September 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Golden Ratio discovered in uterus
by Bill Gosper 09 Sep '12

09 Sep '12

Here's a piecewise linear spiral approximation to a nautilus (shown me by Jack Holloway): http://www.amazon.com/The-Irrationals-Story-Numbers-Count/dp/0691143420/ref… --rwg Also, quarter-circles (in "four point ellipses") do interesting things<http://gosper.org/pumpstill.gif> . ----- Hi Jim, Yes, there is more to be said, but I had to use that curve (consisting of quarter circles) because the point I wanted to make is specifically about that curve. The curve is only a ruler-and-compass approximation to an exponential spiral and the differences may be important in some contexts. But it is a much larger error to claim that curve is a model for the nautilus, and that big error is continuously being copied and re-published. E.g., they both appear on the cover of Livio's popular book The Golden Ratio, and inside, his text about them confusingly does not clarify the difference. As to the curve not being a "true spiral" I don't know of a standard definition of "true spiral" that includes the many types of spiral that people often study, but doesn't include that assemblage of quarter circles. As far as I am concerned, it is a "true spiral" that happens not to be C2 continuous. Georgehttp://georgehart.com/ On 8/15/2012 11:02 PM, James Propp wrote:>>From 0:20 to 0:40 in his video, George shows a famous picture of the> "golden spiral". But as many of you probably know (no doubt including> George), this picture is bogus. Specifically, it conflates two different> curves: a curve made up of quarter-circular arcs (which isn't a true> spiral) and a curve made of true spiral arcs (which isn't tangent to the> rectangles).>> George doesn't discuss this in his video, since minor issues like this> would distract from his main point. But still, I wish the video hadn't> used the inaccurate picture.>> Yes, I'm being a bit picky. At times like this I am reminded of a motto I> saw on a tee-shirt once: "I'm not pompous. I'm pedantic. There's a> difference." :-)>> Jim>> On Tue, Aug 14, 2012 at 4:22 PM, George Hart <george(a)georgehart.com <http://gosper.org/webmail/src/compose.php?send_to=george%40georgehart.com>> wrote:>>> Hi Alex,>>>> Unfortunately, as a consequence of this being hosted by the Guardian, I>> expect all the folks who uncritically see the golden ratio everywhere will>> now be blathering about golden uteri. I've started making a series of math>> videos and here's one rebuttal to the golden ratio cult:>>>> http://www.youtube.com/watch?**v=_gxC8OjoQkQ<http://www.youtube.com/watch?v=_gxC8OjoQkQ>>>>> As to the image on that page, I hope you don't mind if I interpret your>> use of the middle finger as an editorial comment. I feel it's misleading>> to show precise metal calipers and print 1.618, as if finger measurements>> were precise enough to support four significant digits.>>>> George>> http://georgehart.com/>>>>>>>>>> On 8/14/2012 10:23 AM, Alex Bellos wrote:>>>>> I've just launched a maths blog for the Guardian>>>>>> Here's my first post: the research is serious, but any conclusions need>>> to be taken with a medium to large dosage of salt.>>>>>> http://www.guardian.co.uk/**science/alexs-adventures-in-**>>> numberland/2012/aug/14/golden-**ratio-uterus<http://www.guardian.co.uk/science/alexs-adventures-in-numberland/2012/aug/1…>>>>>>> alex>>> ______________________________**_________________

3 8

[math-fun] No more posts on 'Junk DNA'
by rcs＠xmission.com 09 Sep '12

09 Sep '12

I am by fiat hereby declaring that JUNK DNA IS OFF TOPIC for the Math-Fun list for 30 days, until October 8. Somehow, this topic brings out a lurking nastiness in our posters that is not awakened by everted spheres and truncated tetrahedra. If you are in doubt, don't post it. Run it by me first. Rich Schroeppel Moderator, Math-fun list rcs(a)xmission.com

2 1

[math-fun] Junk DNA allegedly isn't. Big change in standard picture of life.
by Warren Smith 09 Sep '12

09 Sep '12

> From: Henry Baker <hbaker1(a)pipeline.com> > Re biological Rube Goldberg: > I disagree vehemently. The more I learn of DNA, RNA, proteins, signalling, > etc., the more impressed I am. > For example, the process of mere _copying_ of DNA operates very close to the > thermodynamic limit -- i.e., it would be very difficult to design _any_ > system for copying DNA that produced less waste heat than our own DNA > copying. --Exactly wrong. DNA copying is done with rechecking and with use of ATP->AMP not ADP so it can get extra certainty. In other words it intentionally uses both extra energy and extra chemical steps. Many biochemical reactions in fact ARE reversible (e.g. fructose <--> glucose interconversion) and hence consume zero energy. But DNA copying is not one of them. > If our DNA copying were any less thermodynamically efficient, and we > continued to copy at the same rate, our dividing cells would fry themselves > like a fried egg. --utter bull. DNA copying is a very small expenditure in human cells compared to other stuff. > Most evolutionary biologists will tell you that _none_ of the DNA is truly > "junk". The proof: if it weren't important, it would relatively quickly > disappear after a few generations. --complete bull. DNA actually is expensive in bacteria and viruses, and they have little or no junk. But in eukaryotes like us, DNA is a tiny expenditure relatively speaking so we can afford lots of junk. By the way, you just "proved" microsoft windows is small and compact, not bloated. > Several biologists have proposed that complex organisms have lots of DNA as > a form of "diagonalization": all of its siblings with less DNA didn't > survive. This is akin to comparing the power of Turing Machines with more > or less tape: those with "more" tape can simulate _all_ Turing Machines with > "less" tape and do something different. --you are totally confused. (And if you are right re "several biologists", they are confused too.) > For example, some of our DNA encodes experiences with viruses, some of which > may be long extinct or at least dormant. Yet if that virus suddenly shows > up, there are at least some fraction of humans that will survive. --some of our DNA encodes viruses that incorporated themselves into our DNA and are now carried along for the ride for free inside us, where they are incurable since part of us. Further, some our DNA is "self replicating elements" which in fact multiply WITHIN our DNA; they are "out for themselves" not you. Somewhat like cancer but at a smaller size and longer time scale. This is (I suspect) the main reason closely related species have way different amounts of junk and that much of our junk is repeating sequences. > Plants (which typically don't move) seem to require more DNA than animals > (which do move). One can speculate that plants have to sit there and take > every insult, while animals can avoid the insult by moving and even > emigrating when necessary. --plants do a lot more chemistry than animals, so they need more genes (which describe chemistry). You can't synthesize many amino acids, fats, and vitamins you need because you lack the genes. Plants do not have that luxury. Animals can't do photosynthesis, even though it'd clearly be beneficial if they could. --In this post I am repeating the usual bio- philosophy & refuting weird myths posted by HB. The alleged new "it aint junk" philosophy to some extent must overthrow the old bio philosophy, but I'd like to know what is wrong with the old arguments, because they sure mostly are not wrong.

9 19

Re: [math-fun] junk DNA, mouse zap experiments
by Henry Baker 09 Sep '12

09 Sep '12

I think an appropriate model for non-junk/junk status would be a long-tailed distribution, where the "non-junkness" of some DNA portion never gets to zero, and life is always a little bit better with a little bit more of the distribution (i.e., the area under the curve approaches infinity). These long-tailed distributions have been heavily discussed in relation to retailers like Amazon, which can have 1 million book titles in their "inventory", whereas a local retail bookstore might have perhaps 20,000 titles (I'm guessing as to the exact number, but they certainly don't have 1 million). Yes, there is human life when the number of books available is only 100 or 1000, but life is much richer with 100,000 or 1 million books available. So, although Sturgeon is "correct", when he says that 95% of everything is crap, he's discounting the value of having 20x the number of books to choose from. Another long-tailed distribution is that of patents in the patent system. If we're being charitable, we assume that every patent is valid, and that there is a perceived use for every patent. However, the technology for some patents "masks" the need for other patents, because so long as patent A technology is available, it may be strictly better than patent B, so patent B will see _zero_ use. However, if for some reason the technology in patent A becomes non-viable -- e.g., it depends upon some material which has suddenly become unavailable, patent B receives all of the attention and resources that were previously given to patent A. Whether patent B was "junk" before becomes merely a matter of definition. At 06:50 PM 9/8/2012, you wrote: >>Further it seems that this test has been done, though removing only a >portion of the junk DNA: >http://www.ncbi.nlm.nih.gov/pubmed/15496924 > >--that was interesting. If the New York Timesy claim that our 3 >billion base-pairs of >"junk DNA" is filled with a "switchboard" with 4 million switches... which would >be on average one switch per 750 base pairs... then this experiment where they >removed a 1.5 Mbp and an 0.85 MBp region from junk mouse DNA, then the >resulting mice grew up and seemed to be fine... would have removed an >expected number of >over 2000+1000 of those "switches." > >Kind of like if you randomly zapped two swaths of 2000 and 1000 transistors >from an early computer with 4 million in all. Would that leave the >computer apparently >working fine? Even if those computer designers had tried to design it to >be robust against faults (which they mostly did not), still, this amount >of fault tolerance would be pretty impressive. > >So the 4M switches in our DNA if there, must either be very >nonuniformly distributed, or very unimportant, or part of an extremely >fault-tolerant circuit. > >If, in contrast (to compare with the genes in DNA, traditionally >regarded as non-junk), you randomly zapped one gene per thousand in >your set of about 25K genes (total zappage: 25 of your genes) then >what? Would you be fine?

1 0

[math-fun] junk DNA, mouse zap experiments
by Warren Smith 09 Sep '12

09 Sep '12

>Further it seems that this test has been done, though removing only a portion of the junk DNA: http://www.ncbi.nlm.nih.gov/pubmed/15496924 --that was interesting. If the New York Timesy claim that our 3 billion base-pairs of "junk DNA" is filled with a "switchboard" with 4 million switches... which would be on average one switch per 750 base pairs... then this experiment where they removed a 1.5 Mbp and an 0.85 MBp region from junk mouse DNA, then the resulting mice grew up and seemed to be fine... would have removed an expected number of over 2000+1000 of those "switches." Kind of like if you randomly zapped two swaths of 2000 and 1000 transistors from an early computer with 4 million in all. Would that leave the computer apparently working fine? Even if those computer designers had tried to design it to be robust against faults (which they mostly did not), still, this amount of fault tolerance would be pretty impressive. So the 4M switches in our DNA if there, must either be very nonuniformly distributed, or very unimportant, or part of an extremely fault-tolerant circuit. If, in contrast (to compare with the genes in DNA, traditionally regarded as non-junk), you randomly zapped one gene per thousand in your set of about 25K genes (total zappage: 25 of your genes) then what? Would you be fine? 1. This question was examined for a bacterium, not a human: http://en.wikipedia.org/wiki/Mycoplasma_laboratorium this bacterium, one of the smallest genomes known, had 475 genes but they synthesized replacement DNA with fewer genes trying to find a minimal survivable subset. (Actually this bacterium is an obligate parasite i.e. cannot really survive on own without help, but whatever.) They got it down to 382. So it seems 80% of this bacterium's genes are vital. Another measure is this: apparently almost every (apparently over 90%) mouse gene, has an analogous gene inside humans. This suggests that most mammal genes are important, otherwise they'd have vanished & appeared a lot more during mammal evolution. It suggests randomly zapping 25 of your genes, would likely kill you. 2. In http://en.wikipedia.org/wiki/Knockout_mouse it is stated that "While knockout mouse technology represents a valuable research tool, some important limitations exist. About 15 percent of gene knockouts are developmentally lethal, which means that the genetically altered embryos cannot grow into adult mice." So about 1 in 7 of mouse genes are so important that zapping just that one gene, will kill it. Zapping 25 genes would almost certainly kill it. So it looks like genes really are way more crucial than junk. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] Dan Weinreb
by rcs＠xmission.com 08 Sep '12

08 Sep '12

<from the grapevine -- long twisty path omitted -- rcs> I am very sorry to report that Dan Weinreb passed away this morning. Dan was a founder of our local technology community, working on the Lisp Machine at MIT in the 1970s, as a founder of Symbolics in the early 1980s, later as founder of the object database company Object Design, and as our colleague at ITA Software and Google since 2006. Dan was a wonderful man, and will be missed.

3 2

Re: [math-fun] gaps between primes, P2s, P3s, etc
by Phil Carmody 08 Sep '12

08 Sep '12

From: Gareth McCaughan <gareth.mccaughan(a)pobox.com> > On Friday 07 September 2012 20:47:08 Warren Smith wrote: > > A "P2" is the product of exactly 2 primes. > ... > > > The most that is rigorously known about upper-bounding prime gaps is, > ... > > A well-known mathematician suggested to me that perhaps a much > > stronger upper bound would be possible (or was already known), [...] > > if we considered gaps between, not "primes," but rather "primes U P2's." > > > > I counter-argued that since P2s are much rarer than primes, that seemed > > silly; [...] > > Aside from what's already been pointed out, namely that P2s > are commoner than primes rather than rarer, note the difference > between (1) what the biggest gaps *are*, (2) what typical gaps > are, and (3) the best bound we can prove on either. I'm glad someone raised these points, as there's a huge disconnect between what we suspect to be true (that can be supported using believable heuristics) and what we can prove to be true. Life would be more satisfying once ERH had its i's crossed and its t's dotted, but even then the provable bounds still look dreadfully slack at omega(sqrt(p)), assuming Cramer's still the tightest ERH-conditional bound. > For instance: the numbers floor(n (log n)^2) are rarer than > the primes "by about a factor log n" but the gap between two > of these of size ~n is only on the order of (log n)^2, much > smaller than the biggest gaps between consecutive primes; > and if we take these numbers and perturb them in a suitable > (well controlled) way we can make the biggest gaps between > them *provably* about as big as the biggest gaps between primes > are conjectured to be. The fact that these numbers are rarer > than the primes, and therefore have bigger gaps on average, > is neither here nor there. Clearly your original set is the corner case with specifically chosen gaps and density, but of course it displays none of the traits of randomness that the primes have. Stirring in "suitable" perturbations still makes it a highly contrived set with all th properties you desire "just so". So I don't believe it helps give any insight into the primes at all, it's just an example of what the primes can never be like. Instead. What happens if we take the lucky (or ludic) numbers - what's known about their gaps? Phil

1 0

[math-fun] Computing with water droplets
by rcs＠xmission.com 08 Sep '12

08 Sep '12

This might make a nice fountain. --Rich --- http://www.sciencedaily.com/releases/2012/09/120907082027.htm Towards Computing With Water Droplets: Superhydrophobic Droplet Logic ScienceDaily (Sep. 7, 2012) ? Researchers in Aalto University have developed a new concept for computing, using water droplets as bits of digital information. This was enabled by the discovery that upon collision with each other on a highly water-repellent surface, two water droplets rebound like billiard balls. http://www.geekosystem.com/water-drop-computing/ [an ad-heavy page, but includes a decent video of a 1-bit counter]

2 1

Re: [math-fun] Dan Weinreb
by Bill Gosper 08 Sep '12

08 Sep '12

Quad> On the internet, he was always willing to share his opinions, his stories, etc. in great detail, and remained respectful about it. A very unfortunate loss, especially to the Lisp community. -Robert On 9/7/2012 8:32 PM, rcs(a)xmission.com <http://gosper.org/webmail/src/compose.php?send_to=rcs%40xmission.com> wrote:> <from the grapevine -- long twisty path omitted -- rcs>>> I am very sorry to report that Dan Weinreb passed away this morning.> Dan was a founder of our local technology community, working on the> Lisp Machine at MIT in the 1970s, as a founder of Symbolics in the> early 1980s, later as founder of the object database company Object> Design, and as our colleague at ITA Software and Google since 2006.> Dan was a wonderful man, and will be missed.> Didn't he originate the recursive Editor acronyms EINE and ZWEI? EINE Is Not Emacs and ZWEI Was EINE Initially ? --rwg

1 0

[math-fun] gaps between primes, P2s, P3s, etc
by Warren Smith 08 Sep '12

08 Sep '12

A "P2" is the product of exactly 2 primes. A "P3" ditto for 3 primes etc. The most that is rigorously known about upper-bounding prime gaps is, there is always a prime between X and X+X^(21/40+epsilon) if X is sufficiently large, for any fixed epsilon>0. A well-known mathematician suggested to me that perhaps a much stronger upper bound would be possible (or was already known), e.g. bringing the exponent 21/40 way down, likely below 1/2, if we considered gaps between, not "primes," but rather "primes U P2's." I counter-argued that since P2s are much rarer than primes, that seemed silly; adding P2's to the primes should have almost no effect on typical gap sizes asymptotically. And for that matter adding P3s, P4s, .. etc up to any fixed number of allowed factors also should have almost no effect. A random large number N has lnlnN prime factors on average, with standard deviation sqrt(lnlnN), see http://en.wikipedia.org/wiki/Erdos-Kac_theorem . Comments from math-fun number theorists?

4 3