October 2017 - mathfun@mailman.xmission.com

[math-fun] Fabricating a glass/crystal sphere circa 1500
by Bill Gosper 19 Nov '17

19 Nov '17

I watched a transit of Venus through an ~8" reflector made by Aerobie|Aeropress inventor Alan Adler, who precisely strained spherical mirrors. The editor of Sky & Telescope was very enthused, but the technique never caught on, apparently because CNC grinding of paraboloids got cheap. Alan is now using his extra mirrors under homemade tungsten tops that spin for twenty minutes. What is the optimal shape to trade air drag for moment of inertia? It turns out to be hard to model air drag. --rwg On 2017-10-23 14:37, Richard Howard wrote: > Large telescope mirrors start as extremely accurate sections of a sphere > made by carefully randomized grinding of two surfaces against each other. > The only two surfaces that fit together in all orientations are spheres. > > Starting with a rough glass sphere and a rough hemispherical hole in a > plate, continued random grinding produces a perfect sphere (and a perfect > hemispherical hole). > > BTW, the thermal approaches have the difficulty of dealing with the thermal > expansion of glass. A 8" sphere could take many months to anneal without > shattering. Corning museum has a nice video of making of the largest glass > paperweight (~13" diameter and over 109 lbs). > > --R > > On Mon, Oct 23, 2017 at 1:58 PM, Eugene Salamin via math-fun < > math-fun(a)mailman.xmission.com> wrote: > >> It was perhaps 58 years ago, in our High School auditorium, the speaker >> gave us a demo of this prestressed glass phenomenon. It was in the shape >> of a chemist's Florence flask, the rounded kind. He first used the flask >> to hammer a nail, no problem. Then he dropped in a speck of carborundum. >> It's harder than glass, and produces a tiny nick on impact. The flask just >> exploded. >> >> -- Gene >> >> >> On Monday, October 23, 2017, 1:29:12 PM PDT, Bill Gosper < >> billgosper(a)gmail.com> wrote: >> >> Interesting 16/sec ball bearing production process: >> https://www.youtube.com/watch?v=19duYMdiXi0 >> Falling raindrops form cabochons: >> https://www.youtube.com/watch?v=a9CRrGvQwe0 >> 1961|2 freshman chemistry lecturer demoed Prince Rupert's Drops, and then >> something >> more interesting. (Prince <othername>'s Drops? Google fails me.) >> Hollow, thick-walled glass blobs, open at one end. He hammered nails with >> one, >> then wrapped it in a towel, dropped in some carborundum crumbs, BAM! --rwg >> >> On 2017-10-22 11:57, Dan Asimov wrote: >> > Henry got that right — in fact, something quite interesting happens, >> > as Wikipedia says: >> > >> > ----- >> > Prince Rupert's Drops (also known as Dutch or Batavian tears) are >> > toughened glass beads created by dripping molten glass into cold >> > water, which causes it to solidify into a tadpole-shaped droplet with >> > a long, thin tail. These droplets are characterized internally by very >> > high residual stresses, which give rise to counter-intuitive >> > properties, such as the ability to withstand a blow from a hammer or a >> > bullet on the bulbous end without breaking, while exhibiting explosive >> > disintegration if the tail end is even slightly damaged. In nature, >> > similar structures are produced under certain conditions in volcanic >> > lava. >> > ----- >> > >> > more at https://en.wikipedia.org/wiki/Prince_Rupert's_Drop. >> > >> > —Dan >> >

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Re: [math-fun] math-fun Digest, Vol 176, Issue 41
by françois mendzina essomba2 31 Oct '17

31 Oct '17

Hi dear James, yes, looking at it, there may be confusion. The approach below with the limits is much clearer. Pi=Limit(2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^(-i-2)))^2)^(2^(i-1)),i,1,n))^(1/2^n))),n=infinity); The same be true for the other products in the list. best regards. Le Mardi 31 octobre 2017 19h00, "math-fun-request(a)mailman.xmission.com" <math-fun-request(a)mailman.xmission.com> a écrit : 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. Some formulas of pi in infinite product: (Fran?ois Mendzina Essomba) ---------------------------------------------------------------------- Message: 1 Date: Tue, 31 Oct 2017 11:26:58 +0100 From: Fran?ois Mendzina Essomba <m_essob(a)yahoo.fr> To: math-fun(a)mailman.xmission.com, m_essob(a)yahoo.fr Subject: [math-fun] Some formulas of pi in infinite product: Message-ID: <5044a46f-24cf-1b86-7252-5d15c807f852(a)yahoo.fr> Content-Type: text/plain; charset=utf-8; format=flowed Hello, Some formulas of pi in infinite product: (4/Pi)=product(1/(1-tan(Pi*2^(-n-3))^2),n=0..infinity); (4/Pi)=product((1/2)*(1+1/cos(Pi*2^(-n-2))),n=0..infinity); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^(-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((1-tan(Pi*2^(-n-2))^2)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product(2^(2^(n-1))/(1/cos(Pi*2^(-n-1))+1)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi/4=sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((1-tan(Pi*2^(-n-2))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)+2^(-4*n-2))); Pi/4=sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((2-1/(cos(Pi*2^(-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)+2^(-4*n-2))); 4/Pi=product(((cos((2*Pi)/8^n)+cos(Pi/8^n))*sec((2*Pi)/8^n))/8+(cos((3*Pi)/(2*8^n))*sec((2*Pi)/8^n))/4+((cos(Pi/8^n)+1)*sec((2*Pi)/8^n))/8+(cos(Pi/(2*8^n))*sec((2*Pi)/8^n))/4,n=1..infinity); Best regards. ------------------------------ 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 176, Issue 41 *****************************************

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[math-fun] Some formulas of pi in infinite product:
by François Mendzina Essomba 31 Oct '17

31 Oct '17

Hello, Some formulas of pi in infinite product: (4/Pi)=product(1/(1-tan(Pi*2^(-n-3))^2),n=0..infinity); (4/Pi)=product((1/2)*(1+1/cos(Pi*2^(-n-2))),n=0..infinity); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^(-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((1-tan(Pi*2^(-n-2))^2)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product(2^(2^(n-1))/(1/cos(Pi*2^(-n-1))+1)^(2^(n-1)),n=1..infinity))^(1/2^n))); Pi/4=sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((1-tan(Pi*2^(-n-2))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)+2^(-4*n-2))); Pi/4=sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((2-1/(cos(Pi*2^(-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)+2^(-4*n-2))); 4/Pi=product(((cos((2*Pi)/8^n)+cos(Pi/8^n))*sec((2*Pi)/8^n))/8+(cos((3*Pi)/(2*8^n))*sec((2*Pi)/8^n))/4+((cos(Pi/8^n)+1)*sec((2*Pi)/8^n))/8+(cos(Pi/(2*8^n))*sec((2*Pi)/8^n))/4,n=1..infinity); Best regards.

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[math-fun] corrective
by François Mendzina Essomba 27 Oct '17

27 Oct '17

The term ''new'' is not appropriate, I just noted the existence of a similar formula at http://functions.wolfram.com/ElementaryFunctions/Log/09/, where we find : ln(x) = Limit(w*(x^(1/w)-1),n=infinity) ; The formula 35 that I proposed could therefore be an improvement of the formula above, in fact: With the formula above: r:=evalf (2^(40)*(2^(1/2^(40))-1),40); r := .69314718056016379415253688694241624546042537621 With the proposed improvement, for x = 2 and n = 40: r1:=evalf (2^(40)*(2^(1/2^(40))-1)/(2^(1/2^(40+1))),40); r1 := .6931471805599453094172321334848075670387 By comparing: Ln(2)-r := -.2184847353047654842396773849e-12 Ln(2)-r1 := -.120266309989632e-25 Which in this case seems to indicate a double gain of decimals obtained.

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[math-fun] new formula of the natural logarithm and curious identities (2)..
by François Mendzina Essomba 27 Oct '17

27 Oct '17

Hello, Formula 40 Limit(2^(n)*((product(x(i),i=1..m))^(1/2^n)-1)/((product(x(i),i=1..m))^(1/(2^(n+1)))),n=infinity)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity),i=1..m); Formula 41 Limit(2^(n)*((product(x(i)/y(i),i=1..m))^(1/2^n)-1)/((product(x(i)/y(i),i=1..m))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity),i=1..m)-sum(Limit(2^(n)*(y(i)^(1/2^n)-1)/(y(i)^(1/(2^(n+1)))),n=infinity),i=1..m); Formula 42 Pi=Limit(2^(n)*(exp(2^(-n)*Pi)-1)/exp(2^(-(n+1)*Pi)),n=infinity); Formula 43 cosh(x)=Limit(2^(n-1)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))),n=infinity); formula 44 sinh(x)=Limit(2^(n-1)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))),n=infinity); formula 45 tanh(x)=Limit(((1/2)*2^(n)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))))/((1/2)*2^(n)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x))))),n=infinity); Formula 46 cotanh(x)= (((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))))/((1/2)*2^(n)*((exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x))))); Formula 47 cos(x)=Limit(2^n*exp(cos(x)/2^n-2^(-n-1)*cos(x))*cos(2^(-n-1)*sin(x))-2^n*exp(-2^(-n-1)*cos(x))*cos(2^(-n-1)*sin(x)),n=infinity); Formula 48 sin(x)=Limit(2^n*exp(cos(x)/2^n-2^(-n-1)*cos(x))*sin(2^(-n-1)*sin(x))+2^n*exp(-2^(-n-1)*cos(x))*sin(2^(-n-1)*sin(x)),n=infinity); Formula 49 Limit(-2^(2*n+1)*cos(sin(x)/2^n)+2^(2*n)*exp(2^(1-n)*cos(x)-cos(x)/2^n)+2^(2*n)*exp(-cos(x)/2^n),n=infinity)= 1; Best regards

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[math-fun] Anti-sphere?
by Leo Broukhis 26 Oct '17

26 Oct '17

Take a sphere with volume V and radius R. Its surface area is minimal among all shapes with volume V. What will happen if we try the opposite - *maximizing* the surface area of a shape with a given V, while restricting the max curvature of the surface (or, equivalently, requiring that the radius of curvature at any point of the surface be no less than a given fraction of R)? Intuitively, for convex shapes, the sphere should be gradually flattening to a pancake rather than stretching to a string. For simply connected shapes and a particular value of the curvature limit, the shape should look like an erythrocyte, I think; but for greater curvatures my intuition fails me. For arbitrary multiply connected shapes, things should be even more interesting. For example, at what value of the curvature the optimum becomes 2-connected, etc. I hope that this problem been considered before, and the only thing I'm missing is the proper search term. Thanks, Leo

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Re: [math-fun] math-fun Digest, Vol 176, Issue 38
by françois mendzina essomba2 26 Oct '17

26 Oct '17

RE: We have noted that Gene's x + sin x HAKMEM recurrence Hi Professor Bill Gosper ... x + sin (x) is the first term obtained by the general formula of October 23, 2017, which I proposed ... There is an undervaluation of the potential of this formula which is rather of convergence (2 * n +1) and not 2 * m ... I realized this, it's a confusion due to the mechanism that gave birth to him whose jumps from one point to another are rather 2 * m Best regards... Le Jeudi 26 octobre 2017 17h17, "math-fun-request(a)mailman.xmission.com" <math-fun-request(a)mailman.xmission.com> a écrit : 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. Fabricating a glass/crystal sphere circa 1500 (Bill Gosper) 2. We have noted that Gene's x+sin x HAKMEM recurrence (Bill Gosper) 3. A new formula of the natural logarithm of a positive real and curious identities .. (Fran?ois Mendzina Essomba) 4. Re: Fabricating a glass/crystal sphere circa 1500 (Richard Howard) ---------------------------------------------------------------------- Message: 1 Date: Wed, 25 Oct 2017 18:05:05 -0700 From: Bill Gosper <billgosper(a)gmail.com> To: math-fun(a)mailman.xmission.com Subject: [math-fun] Fabricating a glass/crystal sphere circa 1500 Message-ID: <CAA-4O0FYVud=ymNZVHtP+kxX+AG4Mp1-kOenZf=AxQi0zc6dAA(a)mail.gmail.com> Content-Type: text/plain; charset="UTF-8" I watched a transit of Venus through an ~8" reflector made by Aerobie|Aeropress inventor Alan Adler, who precisely strained spherical mirrors. The editor of Sky & Telescope was very enthused, but the technique never caught on, apparently because CNC grinding of paraboloids got cheap. Alan is now using his extra mirrors under homemade tungsten tops that spin for twenty minutes. What is the optimal shape to trade air drag for moment of inertia? It turns out to be hard to model air drag. --rwg On 2017-10-23 14:37, Richard Howard wrote: > Large telescope mirrors start as extremely accurate sections of a sphere > made by carefully randomized grinding of two surfaces against each other. > The only two surfaces that fit together in all orientations are spheres. > > Starting with a rough glass sphere and a rough hemispherical hole in a > plate, continued random grinding produces a perfect sphere (and a perfect > hemispherical hole). > > BTW, the thermal approaches have the difficulty of dealing with the thermal > expansion of glass. A 8" sphere could take many months to anneal without > shattering. Corning museum has a nice video of making of the largest glass > paperweight (~13" diameter and over 109 lbs). > > --R > > On Mon, Oct 23, 2017 at 1:58 PM, Eugene Salamin via math-fun < > math-fun(a)mailman.xmission.com> wrote: > >> It was perhaps 58 years ago, in our High School auditorium, the speaker >> gave us a demo of this prestressed glass phenomenon. It was in the shape >> of a chemist's Florence flask, the rounded kind. He first used the flask >> to hammer a nail, no problem. Then he dropped in a speck of carborundum. >> It's harder than glass, and produces a tiny nick on impact. The flask just >> exploded. >> >> -- Gene >> >> >> On Monday, October 23, 2017, 1:29:12 PM PDT, Bill Gosper < >> billgosper(a)gmail.com> wrote: >> >> Interesting 16/sec ball bearing production process: >> https://www.youtube.com/watch?v=19duYMdiXi0 >> Falling raindrops form cabochons: >> https://www.youtube.com/watch?v=a9CRrGvQwe0 >> 1961|2 freshman chemistry lecturer demoed Prince Rupert's Drops, and then >> something >> more interesting. (Prince <othername>'s Drops? Google fails me.) >> Hollow, thick-walled glass blobs, open at one end. He hammered nails with >> one, >> then wrapped it in a towel, dropped in some carborundum crumbs, BAM! --rwg >> >> On 2017-10-22 11:57, Dan Asimov wrote: >> > Henry got that right ? in fact, something quite interesting happens, >> > as Wikipedia says: >> > >> > ----- >> > Prince Rupert's Drops (also known as Dutch or Batavian tears) are >> > toughened glass beads created by dripping molten glass into cold >> > water, which causes it to solidify into a tadpole-shaped droplet with >> > a long, thin tail. These droplets are characterized internally by very >> > high residual stresses, which give rise to counter-intuitive >> > properties, such as the ability to withstand a blow from a hammer or a >> > bullet on the bulbous end without breaking, while exhibiting explosive >> > disintegration if the tail end is even slightly damaged. In nature, >> > similar structures are produced under certain conditions in volcanic >> > lava. >> > ----- >> > >> > more at https://en.wikipedia.org/wiki/Prince_Rupert's_Drop. >> > >> > ?Dan >> > ------------------------------ Message: 2 Date: Thu, 26 Oct 2017 02:12:52 -0700 From: Bill Gosper <billgosper(a)gmail.com> To: math-fun(a)mailman.xmission.com Subject: [math-fun] We have noted that Gene's x+sin x HAKMEM recurrence Message-ID: <CAA-4O0EknOVRyUmDQaRqVTqPrpgzZrgDm6gGf6XnB0BHv5enqg(a)mail.gmail.com> Content-Type: text/plain; charset="UTF-8" converges cubically to odd multiples of ?. E.g., In[539]:= Series[x+Sin@x,{x,3?,3}] Out[539]= 3 ?+1/6 (x-3 ?)^3+O[x-3?]^4 Rohan notes that x-sin x converges cubically to the even multiples: In[523]:= Series[x-Sin@x,{x,2?,3}] Out[523]= 2 ?+1/6 (x-2 ?)^3+O[x-2 ?]^4 and x+cos x converges cubically to (2n+1/2)?, In[519]:= Series[Cos[x]+x,{x,5?/2,3}] Out[519]= 5 ?/2+1/6 (x-(5?)/2)^3+O[x-5 ?/2]^4 In[540]:= Series[Cos[x]+x,{x,?/2,3}] Out[540]=?/2+1/6 (x-?/2)^3+O[x-?/2]^4 and x-cos x converges to (2n+3/2)?: In[522]:= Series[x-Cos[x],{x,3?/2,3}] Out[522]= (3?)/2+1/6 (x-(3?)/2)^3+O[x-(3?)/2]^4 --rwg "... given the difficulty related to the calculation of sinuses." Indeed few things are as difficult as pushing stones up your nose. ------------------------------ Message: 3 Date: Thu, 26 Oct 2017 13:47:07 +0200 From: Fran?ois Mendzina Essomba <m_essob(a)yahoo.fr> To: math-fun(a)mailman.xmission.com, m_essob(a)yahoo.fr Subject: [math-fun] A new formula of the natural logarithm of a positive real and curious identities .. Message-ID: <ae6c7966-c2c9-8df8-9c8b-e6f2813e101b(a)yahoo.fr> Content-Type: text/plain; charset=utf-8; format=flowed Hello, I found this formula which makes it possible to make an approximation of the natural logarithm of a number. Formula 35 ln(x)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right); I have deduced the following identities: Formula 36 Limit(2^(n)*((x*y)^(1/2^n)-1)/((x*y)^(1/(2^(n+1)))),n=infinity,right)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right)+Limit(2^(n)*((y)^(1/2^n)-1)/((y)^(1/(2^(n+1)))),n=infinity,right); Formula 37 Limit(2^(n)*((x/y)^(1/2^n)-1)/((x/y)^(1/(2^(n+1)))),n=infinity,right)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right)-Limit(2^(n)*((y)^(1/2^n)-1)/((y)^(1/(2^(n+1)))),n=infinity,right); Formula 38 Limit(2^(n)*((product(x(i),i=1..infinity))^(1/2^n)-1)/((product(x(i),i=1..infinity))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity); Formula 39 Limit(2^(n)*((product(x(i)/y(i),i=1..infinity))^(1/2^n)-1)/((product(x(i)/y(i),i=1..infinity))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity) - sum(Limit(2^(n)*(y(i)^(1/2^n)-1)/(y(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity); However, in the numerical results, formula 35 intrigues me about its convergence. It seems that for a given n, we get approximately the same number of exact decimals despite the size of the numbers. by example: X=55 ; u := ln(55) ~? 1048576/55*(55^(1/1048576)-1)*55^(2097151/2097152) 4.00733318523247091866270291119 = 4.00733318523490960132292657729 Exact number of decimal places : 11 X=5555555555555 u := ln(5555555555555) ~ 1048576/5555555555555*(5555555555555^(1/1048576)-1)*5555555555555^(2097151/2097152) 29.3458195440203748840441577653 = 29.3458195449780708532896646379 Exact number of decimal places : 9 Moreover, its convergence is not linear with n. Best Regards ------------------------------ Message: 4 Date: Thu, 26 Oct 2017 11:16:27 -0400 From: Richard Howard <rich(a)richardehoward.com> To: math-fun <math-fun(a)mailman.xmission.com> Subject: Re: [math-fun] Fabricating a glass/crystal sphere circa 1500 Message-ID: <CABkJZAZP9Mf5sMRC6MjLGYmRHB47KWpTOkKEGtCBb_DcUCUfwQ(a)mail.gmail.com> Content-Type: text/plain; charset="UTF-8" If you want to cheat and make a perfect parabola for a telescope mirror, it is hard to beat the spinning liquid trick. Mercury work well, if you want to only look up, but epoxy is more versatile (and less toxic). In the 60's I made a 120" f/1 mirror at Cal Tech out of epoxy poured on a roughly parabolic form and spun on air bearings. This mirror and its 60" brothers did the first IR sky surveys looking for dim stars, molecular clouds, etc. The parabola you get by spinning is not quite perfect enough for first-class optical imaging, but just fine for 2-10 micron IR. --R On Wed, Oct 25, 2017 at 9:05 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > I watched a transit of Venus through an ~8" reflector made by > Aerobie|Aeropress inventor Alan Adler, > who precisely strained spherical mirrors. The editor of Sky & Telescope > was very enthused, but the > technique never caught on, apparently because CNC grinding of paraboloids > got cheap. Alan is > now using his extra mirrors under homemade tungsten tops that spin for > twenty minutes. > What is the optimal shape to trade air drag for moment of inertia? It > turns out to be hard > to model air drag. --rwg > > On 2017-10-23 14:37, Richard Howard wrote: > > Large telescope mirrors start as extremely accurate sections of a sphere > > made by carefully randomized grinding of two surfaces against each other. > > The only two surfaces that fit together in all orientations are spheres. > > > > Starting with a rough glass sphere and a rough hemispherical hole in a > > plate, continued random grinding produces a perfect sphere (and a perfect > > hemispherical hole). > > > > BTW, the thermal approaches have the difficulty of dealing with the > thermal > > expansion of glass. A 8" sphere could take many months to anneal > without > > shattering. Corning museum has a nice video of making of the largest > glass > > paperweight (~13" diameter and over 109 lbs). > > > > --R > > > > On Mon, Oct 23, 2017 at 1:58 PM, Eugene Salamin via math-fun < > > math-fun(a)mailman.xmission.com> wrote: > > > >> It was perhaps 58 years ago, in our High School auditorium, the speaker > >> gave us a demo of this prestressed glass phenomenon. It was in the > shape > >> of a chemist's Florence flask, the rounded kind. He first used the > flask > >> to hammer a nail, no problem. Then he dropped in a speck of > carborundum. > >> It's harder than glass, and produces a tiny nick on impact. The flask > just > >> exploded. > >> > >> -- Gene > >> > >> > >> On Monday, October 23, 2017, 1:29:12 PM PDT, Bill Gosper < > >> billgosper(a)gmail.com> wrote: > >> > >> Interesting 16/sec ball bearing production process: > >> https://www.youtube.com/watch?v=19duYMdiXi0 > >> Falling raindrops form cabochons: > >> https://www.youtube.com/watch?v=a9CRrGvQwe0 > >> 1961|2 freshman chemistry lecturer demoed Prince Rupert's Drops, and > then > >> something > >> more interesting. (Prince <othername>'s Drops? Google fails me.) > >> Hollow, thick-walled glass blobs, open at one end. He hammered nails > with > >> one, > >> then wrapped it in a towel, dropped in some carborundum crumbs, BAM! > --rwg > >> > >> On 2017-10-22 11:57, Dan Asimov wrote: > >> > Henry got that right ? in fact, something quite interesting happens, > >> > as Wikipedia says: > >> > > >> > ----- > >> > Prince Rupert's Drops (also known as Dutch or Batavian tears) are > >> > toughened glass beads created by dripping molten glass into cold > >> > water, which causes it to solidify into a tadpole-shaped droplet with > >> > a long, thin tail. These droplets are characterized internally by very > >> > high residual stresses, which give rise to counter-intuitive > >> > properties, such as the ability to withstand a blow from a hammer or a > >> > bullet on the bulbous end without breaking, while exhibiting explosive > >> > disintegration if the tail end is even slightly damaged. In nature, > >> > similar structures are produced under certain conditions in volcanic > >> > lava. > >> > ----- > >> > > >> > more at https://en.wikipedia.org/wiki/Prince_Rupert's_Drop. > >> > > >> > ?Dan > >> > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > ------------------------------ 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 176, Issue 38 *****************************************

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[math-fun] A new formula of the natural logarithm of a positive real and curious identities ..
by François Mendzina Essomba 26 Oct '17

26 Oct '17

Hello, I found this formula which makes it possible to make an approximation of the natural logarithm of a number. Formula 35 ln(x)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right); I have deduced the following identities: Formula 36 Limit(2^(n)*((x*y)^(1/2^n)-1)/((x*y)^(1/(2^(n+1)))),n=infinity,right)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right)+Limit(2^(n)*((y)^(1/2^n)-1)/((y)^(1/(2^(n+1)))),n=infinity,right); Formula 37 Limit(2^(n)*((x/y)^(1/2^n)-1)/((x/y)^(1/(2^(n+1)))),n=infinity,right)=Limit(2^(n)*(x^(1/2^n)-1)/(x^(1/(2^(n+1)))),n=infinity,right)-Limit(2^(n)*((y)^(1/2^n)-1)/((y)^(1/(2^(n+1)))),n=infinity,right); Formula 38 Limit(2^(n)*((product(x(i),i=1..infinity))^(1/2^n)-1)/((product(x(i),i=1..infinity))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity); Formula 39 Limit(2^(n)*((product(x(i)/y(i),i=1..infinity))^(1/2^n)-1)/((product(x(i)/y(i),i=1..infinity))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity) - sum(Limit(2^(n)*(y(i)^(1/2^n)-1)/(y(i)^(1/(2^(n+1)))),n=infinity,right),i=1..infinity); However, in the numerical results, formula 35 intrigues me about its convergence. It seems that for a given n, we get approximately the same number of exact decimals despite the size of the numbers. by example: X=55 ; u := ln(55) ~ 1048576/55*(55^(1/1048576)-1)*55^(2097151/2097152) 4.00733318523247091866270291119 = 4.00733318523490960132292657729 Exact number of decimal places : 11 X=5555555555555 u := ln(5555555555555) ~ 1048576/5555555555555*(5555555555555^(1/1048576)-1)*5555555555555^(2097151/2097152) 29.3458195440203748840441577653 = 29.3458195449780708532896646379 Exact number of decimal places : 9 Moreover, its convergence is not linear with n. Best Regards

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[math-fun] We have noted that Gene's x+sin x HAKMEM recurrence
by Bill Gosper 26 Oct '17

26 Oct '17

converges cubically to odd multiples of π. E.g., In[539]:= Series[x+Sin@x,{x,3π,3}] Out[539]= 3 π+1/6 (x-3 π)^3+O[x-3π]^4 Rohan notes that x-sin x converges cubically to the even multiples: In[523]:= Series[x-Sin@x,{x,2π,3}] Out[523]= 2 π+1/6 (x-2 π)^3+O[x-2 π]^4 and x+cos x converges cubically to (2n+1/2)π, In[519]:= Series[Cos[x]+x,{x,5π/2,3}] Out[519]= 5 π/2+1/6 (x-(5π)/2)^3+O[x-5 π/2]^4 In[540]:= Series[Cos[x]+x,{x,π/2,3}] Out[540]=π/2+1/6 (x-π/2)^3+O[x-π/2]^4 and x-cos x converges to (2n+3/2)π: In[522]:= Series[x-Cos[x],{x,3π/2,3}] Out[522]= (3π)/2+1/6 (x-(3π)/2)^3+O[x-(3π)/2]^4 --rwg "... given the difficulty related to the calculation of sinuses." Indeed few things are as difficult as pushing stones up your nose.

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Re: [math-fun] Fabricating a glass/crystal sphere circa 1500
by Keith F. Lynch 25 Oct '17

25 Oct '17

Richard Howard <rich(a)richardehoward.com> wrote: > Large telescope mirrors start as extremely accurate sections of > a sphere made by carefully randomized grinding of two surfaces > against each other. The only two surfaces that fit together in > all orientations are spheres. Nitpick: Flat surfaces also fit together in all orientations. I suppose a flat surface can be considered part of a sphere of infinite radius. Indeed, the method you describe is also used to make extremely flat mirrors, except that three surfaces are used, two at a time of which are ground against each other. I recently mentioned this in another forum in which we were discussing whether heliographs could be used for long distance communication on the Ringworld (an immense fictional solid ring around a sun-like star) without high technology. I argued that it should be possible to send something like Morse code over distances up to about 100 million kilometers if the recipient has good binoculars, even if the heliograph mirror is no larger than the Great Pyramid, a size that pre-technological cultures could obviously build things. > Starting with a rough glass sphere and a rough hemispherical hole in > a plate, continued random grinding produces a perfect sphere (and a > perfect hemispherical hole). How was the spherical gyro in Gravity Probe B, supposedly the roundest object ever made, constructed? My understanding is that it wasn't round enough, and the measurement of the frame dragging effect was below the noise floor, though some claim the noise could be compensated for. It's undisputed that it did measure the much larger space-time curvature effect. (When Magellan's crew returned to Europe after sailing consistently west, they thought that was because the world was round. They were mostly right, but if they had kept very close track of the curvature, they'd have noticed that they were short by about an inch. The missing inch was because of the space-time curvature around the Earth.) (Copernicus, Galileo, Kepler et al believed that the celestial sphere stood still and Earth rotated. They were mostly right, but if they had kept close track, they would have noticed that the celestial sphere rotates by about 0.04 seconds of arc per year due to frame dragging.) As for Rupert's drops, you can find high-speed videos on YouTube. Some show that the drops aren't harmed by being shot by a rifle on their thick part. Others show that the slightest tap on the narrow part makes them explode, and that the shock travels up the drop at several kilometers per second. Never watch this in person without safety goggles!

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