January 2016 - mathfun@mailman.xmission.com

[math-fun] Hi-Q (peg solitaire) nostalgia
by Bill Gosper 21 Jan '16

21 Jan '16

Foo. Last night I told friends (the Ridenours) about once memorizing a ridiculously improbable solution merely for the purpose of pretending to discover it while sitting next to a stranger on an airplane, e.g.. I completely forgot about http://www.inwap.com/pdp10/hbaker/hakmem/games.html (Item 75). (Reminder: Item 70 should say bishop at KB8 (not KB7!)). --rwg Shortly prior, young Rohan Ridenour tweeted a peculiar optical illusion https://twitter.com/wolframtap/status/689927650245087232?lang=en Mystery: Notice the blank in (White .Black.Gray), sine qua "Tweet is too long" !?

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Re: [math-fun] Contacting Ray Smullyan
by Dan Asimov 21 Jan '16

21 Jan '16

I get that Rich was trying to kiddingly demonstrate a counterexample to my post that he quoted. Touché. But I'd have to say that "X does not exist" statements are in a different category. Obviously, "There exists a king of France, who is unique, and who doesn't exist" would be be hard to parse. But "There exists the concept of the King of France, and there is nothing in the world corresponding to that concept" is what is really meant by Rich's statement (normally; perhaps Rich would differ with this). —Dan > On Jan 20, 2016, at 3:23 PM, rcs(a)xmission.com wrote: > > The King of France does not exist. :-) > > ----- > Quoting Dan Asimov <asimov(a)msri.org>: > >> I may be wrong, but I think philosophers of language tend to >> interpret "the" statements as saying, first of all, that >> what follows the "the" exists and is unique, and then whatever >> further is said the "the" thing. >> >> So if we agree to this, any statement about "the" King of France >> (presuming a current time frame) is just false. >> >> ?Dan >> >> >>> On Jan 20, 2016, at 2:19 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: >>> >>> These discussions about properties of a nonexistent object require >>> nontrivial interpretation, which can't be done mathematically. Better, I >>> think, to say that some "propositions" have no truth value (perhaps because >>> they are not actually propositions). The classic example is "The present >>> King of France is bald." Reasoning about the empty set cannot help us here. >>> One can defend the proposition by saying, "Show me one hair from the >>> present King of France's head!" One can attack it by saying, "Show me his >>> shiny scalp!" >>> >>> I think "propositions" like "So-and-so's husband is not married" have the >>> same problem. The statement "Chris has no husband" does not >>> straightforwardly imply "Chris's husband is unmarried"; it can only have >>> that implication under some fairly gnarly rule of interpretation, which I >>> challenge anybody to verbalize. >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >

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Re: [math-fun] Tetrahedron reality
by Henry Baker 21 Jan '16

21 Jan '16

Off the top of my head, I don't have answers to any of Rich's questions. However, the computer graphics people have done a lot of work on similar issues having to do with things like triangles intersecting triangles (in 3D !) and tetrahedra intersecting tetrahedra. Amazingly, the best algorithms aren't as clever as one might expect; they look a lot more like multiple-precision brute force. I don't know whether there are provable lower bounds on any of these algorithms, so there may still be room for improvement. (A lower bound might show that computing some feature of a complex polyhedron might solve some other known problem -- e.g., satisfiability or colorability.) I'm also interested in whether there are "modular analogs" to some of these questions in a modular space. I.e., a classical sort of algorithm, but computed modulo some integer or some polynomial. At 04:52 PM 1/20/2016, rcs(a)xmission.com wrote: >When does a set of edge lengths make a triangle? > >Assume that the lengths are non-negative. >Then we need the triangle inequality, A+B <= C. >We can either check all three permutations, or >require the lengths are sorted, with C maximal. >When the inequality is an equality, then A+B-C >is zero, and Hero(n?)'s Formula for the area, >S = 1/4 * sqrt((A+B+C)(A+B-C)(A-B+C)(-A+B+C)) >is 0. Moreover, if A,B,C >=0, at most one of the >factors in the sqrt can be negative, so looking >at the sign of the radicand signals the reality >of the triangle. > >Puzzle: Same problem, in 3D. >When do six edges make a real tetrahedron? > >Assume the base triangle edges are A,B,C, with >opposite edges D,E,F. >The four faces must be real triangles; >moreover the area of the largest face must be >no greater than the sum of the areas of the other >three faces. This constraint is obviously >true for a real tetrahedron. Necessity follows >from the example A=B=C=7, D=E=F=4, where the >area of the base ABC exceeds the sum of areas >of the three skinny 744 triangles. >(In a 777 triangle, the distance from the center >to a vertex is 7/sqrt3 = 4.04...) > >Another possible geometry failure: Build the >tetrahedron as a planar quadrilateral, with >the diagonals TBD. If we choose one diagonal, >its length is restricted (both min & max) by the >triangle inequality. Now imagine the quadrilateral >as two triangles with the (first) diagonal as a >hinge: The second diagonal must be long enough >to connect the two remaining vertices, but not >too long. I can make a too-long example with >the quadrilateral (a rhombus) having sides 2,2,2,2. >If arranged as a square, the diagonals would be >2sqrt2 = 2.828...; requiring each diagonal to be >3 gives an impossible tetrahedron which satisfies >the various triangle inequalities, and the sum-of- >areas constraints. > >One puzzle is to find something analogous to the >sum-of-areas constraint that captures this >second-diagonal-not-too-long restriction. > >I think that the second-diagonal-too-short case >is covered by the sum-of-areas constraint: If SOA >is true, then 2nd-diag-too-short can't happen. > >Another puzzle is to find something analogous to >Hero(n?)'s formula, for the volume. One might hope >that the volume could be expressed in terms of the >areas of the four faces, but I think this is false. >It is true in the case of a "right-tetrahedron", >where one vertex is the corner of three right >triangles: The volume is the (sqrt2)/3 of the square >root of the product of the areas of the three right >triangles. The area of the "hypotenuse" face is >fixed in this case, by a 3D version of Pythagoras: >square-of-hyp-area = sum of squares of other three >face areas. > >There's a volume formula in terms of the six edges, >as the square-root of a sixth-degree polynomial. >I don't know of any formula which uses the area- >excesses as factors, analogous to the appearance >of the edge-excesses (i.e. A+B-C &c) in Hero(n?)'s >area formula. > >Rich

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[math-fun] Contacting Ray Smullyan
by James Propp 21 Jan '16

21 Jan '16

Does anyone on the list have contact info for Smullyan? I want to send him my paradoxical pair of sentences "If the premise of this implication does not contain the letter p, then two is not even" and its formal contrapositive "If two is even, then the premise of this implication contains the letter p" (the first is true whereas the second is false); more importantly, I want to find out if he knows of other, better sentences of this kind. (I haven't read all his books; it's possible that he had this idea years before I did, and implemented it in a better way.) I should probably send my question to Hofstadter too, but that seems less urgent. (The last time I saw Smullyan, he did not look that well.) Jim Propp

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[math-fun] carbon oxygen bogosity
by Warren D Smith 21 Jan '16

21 Jan '16

> 1. In Argentina, Largest Titanosaur (Henry Baker) > FYI -- Time to bring back my theory that pre-K-T oxygen levels were far in > excess of today's oxygen levels. Animals this large wouldn't be able to > survive in today's <21% O2 levels; they would feel like we do today at > 20,000' altitudes (~50% O2 saturation levels). These excessively high > levels of O2 pre-K-T were created by a billion+ years of plant-life CO2 > reduction, leading to *huge* amounts of surface carbon in the form of > coal/oil/etc. --actually, today's blue whales are larger than titanosaurs, and probably more active and higher metabolisms too, so animals this large have no trouble surviving our oxygen levels. And I think the limits on large land animals are not oxygen, but rather mechanical strength. > The K-T event 66 million years ago set of a planet-wide fire that burned for > years and years, and finally brought atmospheric O2 levels down to nearly > today's levels. --I don't believe you. If oxygen levels were high enough to sustain a many year long planet wide fire, then one would have got started without need of a meteor. Lightning, etc. I also claim oxygen levels are self regulating in the sense that if they get too high, there are more fires. This prevents oxygen levels from ever exceeding some amount, which one author estimated to be 25%. > (Tiny models of such fires burn today in West Virginia, > some of which have been burning for >100 years.) --those are in sheltered conditions inside coal mines. Where, incidentally, the oxygen levels are low. Anyhow, despite criticizing your argument, oxygen levels were indeed higher and lower in the past. https://en.wikipedia.org/wiki/Geological_history_of_oxygen and I do not know how solid their reconstruction of the past is, but they have one. Originally there was no O2. "The maximum of 35% was reached towards the end of the Carboniferous period (about 300 million years ago), a peak which may have contributed to the large size of insects and amphibians at that time." Note insects have no lungs and hence their size might be oxygen-limited, although the wikipedia article claims it isn't. Also note the dinosaur-killing meteor hit was 66 million years ago, much later than the carboniferous which ended 300 Myr ago. "At current rates of primary production, today's concentration of oxygen could be produced by photosynthetic organisms in 2000 years. In the absence of plants, the rate of oxygen production by photosynthesis was slower in the Precambrian, and the concentrations of O2 attained were less than 10% of today's and probably fluctuated greatly" which if true also would seem to kill Baker's coal theory, since a mere 2000 years to generate all oxygen, seems peanuts compared to the timescale of the carboniferous, 60 Myr long.

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[math-fun] Tetrahedron reality
by rcs＠xmission.com 21 Jan '16

21 Jan '16

When does a set of edge lengths make a triangle? Assume that the lengths are non-negative. Then we need the triangle inequality, A+B <= C. We can either check all three permutations, or require the lengths are sorted, with C maximal. When the inequality is an equality, then A+B-C is zero, and Hero(n?)'s Formula for the area, S = 1/4 * sqrt((A+B+C)(A+B-C)(A-B+C)(-A+B+C)) is 0. Moreover, if A,B,C >=0, at most one of the factors in the sqrt can be negative, so looking at the sign of the radicand signals the reality of the triangle. Puzzle: Same problem, in 3D. When do six edges make a real tetrahedron? Assume the base triangle edges are A,B,C, with opposite edges D,E,F. The four faces must be real triangles; moreover the area of the largest face must be no greater than the sum of the areas of the other three faces. This constraint is obviously true for a real tetrahedron. Necessity follows from the example A=B=C=7, D=E=F=4, where the area of the base ABC exceeds the sum of areas of the three skinny 744 triangles. (In a 777 triangle, the distance from the center to a vertex is 7/sqrt3 = 4.04...) Another possible geometry failure: Build the tetrahedron as a planar quadrilateral, with the diagonals TBD. If we choose one diagonal, its length is restricted (both min & max) by the triangle inequality. Now imagine the quadrilateral as two triangles with the (first) diagonal as a hinge: The second diagonal must be long enough to connect the two remaining vertices, but not too long. I can make a too-long example with the quadrilateral (a rhombus) having sides 2,2,2,2. If arranged as a square, the diagonals would be 2sqrt2 = 2.828...; requiring each diagonal to be 3 gives an impossible tetrahedron which satisfies the various triangle inequalities, and the sum-of- areas constraints. One puzzle is to find something analogous to the sum-of-areas constraint that captures this second-diagonal-not-too-long restriction. I think that the second-diagonal-too-short case is covered by the sum-of-areas constraint: If SOA is true, then 2nd-diag-too-short can't happen. Another puzzle is to find something analogous to Hero(n?)'s formula, for the volume. One might hope that the volume could be expressed in terms of the areas of the four faces, but I think this is false. It is true in the case of a "right-tetrahedron", where one vertex is the corner of three right triangles: The volume is the (sqrt2)/3 of the square root of the product of the areas of the three right triangles. The area of the "hypotenuse" face is fixed in this case, by a 3D version of Pythagoras: square-of-hyp-area = sum of squares of other three face areas. There's a volume formula in terms of the six edges, as the square-root of a sixth-degree polynomial. I don't know of any formula which uses the area- excesses as factors, analogous to the appearance of the edge-excesses (i.e. A+B-C &c) in Hero(n?)'s area formula. Rich

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[math-fun] Poncelet's porism puzzle
by Fred Lunnon 20 Jan '16

20 Jan '16

Recapitulating "Euler's theorem", badly garbled in an earlier post of mine: Relates circumradius R , inradius r , centre displacement d of triangle by R^2 - 2 R r - d^2 = 0 . Actually due to Chapple, according to http://mathworld.wolfram.com/EulerTriangleFormula.html . Synthetic proof at https://en.wikipedia.org/wiki/Euler's_theorem_in_geometry#/media/File:GeometryEulerTheorem.png . Provides foundation of Poncelet's porism for triangles in Euclidean 2-space; see http://mathworld.wolfram.com/PonceletsPorism.html . My Scrooge's Christmas present (to myself, naturally) was to investigate the extent to which Chapple's result generalises to higher dimensions: specifically, *** What can be established about the relation between circumradius, *** *** inradius and displacement of a tetrahedron in Euclidean 3-space? *** I'll post more about this problem after a few days, in case other social misfits out there are motivated to tackle it themselves, or alternatively can disabuse me of any delusional claim to originality. Incidentally, Poncelet's porism now has an entire book devoted to it, Vladimir Dragović, Milena Radnović "Integrable Billiards" Birkhäuser 2011 ; chapter 1 free online at http://www.springer.com/cda/content/document/cda_downloaddocument/978303480… . Fred Lunnon

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[math-fun] carbon oxygen bogosity
by Warren D Smith 20 Jan '16

20 Jan '16

>Keith F. Lynch kfl at KeithLynch.net I've long been baffled by the claim that the free oxygen in the atmosphere came from photosynthesis. If the atmosphere started out neutral, then oxygen in the atmosphere must be balanced by reduced matter (biomass, peat, coal, oil, natural gas, etc.). It seems implausible that there's enough combustible stuff that burning it all would consume all the oxygen. --WDS: Interesting. But I think you're wrong, see below. First of all, the earth's overall elemental composition clearly is reductant-rich, not oxidant-rich -- just consider the iron core. So on a lifeless earth there would be no free O2. It is ludicrous to contend that there just happens to be a teeny tiny amount more O than it takes the oxidize the rest, left over, just enough to form the atmosphere. There also is no question the earth atmosphere was originally oxygen-free and we know from geological evidence roughly when the "great oxygenation event" occurred yielding massive iron oxide ore deposits, etc. There is about 235 grams of free O2 per square cm of earth surface. (If atmospheric pressure is 1 kg per square cm.) This amounts to about 10^18 kg = 10^6 gigatons of free O2. So the question of whether there is enough combustible stuff to get rid of all the O2 comes down to: if you took a square cm of earth and went down, say, 1 km, would that long prism of soil, rock (or ocean floor material) contain enough combustible stuff to consume 235 grams of O2? Which if it were carbon would mean >=88 grams? Wikipedia: "Soil plays a major role in the global carbon cycle, with the global soil carbon pool estimated at 2500 gigatons. This is 3.3 times the size of the atmospheric pool (750 gigatons) and 4.5 times the biotic pool (560 gigatons)." Wikipedia: The ocean floors are covered with ooze and sediments, ranging from 0.6 to 9 km thick, largely made of dead microbes. "Ooze is pelagic sediment that consists of at least 30% of microscopic remains of either calcareous or siliceous planktonic debris organisms. The remainder typically consists almost entirely of clay minerals." It has been found that ocean siliceous ooze will react with hot oxygen and will thereby lose mass from 3 to 16.5%. See tables in book "THE MINERAL RESOURCES OF THE SEA" Elsevier 1965 "loss on ignition." So: those numbers imply soil carbon is totally inadequate to use up all the free oxygen; burning it all it would only use up about 0.67% of the O2. However, ocean ooze combustible material seems more than enough to consume all oxygen, in fact it looks to me like enough to use up at least 5 times all free oxygen. So, that's where it went. The geologic "carbon cycle" is that carbonate rocks are baked in volcanos and emit CO2, which then gets grabbed by life and ends up as ocean ooze, which forms into carbonate rocks, etc. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] carbon oxygen bogosity
by Keith F. Lynch 20 Jan '16

20 Jan '16

I've long been baffled by the claim that the free oxygen in the atmosphere came from photosynthesis. If the atmosphere started out neutral, then oxygen in the atmosphere must be balanced by reduced matter (biomass, peat, coal, oil, natural gas, etc.). It seems implausible that there's enough combustible stuff that burning it all would consume all the oxygen. Note that carbon in carbonate rocks isn't reduced -- it's already combined with as much oxygen as it can be. Carbonate rocks can explain the disappearance of carbon dioxide from the atmosphere, but not the appearance of free oxygen. The obvious alternative is that over many millions of years solar UV split water vapor into oxygen and hydrogen, and the hydrogen escaped into space. This would drop the sea level, but only by a few meters. Yes, this is happening too slowly today to account for how much free oxygen there is. That's because the upper atmosphere, where there's plenty of solar UV, is very dry, due to the stratospheric cold trap. But the "iceball earth" theory says the ground-level climate oscillated for most of the Proterozoic eon between about -50 C and +50 C. During the warm periods, it seems likely that the upper stratosphere would have been warm and moist. And that's just when the oxygen appeared. Am I really the first to think of this? If I'm right, discovery of a planet with an oxygen atmosphere wouldn't necessarily imply that there is, or ever was, life on that planet. As for Henry Baker's conjecture that massive fires at the end of the Cretaceous consumed much of the world's free oxygen, that would have raised carbon dioxide to lethal levels. It's also not plausible that the fires would burn for "years and years." Why wouldn't everything on the surface that could burn all burn within at most a week? Stuff underground in oxygen-poor caves and crevices might smolder for a long time, as with today's underground coal fires, but wouldn't remove much oxygen or add much CO2. That being said, it does seem likely that there were global fires at the end of the Cretaceous. When the secondary meteors kicked up by the main impactor re-entered all over the world, the whole sky would have lit up as if H-bombs were going off everywhere. But I doubt there was much on the surface to burn except live plants and animals. As Warren D Smith pointed out, any lakes of petroleum couldn't have lasted past the first lightning storm, especially not if oxygen levels were higher than today's.

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Re: [math-fun] carbon oxygen bogosity
by Henry Baker 19 Jan '16

19 Jan '16

At 09:13 AM 1/19/2016, Warren D Smith wrote: >--I don't believe you. If oxygen levels were high enough to sustain >a many year long planet wide fire, then one would have got started >without need of a meteor. Lightning, etc. True, but there are fires, and there are FIRES. Dresden-type fires are substantially different from regular fires, in that they create their own winds & weather; the scale of the fire is important -- a planetary scale fire would be substantially different from even a Dresden-type fire. The impact 66mya lowered sea levels around the world by at least a foot, thus exposing a lot more carbon that had previously been covered with water. Anyhow, despite criticizing your argument, oxygen levels were indeed higher and lower in the past. > https://en.wikipedia.org/wiki/Geological_history_of_oxygen >and I do not know how solid their reconstruction of the past is, but they have one. > >Originally there was no O2. >"The maximum of 35% was reached towards the end of the Carboniferous >period (about 300 million years ago), a peak which may have >contributed to the large size of insects and amphibians at that time." I'm disagreeing with these conclusions, which are sketchy, at best.

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