Re: [math-fun] math-fun Digest, Vol 176, Issue 38
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@mailman.xmission.com" <math-fun-request@mailman.xmission.com> a écrit : Send math-fun mailing list submissions to math-fun@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@mailman.xmission.com You can reach the person managing the list at math-fun-owner@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@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Fabricating a glass/crystal sphere circa 1500 Message-ID: <CAA-4O0FYVud=ymNZVHtP+kxX+AG4Mp1-kOenZf=AxQi0zc6dAA@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@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@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@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] We have noted that Gene's x+sin x HAKMEM recurrence Message-ID: <CAA-4O0EknOVRyUmDQaRqVTqPrpgzZrgDm6gGf6XnB0BHv5enqg@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@yahoo.fr> To: math-fun@mailman.xmission.com, m_essob@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@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@richardehoward.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Fabricating a glass/crystal sphere circa 1500 Message-ID: <CABkJZAZP9Mf5sMRC6MjLGYmRHB47KWpTOkKEGtCBb_DcUCUfwQ@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@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@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@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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