June 2013 - mathfun@mailman.xmission.com

[math-fun] Scientists find proof plants are capable of complex arithmetic
by Henry Baker 24 Jun '13

24 Jun '13

FYI -- I saw the headline & was hoping for square roots of negative numbers, something which is beyond the capabilities of most hand calculators... BTW, NPR last Friday talked about plants continuing their circadian rhythms after they have been picked & are in the produce aisle at the grocery store; apparently, the grocery store lighting is upsetting the circadian rhythms of the fresh fruits & vegetables! http://www.telegraph.co.uk/science/science-news/10134715/Scientists-find-pr… Scientists find proof plants are capable of complex arithmetic Plants carry out arithmetic calculations each night to prevent them starving before dawn, scientists have discovered for the first time. By Nick Collins, Science Correspondent 7:30AM BST 23 Jun 2013 In order to keep themselves going in the absence of sunlight, plants perform division equations throughout the night to ration their stores of starch until the moment the sun reappears. By counting their starch and dividing it by the number of hours left until morning they ensure they do not run out until the crack of dawn, and can even adjust their calculations during the night. The discovery is the first known example in nature of sophisticated arithmetic being carried out at a fundamental level, as opposed to in the brain cells of animals like humans, scientists said. It plays a crucial role in plants' survival because they are dependant on starch, which is produced from carbon dioxide and sunlight during the day, to provide energy during the night. If they were to find themselves without starch during the hours of darkness they would quickly begin to starve, would stop growing and take several hours to recover even after light returned. "The capacity to perform arithmetic calculation is vital for plant growth and productivity," Prof Alison Smith, a metabolic biologist who helped make the discovery, explained. "The calculations are precise so that plants prevent starvation but also make the most efficient use of their food. If the starch store is used too fast, plants will starve and stop growing during the night. If the store is used too slowly, some of it will be wasted." It was already understood that plants have a mechanism which ensures their starch is not used up until the end of the night, but scientists previously had no idea how they managed it. Researchers from the John Innes Centre in Norwich discovered the hidden ability after studying Arabidopsis, a small flowering plant in the mustard family. They found that the plants used their starch at a steady rate throughout the night, so that about 95 per cent of their stock had been used up by dawn each day. When researchers changed light conditions to make night arrive unexpectedly early or late, the plants adapted by slowing or increasing their rate of starch usage to ensure their reserves lasted until dawn. This indicated that they had made a new calculation based on their internal clock which knows the amount of time left until morning, Prof Smith explained. When the plants' light exposure was shortened so that they had a smaller batch of starch to begin with, they consumed it more slowly, demonstrating that the equation could also change based on the size of their reserves. The scientists even attempted to trick the plants by introducing windows of sunlight during the night but each time they were able to adjust their starch use accordingly, demonstrating that the balance was being continuously computed throughout he night. They suggested that information about the size of the plants' starch stores and the time left until dawn must be based on the levels of two types of molecules, dubbed "S" for starch and "T" for time. "We propose there is a molecule called S which tracks the amount of starch in the plant, and a molecule called T which tracks the time until dawn," Prof Smith explained. "The closer to dawn you get, the less of T and S you have." If the "S" molecule prompts the use of starch and the "T" molecule prevents it, then dividing the number of "S" molecules by "T" molecules would enable the plant to use up its reserves at a steady rate and expire at dawn, she explained. Professor Martin Howard, who came up with the equation, said the study provides "The first concrete example in biology of such a sophisticated arithmetic calculation" at a fundamental biological level. But similar equations could be responsible for other natural phenomena where reserves are built up ahead of long periods without food, and run out just as the period of fasting ends. Migrating little stints arrive at their Arctic breeding grounds following a 5,000km journey with fat reserves so low that they could only survive another 14 hours, they wrote in the eLife journal. Similarly, during the four-month period in which they incubate eggs, male emperor penguins' fat levels reach a critically low level just at the point their partner is expected to return.

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Re: [math-fun] how to make a rhombic dodecahedon
by Adam P. Goucher 23 Jun '13

23 Jun '13

> ----- Original Message ----- > From: Dan Asimov > Sent: 06/23/13 01:18 AM > To: math-fun > Subject: Re: [math-fun] how to make a rhombic dodecahedon > > Just to add to this, Wikipedia reminds me that there are two distinct lattice packings Only the face-centred cubic (a b c a b c ...) is a lattice packing; the hexagonal close packing (a b a b a b ...) is not a lattice. (A lattice in R^n requires that the group of translational symmetries act transitively on the points, whereas HCP has two orbits.) Nevertheless, these are certainly the two most interesting of these 2^aleph_0 packings, by virtue of naturally occurring in crystallography. I prefer FCC due to its substantially larger point symmetry group. > that are special cases of the 2^aleph_0 packings that Adam refers to below: the cubic close-packing and the hexagonal close-packing: < http://en.wikipedia.org/wiki/Sphere_packing >. These are defined, respectively, by whether the alternation of the layers Adam mentions is in the pattern a b c a b c ... or a b a b a b .... > > Also, minor nitpicking point: the enumeration of the distinct not-necessarily-lattice ways the layers can be alternated should involve the equivalence of any shifted sequence of the 3 types of layers, and also possible equivalence by rotations between two arrangements. Good point. That doesn't affect the fact that there are uncountably many such packings, though. Sincerely, Adam P. Goucher

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Re: [math-fun] how to make a rhombic dodecahedon
by Adam P. Goucher 23 Jun '13

23 Jun '13

Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings. (Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.) Sincerely, Adam P. Goucher http://cp4space.wordpress.com > ----- Original Message ----- > From: Bill Gosper > Sent: 06/21/13 11:43 AM > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] how to make a rhombic dodecahedon > > I asked Neil if Kepler's sphere-stacking conjecture was about tetrahedral > or square > pyramids of cannonballs. He fooled with Mathematica briefly and said it > doesn't matter <http://gosper.org/kepvor.png>. > --rwg > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] how to make a rhombic dodecahed[r]on
by Bill Gosper 22 Jun '13

22 Jun '13

AdamG>Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings. (Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.) Sincerely, Adam P. Goucher http://cp4space.wordpress.com ----- Original Message ----- From: Bill Gosper Sent: 06/21/13 11:43 AM To: math-fun(a)mailman.xmission.com Subject: [math-fun] how to make a rhombic dodecahedon I asked Neil if Kepler's sphere-stacking conjecture was about tetrahedral or square pyramids of cannonballs. He fooled with Mathematica briefly and said it doesn't matter <http://gosper.org/kepvor.png>. --rwg http://mathworld.wolfram.com/KeplerConjecture.html says that Hales's proof will require 20 man-years to check. I rewrote http://gosper.org/kepvor.png to picture three of the aleph-null hexagonal packings (tp, tp1, and tp2). Apparently, only one is congruent to the octahedral packing. I expected two. With the staggered (tp2) packing, the Voronoi dodecahedron has six faces trapezoidal, but presumably tessellates 3space. (Twist a rhombic dodecahedron pi/3 about a hexagonal equator.) --rwg

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[math-fun] 70 million is now 12012
by Robert Baillie 22 Jun '13

22 Jun '13

Do you recall the remarkable paper by Zhang proving that there are infinitely many pairs of primes that differ by at most 70 million? Well, thanks to work by Terence Tao and others, that bound of 70 million has been pushed down to just 12012. This website tracks the progress on that upper bound: http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_pr…

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Re: [math-fun] Mathematica woes
by Bill Gosper 22 Jun '13

22 Jun '13

It will probably show up on your machine after an hour... --Michael (It will probably take *two* hours for my copy of this message to show up on my mail host. Something to do with graylisting.) Dan, if you can write files, you can Export the Graphics object. --rwg On Fri, Jun 21, 2013 at 3:38 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: Yes! I can use a command-line interface with the command math (which didn't work until this morning, just a minute ago). However, despite logging in with ssh -X I seem to get no graphical output whatsoever on my home machine. E.g., I type Plot[x^2,{x,1,4}] and Mma responds with the character string -Graphics- but it ain't on my machine! --Damn On 2013-06-21, at 12:30 PM, Neil Sloane wrote: Dan, Can you log in remotely to that system? Then open an x-terminal window Then type math (on most machines it is /usr/common/bin/math ) That should start a command-line version of mathematica On Fri, Jun 21, 2013 at 3:07 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: Sheesh, I can't even *get* the text-only interface. When the options box finally gets around to opening, it offer +Notebook, +Slide Show, +Other. Clicking on any one of these, if it has any effect at all, takes to long to see a response that any further messing with this setup is clearly a waste of time. On 2013-06-21, at 11:44 AM, Michael Kleber wrote: Use the text-only interface :-/.

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[math-fun] unsubsribe
by Marek Wolf 21 Jun '13

21 Jun '13

unsubsribe -- Marek Wolf

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[math-fun] how to make a rhombic dodecahedon
by Bill Gosper 21 Jun '13

21 Jun '13

I asked Neil if Kepler's sphere-stacking conjecture was about tetrahedral or square pyramids of cannonballs. He fooled with Mathematica briefly and said it doesn't matter <http://gosper.org/kepvor.png>. --rwg

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[math-fun] HoTT foundations of maths
by Warren D Smith 21 Jun '13

21 Jun '13

Well, I wish somebody had a bit to say about applications or motivations. I mean, was there something broke that needed to be fixed? Can it do stuff you could not do before, clears up issues, what? Is the point that it meshes better with attempts to build automated theorem provers? It sounds vaguely reasonable.

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[math-fun] A foundations of mathematics for the 21st century
by Hans Havermann 21 Jun '13

21 Jun '13

John Baez on Google+: It's here! For decades, mathematicians been dreaming of an approach to math where different proofs that x = y can be seen as different paths in a space. It's finally been made precise, thanks to Vladimir Voevodsky and a gang of mathematicians who holed up for a year at the Institute for Advanced Studies, at Princeton. I won't try to explain it, since that's what the book does. I'll just mention a few of the radical new features: * It includes set theory as a special case, but it's founded on more general things called 'types'. Types include sets, but also propositions. Proving a proposition amounts to constructing an element of a certain type. So, proofs are no longer 'outside' the mathematics being discussed, they're inside it just like everything else. * The logic is 'constructive', meaning that to prove something exists amounts to giving a procedure for constructing it. As a result, the whole system can be and is being computerized with the help of programs like COQ and AGDA. * Types can be seen as 'spaces', and their elements as 'points'. A proof that two elements of a type are equal can be seen as constructing a path between two points. Sets are just a special case: the '0-types', which have no interesting higher-dimensional aspect. There are also types that look like spheres and tori! Technically speaking, the branch of topology called homotopy theory is now a part of logic! That's why the subject is called homotopy type theory. * Types can also be seen as infinity-groupoids. Very roughly, these are structures with elements, isomorphisms between elements, isomorphisms between isomorphisms, and so on ad infinitum. So, a certain chunk of the important new branch of math called 'higher category theory' is now part of logic, too. * The most special contribution of Voevodsky is the univalence axiom. Very very roughly, this expands the concept of 'equality' so that it's just as general as the hitherto more flexible concept of 'isomorphism' - or, if you know some more fancy math, 'equivalence'. Mathematicians working on homotopy theory and higher category theory have known for decades that equality is too rigid a concept to be right - for certain applications. The univalence axiom updates our concept of equality so that it's good again! Since this is all about foundations, and it's all quite new, please don't ask me yet what its practical applications are. Ask me in a hundred years. For now, I can tell you that this is the 'upgrade' that the foundations of math has needed ever since the work of Grothendieck. It's truly 21-century math. It's also a book for the 21st century, because it's escaped the grip of expensive publishers! While it's 600 pages long, a hardback copy costs less than $27. Paperback costs less than $18, and an electronic copy is free! http://homotopytypetheory.org/book/

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