bigger supercluster. This suggests a kind of fractal structure to the universe. In other words, the universe is made up of clusters regardless of the scale at which you look at it. The problem with this is that it contradicts one of the basic ideas of modern cosmologyÂ-the Hubble law. This is the observation that the cosmological red-shift of an object is linearly proportional to its distance from Earth. It is so profoundly embedded in modern cosmology that most currently accepted theories of universal expansion depend on its linear nature. ThatÂs all okay if the universe is homogeneous (and therefore linear) on the largest scales. But the evidence is paradoxical. Astrophysicists have measured the linear nature of the Hubble law at distances of a few hundred megaparsecs. And yet the clusters visible on those scales indicate the universe is not homogeneous on the scales. And so the argument that the Hubble lawÂs linearity is a result of the homogeneity of the universe (or vice versa) does not stand up to scrutiny. Once again this is an embarrassing failure for modern cosmology. It is sometimes tempting to think that astrophysicists have cosmology more or less sewn up, that the Big Bang model, and all that it implies, accounts for everything we see in the cosmos. Not even close. Cosmologists may have successfully papered over the cracks in their theories in a way that keeps scientists happy for the time being. This sense of success is surely an illusion. And that is how it should be. If scientists really think they are coming close to a final and complete description of reality, then a simple list of paradoxes can do a remarkable job of putting feet firmly back on the ground. Ref: arxiv.org/abs/1501.01919 : Paradoxes Of Cosmological Physics In The Beginning Of The 21-St Century http://arxiv.org/abs/1501.01919

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[math-fun] Human body shape, size, mass
by Warren D Smith 22 Jan '15

22 Jan '15

> H.Baker: Fascinating! If you're going to include mortality, shouldn't you include the data suggesting that near-starvation ("CR") diets may improve longevity? I.e., I'm 6'0" and 165#, but I've been told by CR fanatics that I'd live a lot longer if I weighed 130-135# instead. (I might also run faster & climb hills on my bike faster). --well, you probably won't live longer at 130-135#. The MetLife tables say you want to be 157-171 lb for minimum mortality. (The running and biking, maybe.) There is the slight escape hatch about correlation is not causality. For example, maybe if you are richer, you live longer, but then you also are fatter... whereas really, you'd live longer if rich & thin, so the metlife tables are deceptive. (I just made all that up, of course.) I would be interested to see if anybody else can cook up yet more hokey models that succeed/fail in predicting the correct exponents. Perhaps some other thermal model will do it, for example; I did not look into that literature much.

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Re: [math-fun] Human body shape, size, mass
by Henry Baker 22 Jan '15

22 Jan '15

Fascinating! If you're going to include mortality, shouldn't you include the data suggesting that near-starvation ("CR") diets may improve longevity? I.e., I'm 6'0" and 165#, but I've been told by CR fanatics that I'd live a lot longer if I weighed 130-135# instead. (I might also run faster & climb hills on my bike faster). Of course, I haven't been that light since junior high school, so it's getting a little late for such a drastic change. At 01:08 PM 1/22/2015, Warren D Smith wrote: >Tall people are *not* just scaled up versions of short people. >Hence, human body weight does *not* scale proportionally to >height^3. > >The so-called BMI (body mass index) suggested by health and fitness >scumballs, would indicate that the mass of "healthy weight" adults >should scale proportionally to height^2. > >For children of both genders, median heights are related to median >weights at all >ages 2-19 by the following empirical law found by Nick Korevaar > http://www.math.utah.edu/~korevaar/ACCESS2003/bmi.pdf >for median height and median weight USA boys and girls ages 2-19: > MedianWeightInPounds = 0.002384649 * MedianHeightInInches^2.593828 >fits extremely well. So for example I'm 68.5 inches = 174 cm >so if I were age 19 and median among both genders, I "should" be weight > 137.7 pounds = 0.002384649 * 68.5^2.593828 = 62.5 kg. >Unfortunately I'm considerably older than 19 and currently >weigh 155 lb = 70.3 kg. >In metric, Korevaar's formula is > MedianWeightInKg = 14.848467 * MedianHeightInMeters^2.593828. >Other fitters using other data sets for people below age 20 in >different countries have found p's throughout the interval [2.59, >2.93]. > >Why the exponent p=2.59383? >Korevaar's formula also would predict (at least naively) that waist >circumference >should be proportional to Height^q where q=(p-1)/2=0.79691. >And empirical fits to data for people below age 20 indeed have found >q's throughout the interval 0.68<q<0.82. > >FAILURE #1: >If a human were shaped like a height=H, radius=R cylinder of constant material, >and wished to just-survive "thin column Euler buckling" >under a load proportional to her weight >and hence to H*R^2, then >H*R^2 = const * R^4/H^2 >hence R is proportional to H^1.5, hence q=1.5. > >FAILURE #2: >Same, but human sprints at stride duration proportional to height^(1/2) >[ideal pendulum] and hence ground speed also proportional to height^(1/2) >if constant leg angle. >This agrees with the fact the fastest sprinters are tall. Usain Bolt is 6'5". >Each stride the force exerted scales like >weight*height/height^(1/2) = H^(3/2)*R^2, >not weight = H*R^2. >Hence redoing the Euler buckling argument we find > H^(3/2)*R^2 = const * R^4/H^2 >so H^(7/2) propto R^2 >hence q=7/4=1.75. Well... that only made it worse. >I think your body is in no danger of Euler buckling so those two >models were irrelevant. >We need truly limiting factors for human bodies. > >FAILURE #3: >You generate heat presumably proportional to your mass, which must be >dumped thru your surface. Hence you want your surface area which >behaves like H*R >to scale like your mass H*R^2, hence R=constant, hence q=0. > >FAILURE #4: >Actually, your body heat does not scale proportionally to your mass, >if you believe >animals have smaller metabolic rates. In fact animal's heat >production scales like >mass^(3/4) under a fractal-blood-vessel-network model, see > http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2936637, >which agrees with empirical data on animals. So redoing that we find >H*R scales like (H*R^2)^(3/4) hence H^(1/4) = R^(1/2) hence R=H^0.5 >hence q=0.5. That is better but still too low, outside the observed >interval 0.68<q<0.82 for humans. This q agrees with the "body mass >index" kludge though. > >FAILURE #5: >To prevent your knees and hips breaking, you need their cross sectional area R^2 >to be proportional to your weight H*R^2 (stationary standing) or to H^(3/2)*R^2 >(sprinting model). Both would imply H=constant. Wrong. > >FAILURE #6: >To prevent your pelvis breaking, modeling pelvis as "lintel" type >beam of length R, thickness also proportional to R >(i.e. pelvis assumed to have constant shape and constant material properties), >you need your moment-load = weight*R to scale like R^4 (stationary stand) >implying q=1/2 (wrong, same as in fail#4 and for BMI) -- or instead... > >SUCCESS ON TRY #7: >...with the sprinter load model, >H^(3/2)*R^2 = R^4 for q=3/4=0.75. >Hey, that finally agrees superbly with the empirical data for humans >under age 20, >which says 0.68<q<0.82. >And this q=3/4 would predict p=2.5. > >OK, so we finally were able to predict the correct power law q for >humans in the >growing-age range under a model where the >limiting factor is the risk of your pelvis breaking when you sprint. > >This also explains why women are shorter than men. For birth, women >need to have wider pelvises. Given that is so, they are forced under >our model to be shorter than men. > >Now once you are an adult, you stop growing and your pelvis and height >stay invariant. >But your weight and waist circumference need not stay fixed. >The "optimum" weight is the one at which (among people your height & gender) >mortality rate is minimized. Optimum weights were computed from >mortality statistics by Metropolitan Life Insurance co in 1943 with >revision 1983. >See > http://www.ncbi.nlm.nih.gov/pubmed/6707396 > http://www.ncbi.nlm.nih.gov/pubmed/8332284 > http://www.ncbi.nlm.nih.gov/pubmed/6362514 >Here are the metlife tables, where you are warned they are expected to be noisy >at especially tall or short heights since less data there. >MORTALITY-MINIMIZING WEIGHTS (lb) FOR WOMEN AGES 25-59, >HEIGHT SMALL, MEDIUM, LARGE FRAMES >4'9" 102-111 109-121 118-131 >4'10" 103-113 111-123 120-134 >4'11" 104-115 113-126 122-137 >5'0" 106-118 115-129 125-140 >5'1" 108-121 118-132 128-143 >5'2" 111-124 121-135 131-147 >5'3" 114-127 124-138 134-151 >5'4" 117-130 127-141 137-155 >5'5" 120-133 130-144 140-159 >5'6" 123-136 133-147 143-163 >5'7" 126-139 136-150 146-167 >5'8" 129-142 139-153 149-170 >5'9" 132-145 142-156 152-173 >5'10" 135-148 145-159 155-176 >5'11" 138-151 148-162 158-179 >All weights overstated to include 3lb of clothing. > >MORTALITY-MINIMIZING WEIGHTS (lb) FOR MEN AGES 25-59, >HEIGHT SMALL, MEDIUM, LARGE FRAMES >5'1" 128-134 131-141 138-150 >5'2" 130-136 133-143 140-153 >5'3" 132-138 135-145 142-156 >5'4" 134-140 137-148 144-160 >5'5" 136-142 139-151 146-164 >5'6" 138-145 142-154 149-168 >5'7" 140-148 145-157 152-172 >5'8" 142-151 148-160 155-176 >5'9" 144-154 151-163 158-180 >5'10" 146-157 154-166 161-184 >5'11" 149-160 157-170 164-188 >6'0" 152-164 160-174 168-192 >6'1" 155-168 164-178 172-197 >6'2" 158-172 167-182 176-202 >6'3" 162-176 171-187 181-207 >All weights overstated to include 3lb of clothing. > >Oh wonderful, it seems I am in the min-mortality weight range for my height >(assuming medium frame). Metlife says you can assess your frame size by elbow >measurement: > >"Bend forearm upward at a 90 degree angle. Keep fingers straight and >turn the inside of your wrist toward your body. Place thumb and index >finger of other hand on the two prominent bones on either side of the >elbow. Measure space between your fingers on a ruler.(A physician >would use a caliper.) Compare with tables below listing elbow >measurements in inches for medium-framed men and women. Measurements >lower than those listed indicate small frame. Higher measurements >indicate large frame." >ELBOW MEASUREMENTS FOR MEDIUM FRAME >MenHeight ElbowBreadth WomenHeight ElbowBreadth >5'1"-5'2" 2.5-2.875 4'9"-4'10" 2.25-2.5 >5'3"-5'6" 2.625-2.825 4'11"-5'2" 2.25-2.5 >5'7"-5'10" 2.75-3.0 5'3"-5'6" 2.375-2.625 >5'11"-6'2" 2.75-3.125 5'7"-5'10" 2.275-2.625 >6'3" 2.825-3.25 5'11" 2.5-2.75 >I get 2.8 inches so I'm medium frame. > >Fitting the medium-frame men optimum weights (now with the 3lb >overstatement removed) to a function of height, I find by >log-linear-fit of the data > (61,log(133)), (63,log(137)), (64,log(139.5)), (65,log(142)), (67,log(148)), > (68,log(151)), (70,log(157)), (72,log(164)), (73,log(168)), (75,log(175.5)) >that > WeightLb = 0.4881 * HeightInch^1.3605 >by which formula my ideal weight is allegedly 153.5 lb. Wow, I'm only >1.5lb off! >A good linear fit, but it seems slightly worse than the above power law, is > WeightLb = 2.93*HeightInch - 45.6. > >Fitting the medium frame women optimum weights (now with the 3lb >overstatement removed) to a function of height, I find a linear fit >works very well, > WeightLb = 2.93*HeightInch - 55.9 >which conveniently has the exact same slope as for the men. > >Plots: https://dl.dropboxusercontent.com/u/3507527/IdealHumanWeights.png > >You can't really deduce a good power law from the metife adult data >because it spans only a small domain of heights, unlike the ages 2-19 >data, so I don't see how to deduce >the right exponent without having a huge error bar. > >Now, you can't control your height (well... certain people apparently are >feeding their children HGH to make them taller... which as we'll see >seems to be a really stupid idea) but nevertheless we can be envious >of those whose heights are optimal. The "optimum" height for your >gender is the one that produces the longest lifespan. As far as I can >tell from these tables > http://rangevoting.org/HeightWeightUSA2008.pdf >the optimal heights are about > MEN: 5'5" to 5'6" (about 4 inches below median among USA men ages 20-30) > WOMEN: 5'2" to 5'3" (about 2 inches below median for 20s-ages USA women).

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[math-fun] Galois-conjugate fractals
by James Propp 21 Jan '15

21 Jan '15

Does anyone know of two fractals of respective fractal dimension log alpha and log beta such that alpha and beta are conjugate algebraic numbers? I'd be curious to see whether they have any family resemblance owing to their common algebraic origin. Jim Propp

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[math-fun] Puzzle
by David Wilson 21 Jan '15

21 Jan '15

Choose any real r >= 0. Starting with n = 1, on the first step add r, on subsequent steps either add r or take the reciprocal as you choose. For example, if r = 1/4, we can generate the sequence 1, 5/4, 3/2, 7/4, 2, 1/2, 3/4, 1. For which r is it possible to return to 1 as does this sequence?

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