June 2014 - mathfun@mailman.xmission.com

[math-fun] Soccer balls
by Stuart Anderson 15 Jun '14

15 Jun '14

Not being a big soccer fan, not across the latest on the design, authorised use and reasons for the shape and construction of soccer balls. Looking at the opening game I noticed the truncated icosahedron ball is no longer in use. I found this article which compares the properties of different shaped balls http://www.nature.com/srep/2014/140529/srep05068/full/srep05068.html Does anyone know why the truncated icosahedron is no longer used in the World Cup? Sent from my iPhone

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Re: [math-fun] Soccer balls
by Walter Trump 15 Jun '14

15 Jun '14

New balls have seams that are more contiguous --> cheaper production The seams have a similar importance for soccer balls as the dimples for golf balls. At high speed local turbulences at the seams lead to a laminar air flow of the whole ball. Goalkeepers hated the ball of the world championship 2006 because of large areas without seams. As already mentioned by Dave: At certain speed (transition from laminar to turbulent air flow) the behaviour of a ball (which does not rotate) is unstable. Walter > -----Ursprüngliche Nachricht----- > Von: math-fun [mailto:math-fun-bounces@mailman.xmission.com] > Im Auftrag von math-fun-request(a)mailman.xmission.com > Gesendet: Samstag, 14. Juni 2014 20:35 > An: math-fun(a)mailman.xmission.com > Betreff: math-fun Digest, Vol 136, Issue 27 > > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more > specific than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: Soccer balls (Dan Asimov) > 2. Re: Soccer balls (Dave Dyer) > 3. Re: Soccer balls (Dave Dyer) > 4. Re: Soccer balls (Dan Asimov) > 5. earth has vast internal water reservoir? (Warren D Smith) > 6. Re: Close enough for Government work (Bill Gosper) > 7. Trying to locate Marc Paulhus (Neil Sloane) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Fri, 13 Jun 2014 10:13:04 -0700 > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Soccer balls > Message-ID: <CE6EA8C8-5FD8-4256-A1C6-AE56A837E9BB(a)earthlink.net> > Content-Type: text/plain; charset=us-ascii > > The sense I get from various web pages is that some felt it > was too easy to kick a seam and get unpredictable behavior > from the ball. It's not clear how the new design (as well > described on the first page of the site below) avoids this. > (Six propeller-shaped panels each with 4-fold chiral > symmetry, fitting together like the faces of a cube.) > > I think the improvement isn't just in the geometry of the > panels, but also in the way the seams are sealed, making the > seams a smaller fraction of the total surface area. > > --Dan > > > On Jun 13, 2014, at 12:11 AM, Stuart Anderson > <stuart.errol.anderson(a)gmail.com> wrote: > > > Not being a big soccer fan, not across the latest on the design, > > authorised use and reasons for the shape and construction of soccer > > balls. Looking at the opening game I noticed the truncated > > icosahedron ball is no longer in use. I found this article which > > compares the properties of different shaped balls > > http://www.nature.com/srep/2014/140529/srep05068/full/srep05068.html > > Does anyone know why the truncated icosahedron is no longer > used in the World Cup? > > > > Sent from my iPhone > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > Message: 2 > Date: Fri, 13 Jun 2014 10:51:31 -0700 > From: Dave Dyer <ddyer(a)real-me.net> > To: math-fun <math-fun(a)mailman.xmission.com>, math-fun > <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Soccer balls > Message-ID: <E1WvVeM-0002NF-Ct(a)mx02.mta.xmission.com> > Content-Type: text/plain; charset="us-ascii" > > > New scientist has had several articles on this, both current > and back when the old ball was introduced. The gist I got > was that the seam pattern influences the point at which the > ball becomes unstable as it loses velocity and spin. > > > > > ------------------------------ > > Message: 3 > Date: Fri, 13 Jun 2014 10:51:31 -0700 > From: Dave Dyer <ddyer(a)real-me.net> > To: math-fun <math-fun(a)mailman.xmission.com>, math-fun > <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Soccer balls > Message-ID: <E1WvVeM-0002NG-BA(a)mx02.mta.xmission.com> > Content-Type: text/plain; charset="us-ascii" > > > New scientist has had several articles on this, both current > and back when the old ball was introduced. The gist I got > was that the seam pattern influences the point at which the > ball becomes unstable as it loses velocity and spin. > > > > > ------------------------------ > > Message: 4 > Date: Fri, 13 Jun 2014 11:32:15 -0700 > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Soccer balls > Message-ID: <8FF1C15E-F1DC-4FE9-81D3-406520178CD7(a)earthlink.net> > Content-Type: text/plain; charset=us-ascii > > This non-article in the NY Times gives a nice exploded view > of the three large balls depicted, at least if you click on them. > > Warning: This does not work on my iMac with Mavericks when I > use Firefox, butg works with Safari. > > --Dan > > > On Jun 13, 2014, at 10:51 AM, Dave Dyer <ddyer(a)real-me.net> wrote: > > > > > New scientist has had several articles on this, both > current and back > > when the old ball was introduced. The gist I got was that the seam > > pattern influences the point at which the ball becomes > unstable as it > > loses velocity and spin. > > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > ------------------------------ > > Message: 5 > Date: Sat, 14 Jun 2014 00:58:04 -0400 > From: Warren D Smith <warren.wds(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] earth has vast internal water reservoir? > Message-ID: > > <CAAJP7Y3R4cYE-FEtFuG+R9B9=OVvidLw3ugj9yAjRzjWzyrrLQ(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > http://www.cbc.ca/news/technology/deep-earth-has-oceans-worth- > of-water-10-diamond-reveals-1.2569564 > > >From tiny observations follow huge conclusions: > Somebody found a diamond containing a mineral inclusion, said > mineral 1.5% water. > This"proves" there is a tremendous amount of water in Earth > mantle diamond forming region, from which we extrapolate this > region contains way more water than all oceans. This in turn > suggests a geologic water cycle where ocean water subducted > into the mantle, and outgassed from same... this is rather > revolutionary change in view... > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step) > > > > ------------------------------ > > Message: 6 > Date: Sat, 14 Jun 2014 04:49:17 -0700 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Close enough for Government work > Message-ID: > > <CAA-4O0F-ERNPLvS5TnuHegEU2u7B94-J7XHjUSwPen-Te+fdCg(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > On Mon, Jun 9, 2014 at 7:36 PM, Bill Gosper > <billgosper(a)gmail.com> wrote: > > > Ideally, we might find formulas for a(x), c(x), d(x), and q(x). > > (I keep providing for b(x), but it always comes out > identically 1. ?) > > > > > > On Mon, Jun 9, 2014 at 4:30 AM, Bill Gosper > <billgosper(a)gmail.com> wrote: > > > >> Rejected but juicy http://arxiv.org/abs/math/0703470 (p 9) > describes the > >> remarkable > >> (to me, anyway) *polynomial* valued Somos4 > >> > >> s[0] = 0; s[1] = s[2] = 1; s[3] = -1; s[4] = x; > >> s[n_Integer /; n > 4] := Factor[(s[n - 1]*s[n - 3] + s[n - > 2]^2)/s[n - 4]] > >> > >> In[536]:= s /@ Range[12] > >> > >> Out[536]= {1, 1, -1, x, > >> 1 + x, -1 - x + x^2, -1 - x - x^3, -x (2 + 3 x), > >> 1 + 3 x + 3 x^2 - x^4 + > >> x^5, -(1 + x) (-1 - 2 x + 2 x^2 + 3 x^3 - x^4 + x^5), -1 - 3 x - > >> 3 x^2 - 5 x^3 - 9 x^4 - 3 x^5 + 2 x^6 - > >> x^7, -x (-1 - x + x^2) (3 + 9 x + 9 x^2 + 5 x^3 + 2 x^4 + x^6)} > >> > > > [Clipped: four variable relations] > The polynomial degree goes up like n^2/16 plus a period 8 ripple. > > > > >> For x:=1, s[n] can be expressed in "closed form": > >> a*c^n^2*EllipticTheta[1, d*n, q]: > >> (0.31749282989638009698851538146011901061695 + > >> 0.41577568158982458340525424882529254242763 I) > >> (0.74320667986312908167383113199924669418636 - > >> 0.69294655945321371719182977376188612597500 I)^n^2 > EllipticTheta[ > >> 1, (1.7554385915026838183474896476936322715588657 + > >> 0.050402131298346764198930943819803546234487011 I) n, > >> -0.43035475675354998492420504350604355525329714 - > >> 0.63418111840450730747740547053917541440014778 I] > >> > >> Table[%, {n, -4, 13}] // Chop > >> > >> {-1.000000000000000000000000000000000000000, > >> 1.0000000000000000000000000000000000000000, > >> -1.0000000000000000000000000000000000000000, > >> -1.0000000000000000000000000000000000000000, 0, > >> 1.0000000000000000000000000000000000000000, > >> 1.0000000000000000000000000000000000000000, > >> -1.0000000000000000000000000000000000000000, > >> 1.000000000000000000000000000000000000000, > >> 2.00000000000000000000000000000000000000, > >> -1.000000000000000000000000000000000000000, > >> -3.00000000000000000000000000000000000000, > >> -5.00000000000000000000000000000000000000, > >> 7.0000000000000000000000000000000000000, > >> -4.0000000000000000000000000000000000000, > >> -23.000000000000000000000000000000000000, ...} > >> > >> (Find *those* a,c,d,q in ISC.) The paper suggests that > there are several > >> other such expressions for s[n]. > >> It will be interesting to see how a, c, d, and q vary with x. > >> --rwg > >> > > [Clipped: Numeric difficulties.] > > > > For x = -2/3, we get a slightly messy but exact closed form > 3^(1/16 (-n^2 + (5 + (-1)^n - 2 Sqrt[2] Cos[(n ?)/4] + > 4 Cos[(n ?)/2])^2 Sin[(n ?)/4]^4)) (4 Cos[(n ?)/ > 8] Cos[(n ?)/4] - (-1)^n (Sqrt[2] + (-2 + Sqrt[2]) Cos[(n ?)/4]) > Cos[(n ?)/ > 2]) Sin[(n ?)/8] > > (no ?, no mysterious q), but > 0,1,1,-1,-2/3,1/3,1/3^2,-1/3^3,0,1/3^5,-1/3^6,-1/3^7,2/3^9,1/3^10,... > is no longer a Somos sequence! The problem is that s[8n] is a > multiple of 3x+2, and there's no way to continue the recurrence > more than three steps past those periodic 0s. > > The paper's "closed form" had an eight way case statement. > > There is also an exact ? expression > ((-1)^(1/8) Sqrt[2] EllipticTheta[2, 0, I q])/(3^(1/4) > EllipticTheta[2, 0, > q]) > > where q ->Root[{-((1 + I)/3^(1/4)) + ( > QPochhammer[-1, #1^2] QPochhammer[-(1/#1^2), #1^4])/( > 2 (1 + 1/#1^2)) &, 0.591308037470 + 0.442317013236 I}] > > specifying q, and thus the ? quotient, to arbitrary precision, > but with no hint the values are rational, or even real. > Interestingly, the problematic (0/0) values of the recurrence > are s[8n+4], whereat the ? quotient is conveniently independent > of q. > > The paper mentions finding q -> > Root[{Product[(1 + #^(2*k))*(1 + #^(4*k - > 2)), {k, \[Infinity]}] - (1 + I)/ > 3^(1/4) &, .7241830710727415040344246937315* > I + .5068861260317593704061905537186}] > producing a puzzling algebraic sequence, but > Root can't pick up the scent. > > More generally, if there is some algebraic x for which s[n] vanishes, > then it will periodically vanish for s[k n], and there will be a messy > (non-?) closed form, as above. > --rwg > > > ------------------------------ > > Message: 7 > Date: Sat, 14 Jun 2014 14:34:33 -0400 > From: Neil Sloane <njasloane(a)gmail.com> > To: fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Trying to locate Marc Paulhus > Message-ID: > > <CAAOnSgSE5VZst9JbHiYPV_-29PhH5usbcNSy76caj+RfQE=Tmw(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > There are several references in the OEIS to a preprint: > > Marc Paulhus (paulhus(AT)wanadoo.nl), Pikelets, Discrete Math., (but > apparently unpublished). > > I am trying to get hold of the article - can anyone help? > > Or, can anyone give me his email address? The one I have is broken. > > He has worked with Richard Guy and others on game theory: > For example: > Dan Calistrate, Marc Paulhus and David Wolfe, On the lattice > structure of > finite games, in More Games of No Chance (Berkeley, CA, > 2000), Math. Sci. > Res. Inst. Publ., 42, Cambridge Univ. Press, Cambridge, 2002, > pp. 25-30. > > Thanks > Neil > > > > -- > Dear Friends, I have now retired from AT&T. New coordinates: > > Neil J. A. Sloane, President, OEIS Foundation > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. > Also Visiting Scientist, Math. Dept., Rutgers University, > Piscataway, NJ. > Phone: 732 828 6098; home page: http://NeilSloane.com > Email: njasloane(a)gmail.com > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 136, Issue 27 > *****************************************

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Re: [math-fun] Before I make a fool of myself to Wolfram Support
by Bill Gosper 14 Jun '14

14 Jun '14

Message: 3 Date: Thu, 12 Jun 2014 16:36:30 -0700 From: Bill Gosper <billgosper(a)gmail.com> .. Can anyone justify this? In[1467]:= Table[(Cot[(k \[Pi])/p] Sin[k \[Pi]])/p, {p, 4}] Out[1467]= {Cos[k \[Pi]], 1/2 Cot[(k \[Pi])/2] Sin[k \[Pi]], 1/3 Cot[(k \[Pi])/3] Sin[k \[Pi]], 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]]} In[1468]:= FullSimplify[%, k \[Element] Integers] Out[1468]= {(-1)^k, Cos[(k \[Pi])/2]^2, 0, 0} Millions(?) of users, dozens of years, yet things like In[1471]:= FullSimplify[{I^k - I^(5*k) + 1, I^k - I^(5*k) + one}, k \[Element] Integers] Out[1471]= {1 + I^k - I^(5 k), one} Sometimes it feels like reality rot. --rwg Axel Vogt> I do not understand that Mathematica notations: do you mean cot(k*(Pi)/p)*sin(k*(Pi))/p for p = 1 ... 4, k integer? <AV Exactly. E.g., for p=4, Out[1745]= 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]] In[1746]:= Table[Limit[%, k -> n], {n, 0, 8}] Out[1746]= {1, 0, 0, 0, 1, 0, 0, 0, 1} (which certainly isn't identically 0). Except for needing a Limit, this seems like a handy source of basis elements for period p sequences. But for any specific p, we can TrigExpand and cancel the 0/0: In[1747]:= FullSimplify[TrigExpand[%% /. k -> 4*k] /. k -> k/4] Out[1747]= Cos[(k \[Pi])/4]^2 Cos[(k \[Pi])/2] obviating the Limit: In[1748]:= Table[%, {k, 0, 8}] Out[1748]= {1, 0, 0, 0, 1, 0, 0, 0, 1} (Although the trigexpansion can get cumbersome for large p.) Returning to the question of what would be the correct behavior of In[1468]:= FullSimplify[%, k \[Element] Integers] above, there was a Macsyma convention of yore to always assume, if possible, that the user has not erred (e.g., computed 0/0). This would dictate returning 0, which is needlessly misleading. A non-wrong, but useless alternative would be Indeterminate. Probably Mma's best alternative would be a Piecewise selecting between 0 and Indeterminate. In any event, the following is definitely wrong: Out[1500]= Cot[(\[Pi] x)/3] Sin[\[Pi] x] In[1501]:= Assuming[n \[Element] Integers, Limit[%, x -> n]] Out[1501]= 0 In[1502]:= Limit[%%, x -> 3] Out[1502]= -3 --rwg

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Re: [math-fun] Before I make a fool of myself to Wolfram Support
by Axel Vogt 14 Jun '14

14 Jun '14

> Message: 3 > Date: Thu, 12 Jun 2014 16:36:30 -0700 > From: Bill Gosper <billgosper(a)gmail.com> .. > Can anyone justify this? > In[1467]:= Table[(Cot[(k \[Pi])/p] Sin[k \[Pi]])/p, {p, 4}] > > Out[1467]= {Cos[k \[Pi]], 1/2 Cot[(k \[Pi])/2] Sin[k \[Pi]], > 1/3 Cot[(k \[Pi])/3] Sin[k \[Pi]], 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]]} > > In[1468]:= FullSimplify[%, k \[Element] Integers] > > Out[1468]= {(-1)^k, Cos[(k \[Pi])/2]^2, 0, 0} > > Millions(?) of users, dozens of years, yet things like > In[1471]:= FullSimplify[{I^k - I^(5*k) + 1, I^k - I^(5*k) + one}, > k \[Element] Integers] > > Out[1471]= {1 + I^k - I^(5 k), one} > > Sometimes it feels like reality rot. > --rwg I do not understand that Mathematica notations: do you mean cot(k*(Pi)/p)*sin(k*(Pi))/p for p = 1 ... 4, k integer?

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[math-fun] Trying to locate Marc Paulhus
by Neil Sloane 14 Jun '14

14 Jun '14

There are several references in the OEIS to a preprint: Marc Paulhus (paulhus(AT)wanadoo.nl), Pikelets, Discrete Math., (but apparently unpublished). I am trying to get hold of the article - can anyone help? Or, can anyone give me his email address? The one I have is broken. He has worked with Richard Guy and others on game theory: For example: Dan Calistrate, Marc Paulhus and David Wolfe, On the lattice structure of finite games, in More Games of No Chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ., 42, Cambridge Univ. Press, Cambridge, 2002, pp. 25-30. Thanks Neil -- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane(a)gmail.com

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[math-fun] Close enough for Government work
by Bill Gosper 14 Jun '14

14 Jun '14

Tentative quadrivariate 𝜗₁ identity: Out[742]= EllipticTheta[1, -X + Y, q] EllipticTheta[1, X + Y, q] EllipticTheta[ 1, W - Z, q] EllipticTheta[1, W + Z, q] - EllipticTheta[1, W - Y, q] EllipticTheta[1, W + Y, q] EllipticTheta[ 1, -X + Z, q] EllipticTheta[1, X + Z, q] + EllipticTheta[1, W - X, q] EllipticTheta[1, W + X, q] EllipticTheta[ 1, -Y + Z, q] EllipticTheta[1, Y + Z, q] (= 0?) Testing to 0ᵗʰ order: In[743]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 0}] Out[743]= O[q]^8 Nice. (Too nice? I requested essentially no terms.) OK, try 7ᵗʰ: In[745]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 7}] Out[745]= O[q]^8 Why no improvement? Trying for 8ᵗʰ, In[747]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 8}] // FunctionExpand hits a lacuna in Mma's 𝜗' knowledge: ((1/2 (-x+y)^2 (EllipticThetaPrime^(0,1,0))[1,0,0]+ <big mess>...)) So let's try random exact numerics: In[759]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}, {v, {W, X, Y, Z}}] Out[759]= {W -> -3 + 6 I, X -> -7 I, Y -> 6 - 6 I, Z -> -7 + 9 I} In[760]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}/9/Sqrt[2], {v, {q}}] Out[760]= {q -> -((5 I)/(9 Sqrt[2]))} In[761]:= %742 /. %% /. % Out[761]= EllipticTheta[1, -13 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 - I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 + 13 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -1 + 3 I, -((5 I)/(9 Sqrt[2]))] - EllipticTheta[1, -9 + 12 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 2 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 16 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 3, -((5 I)/(9 Sqrt[2]))] + EllipticTheta[1, -10 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 4 - 3 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 + I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 - 13 I, -((5 I)/(9 Sqrt[2]))] In[762]:= N[%] Out[762]= -2.20279*10^173 + 1.54582*10^173 I I was kinda hoping for 0. How discouraging. But wait! In[764]:= N[%%%, 999] During evaluation of In[764]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating EllipticTheta[1,-13+15 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3-I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3+13 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-1+3 I,-((5 I)/(9 Sqrt[2]))]-<<1>><<1>>EllipticTheta[1,-10+15 I,-((5 I)/(9 Sqrt[2]))] <<13>>[<<1>>] <<1>> <<1>>. >> Out[764]= 0.*10^-856 + 0.*10^-856 I How can people regard symbolic math as more exotic than numerics? --rwg

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[math-fun] earth has vast internal water reservoir?
by Warren D Smith 13 Jun '14

13 Jun '14

http://www.cbc.ca/news/technology/deep-earth-has-oceans-worth-of-water-10-d… >From tiny observations follow huge conclusions: Somebody found a diamond containing a mineral inclusion, said mineral 1.5% water. This"proves" there is a tremendous amount of water in Earth mantle diamond forming region, from which we extrapolate this region contains way more water than all oceans. This in turn suggests a geologic water cycle where ocean water subducted into the mantle, and outgassed from same... this is rather revolutionary change in view... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Before I make a fool of myself to Wolfram Support
by Bill Gosper 12 Jun '14

12 Jun '14

Can anyone justify this? In[1467]:= Table[(Cot[(k \[Pi])/p] Sin[k \[Pi]])/p, {p, 4}] Out[1467]= {Cos[k \[Pi]], 1/2 Cot[(k \[Pi])/2] Sin[k \[Pi]], 1/3 Cot[(k \[Pi])/3] Sin[k \[Pi]], 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]]} In[1468]:= FullSimplify[%, k \[Element] Integers] Out[1468]= {(-1)^k, Cos[(k \[Pi])/2]^2, 0, 0} Millions(?) of users, dozens of years, yet things like In[1471]:= FullSimplify[{I^k - I^(5*k) + 1, I^k - I^(5*k) + one}, k \[Element] Integers] Out[1471]= {1 + I^k - I^(5 k), one} Sometimes it feels like reality rot. --rwg

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[math-fun] Turing test in 10 years?
by Warren D Smith 12 Jun '14

12 Jun '14

I'm much less optimistic than Dan Asimov re machine intelligence. The difference with computer chess is, chess ratings allowed us to estimate progress and extrapolate it 10 (or whatever) years ahead. With the Turing test, I see no analogous thing and no way to extrapolate future progress. I actually invented a vastly superior measure of "intelligence" to the Turing test, which by the way also was invented (essentially the same idea) by Marcus Hutter: http://rangevoting.org/WarrenSmithPages/homepage/works.html paper #93 it to some extent would allow extrapolation of future progress, though it certainly is not as easily employed as chess ratings.

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[math-fun] San Jose Math Semicircle
by Bill Gosper 12 Jun '14

12 Jun '14

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/semizoom.gif <http://gosper.org/decazoom.gif> gives a Mnatsakanianian picture of why the area of a circle is the radius times half the circumference, defines π, and displays the beginning of its continued fraction. --rwg

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