June 2017 - mathfun@mailman.xmission.com

[math-fun] Cantor set overlaps
by Keith F. Lynch 11 Jun '17

11 Jun '17

According to http://www.math.brown.edu/~rkenyon/openprobs/ it is unknown whether the middle-third and middle-half Cantor sets have any irrational number in common. The middle-third Cantor set is equivalent to numbers in the unit interval expressible in ternary with only 0s and 2s. The middle-half Cantor set is equivalent to numbers in the unit interval expressible in quaternary with only 0s and 3s. I considered writing a program to convert every 20-digit (after the radix point) ternary number in the set into quaternary, and see if any of the 2^20 possible sums consist -- within the precision -- entirely of 0s and 3s. My guess is that none would, except for some repeating numbers such as 0.00000000..., 0.02020202..., 0.20202020..., and 0.22222222..., all of which are of course rational. That's because such a small proportion (2^-N) of N-digit quaternary numbers consist only of 0s and 3s. If none did, that would prove nothing, since there could be a solution that stops repeating after 20 places. And if some did, that would prove nothing either, since they could stop working after 20 places (unless I could find some non-repeating pattern of 0s and 2s that would keep working forever -- I assume someone has already looked). So I didn't bother. Similarly if I did it the other way around, converting all 2^20 20-digit quaternary numbers in the set into ternary. My guess is that there's no overlap, since both sets are of measure zero. So an "accidental" overlap would be an astonishing coincidence. But that raises the question of overlaps in Cantor-like sets that are *not* of measure zero. For instance consider the set in which, instead of removing the middle half each time, you remove the middle 8th, then the middle 16th, then the middle 32nd, etc. This set has measure 0.577.... Or the set in which, instead of removing the middle third each time, you remove the middle third, then the middle ninth, then the middle 27th, etc. This set has measure 0.560.... My intuition says that since those two sets are each in the unit interval, which of course has measure 1, and since they each have a measure of more than 1/2, that by the pigeonhole principle overlaps are inevitable. (Again, I'm talking only about irrational numbers. The rationals have measure 0.) But reasoning about infinite sets is tricky. Especially uncountable infinite sets. Wikipedia says the pigeonhole principle is valid for infinite sets, but the only examples it gives are of infinities of different cardinalities, which is not the case here. I think "measure" means that if I choose a large number of random numbers in the unit interval, 58% of them will be in the first set and 56% of them will be in the second set. That certainly seems to imply to me that at least 14% of them must be in both sets. Of course that doesn't mean I'll be able to actually *find* any numbers in both sets. I don't think generating random real numbers in the unit interval is computable, even given a finite-speed perfect random bit generator. However many digits you generate, a later digit may knock the number out of the target set. What do my two sets look like in numbers? In the first set, in binary the first three digits must not be 011 or 100, then the next four digits must not be 0111 or 1000, then the next five digits must not be 01111 or 10000, etc. In the second set, in ternary the first digit must not be 1, then the next two digits must not be 11, then the next three digits must not be 111, etc. You can see how the probability of choosing a random next digit which takes you outside the set drops so quickly that you have a better than even chance of remaining in the set forever. But as a practical matter, you can't actually generate random numbers in the set. Nor can you ever be sure that 1/pi, 1/e, 1/sqrt(2), 1/phi, etc., is in either set, at least not if you're stuck with an algorithm that generates those numbers one digit at a time. (Is Plouffe reading this?) (You can be sure -- eventually -- if it's *not* in the set. This is the halting probem.) To find a number in the set, you need to have some pattern. Some non- repeating pattern, since we're only interested in irrationals. For instance 0.2202002000200002... with the number of 0s increasing by one each time. Then you have to prove there's a similar pattern when you convert to the other base. Good luck with that.

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Re: [math-fun] When I'm 64
by Henry Baker 09 Jun '17

09 Jun '17

At 06:24 PM 6/7/2017, Victor Miller wrote: >I remember an IBM FORTRAN manual stating that arrays were stored in >"natural" order -- backwards! The index registers on the 709/7090/7094 subtracted. Yes! Curiously, IBM saved money on the 7040 (a poor man's 7090) by dispensing with 4 of the 7 index registers -- remember, each bit of an index register was an entire card full of transistor electronics (I seem to recall), so 4x15 = 60 cards saved. So the question is: what happened if you specified index register 5 = 0b101 (which didn't exist on the 7040) ? Answer: the 7040 OR'd the bits of register 1 = 0b001 and register 4 = 0b100, providing a really, really obscure way to OR two 15-bit quantities! Needless to say, the 7040 Fortran compiler didn't make use of this capability, but someone in the medical/statistical computing group that I worked for did: he arranged 3D subarrays and used the 3 index registers for the x,y,z coordinates -- he could index up to a 32x32x32 3-D array this way. (We were simulating the effects of radiation on human tissue, so 3D "voxels" -- although that term came decades later -- were important.) The 7040 never trapped/interrupted on an un-recognized op code; it simply did the best that it could. So we tried to completely map out what all of the non-standard op codes did. Sadly, we never found anything terribly useful -- the best we could do was a block-move-and-complement-every-other-word! Good luck trying to do something useful with that one!

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[math-fun] When I'm 64
by Henry Baker 09 Jun '17

09 Jun '17

FYI -- "Will you still need me, will you still feed me When I'm 64?" I've now lived through: decimal computers (IBM 1620, IBM 1401) 36-bit/15-18bit-address computers (IBM 7040, IBM 7090, GE635, PDP-10, DEC-20) 12-bit computers (PDP-8) Microcoded machines (Xerox Alto, Lisp Machines) 1-bit computers (ICL DAP 32x32x1 "square word", Connection Machine) 4-bit computers (Intel 4004) 8-bit computers (6502, Z-80, 8080, 8051 (still used inside "USB flash drives"!), 6800, etc.) 16-bit computers (PDP-11, Moto 68000, 8086, etc.) 32-bit computers (IBM 360, VAX, 80386, ARM, etc.) 64-bit computers (everyone!) It remains to be seen if I'll make it to 128 or 256... Son of a bits! --- https://apple.slashdot.org/story/17/06/07/0538233/apple-to-phase-out-32-bit… Apple To Phase Out 32-Bit Mac Apps Starting In January 2018 Apple will be phasing out 32-bit apps with iOS 11, and soon the company will make the same changes on its macOS operating system. During its Platform State of the Union keynote at the Worldwide Developers Conference, Apple told developers that macOS High Sierra will be the "last macOS release to support 32-bit apps without compromises."

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[math-fun] Minsky/Trinsky questions
by Bill Gosper 08 Jun '17

08 Jun '17

On 2017-06-06 06:09, Henry Baker wrote: > Now you've lost me. > > The the video starts by talking about the Minsky algorithm over the integers. > > Where did the reals come in? I am genuinely sorry to say: Gotcha. The talk never said x₀ and y₀ were integers. Regrettably, it doesn't exhibit a case where they're not. But they always change by integers--their fractional parts stay constant. Adding small fractions to x₀ and y₀ typically just shifts the first few values by those offsets, but at some point a Floor will be different and the sequences will veer apart. > > Also, if you're calculating with floating point numbers, how can you > be so sure that the algorithm is exactly invertible? Good point. We need x + ⎣y⎦ - ⎣y⎦ to be x. This can fail when |y| > |x| , as when crossing the y axis, or when |δ| or |ε| is large. --rwg > At 01:15 AM 6/6/2017, Bill Gosper wrote: >>Important: >> >>The algorithm described in the talk and the book is over the reals, not the integers. > > > _______________________________________________

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Re: [math-fun] When I'm 64
by Bill Gosper 08 Jun '17

08 Jun '17

On 2017-06-07 18:54, Tom Knight wrote: > One of the machines the Livermore S1 project was supposed to replace > was a weird variant of the ANFSQ-7, a 32 bit machine. But an Admiral > apparently heard about the wonders of parity checking, and insisted > his new machine have parity. The new machine became a 30 bit machine, > with two bits stolen for parity. Tasteful. AN/USQ-17. I was told in St. Paul that it was the last UNIVAC Seymour Cray ever designed. He refused to cooperate with the rest of the designers, who then came up with nothing, so UNIVAC had to build Cray's. It evolved to the UNIVAC 1206, via the similar looking https://en.wikipedia.org/wiki/AN/USQ-20#/media/File:AN_USQ-20.jpg . It was incredibly slow, 115μec to divide, but it was built like a bank vault. Those bay doors had meshing brass teeth. It was air cooled with a deafening scream. You could Bernoulli-levitate a football in the exhaust from the stack. I wrote it a DDT-style interactive debugger called NIP, which stood for nothing in particular. I had to lobby really hard for UNIVAC to disconnect a device built into the FlexoWriter stand that disabled input for many seconds following output, so that no one could waste precious machine time interactively. The console was amazing, availing every flipflop in the processor as an illuminated button. The CPU had a crazy RPT instruction that repeated the succeeding instruction a number of times specified by the address field, optionally bumping the address field of the repeatee. With the machine halted, it was possible to set up a zero-instruction "program" to single-step compute small factorials: RPT ADVANCE MULTIPLY IMMEDIATE 1. Some years ago I war-storied this list about the Big Red Fault Light. These were 1's complement machines that (usually) evaded minus 0 via hardware subtract with end-around borrow, instead of the usual adder with end-around carry. UNIVAC remained unable to design a new processor, so they kept repackaging Seymour's with hilariously marketerized instruction names, concealed by a "Real time, high level language, problem oriented compiler" that was actually a braindead assembler. To index location foo, you had to write RPL·Y+1·W(FOO) . Of course the "compiler" for some reason would not let you add 2 via RPL·Y+2·W(FOO). UNIVAC never told its programmers there was no add 2 instruction. They renamed all the instructions yet again, selling now a "business machine" under the names M460 and M490. 30 bits. Half a billion cents. Do > $5 million in business and suddenly you're $5 million in the red. At my Navy job, the 1206 supplanted the even weirder P(ackard) B(ell) 250, http://www.cca.org/tech/rcs/pb250.html , 22 bits/wd. Totally volatile magnetostrictive delay line (serial) memory. Even the active registers! The divide instruction took 3 MILLISEC, unless you "sequence tagged" it, meaning to skip over as many instructions as the number of specified quotient bits! Depending on signs, there were, I think, seven ways the quotient and remainder could be wrong. But the machine had a completely reliable square root instruction! It is atrocious that subsequent computers didn't. Memory was divided into 256 word loops. Except some for data were 255 and some were 257, so a trivial program could modify a table by repeatedly modifying the "same" location. We programmed in octal. --rwg > >> On Jun 7, 2017, at 8:48 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> The MIT Life hackers once got an absolutely surreal demo of the "PDP-3" in >> the basement of Searle Medidata, out on rte 128, by it's sole caretaker, >> builder(?), >> and programmer Charles Corderman. >> >> If memory serves, some salesman leaked that DEC was building a 36 bit >> challenge >> to IBM's mainframes, and Searle staked much on this belief. When it became >> clear that DEC was abandoning the project, Searle insisted on seeing the >> plans, >> and charged Corderman with completing them. The machine computed parity >> nearly >> everywhere. What a waste of bits. So Corderman changed it to a 37bit >> machine! >> He then built an *entirely* menu-driven OS, where the only pointing device >> was >> a light pen! To make things weirder, the Type 30 display had a purple >> phosphor. >> On the console was a Flexowriter, powered off. We expressed suspicion that >> he had programmed his (utterly peculiar) OS on it. "No, but it does work," >> he >> said, powering it on, and waving his light pen. The Flexowriter typed a >> lowercase >> "r", and he powered it off. >> >> (Kids: The only data from a light pen is a single bit, saying whether or >> not it saw >> the most recent point flashed on the (non rastered) screen. During one >> second, >> the machine was only fast enough to light a very few thousand of the 2^20 >> addressable points. The pen's field of view depended on how close it was >> to >> the screen, but it needed to be pretty close. How can you possibly do >> anything useful with such a thing?) >> >> This is the machine with which Corderman found his famous Switch Engines, >> http://www.conwaylife.com/w/index.php?title=Switch_engine , bizarre c/12 >> diagonal puffers, one of which shoots (Doppler shifted) gliders out the >> front! >> >> Or did I dream all this? --rwg >> >> On 2017-06-07 16:39, Fred Lunnon wrote: >>> Manchester Atlas I with 48-bit words. WFL >>> >>> On 6/8/17, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >> wrote: >>>> My first computer was the IBM 650. It had 2000 10-decimal digit words >>>> stored on a rotating magnetic drum. Senior year in high school class in >>>> applied math in 1959, IBM gave us access to their computer. It was punch >>>> cards, card sorters, and line printers. >>>> >>>> -- Gene >>>> >>>> >>>> On Wednesday, June 7, 2017, 3:52:26 PM PDT, Victor Miller >>>> <victorsmiller(a)gmail.com> wrote: >>>> >>>> And 60 bit (cdc). I also programmed a univac 418 -- 18 bit 1's >> complement >>>> >>>> >>>> On Wed, Jun 7, 2017 at 18:40 Tom Knight <tk(a)mit.edu> wrote: >>>> >>>>> You left out the 40 bit machines… >>>>> >>>>>> On Jun 7, 2017, at 3:50 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>>>>> >>>>>> FYI -- >>>>>> >>>>>> "Will you still need me, will you still feed me >>>>>> When I'm 64?" >>>>>> >>>>>> I've now lived through: >>>>>> >>>>>> decimal computers (IBM 1620, IBM 1401) >>>>>> 36-bit/15-18bit-address computers (IBM 7040, IBM 7090, GE635, PDP-10, >>>>> DEC-20) >>>>>> 12-bit computers (PDP-8) >>>>>> Microcoded machines (Xerox Alto, Lisp Machines) >>>>>> 1-bit computers (ICL DAP 32x32x1 "square word", Connection Machine) >>>>>> 4-bit computers (Intel 4004) >>>>>> 8-bit computers (6502, Z-80, 8080, 8051 (still used inside "USB flash >>>>> drives"!), 6800, etc.) >>>>>> 16-bit computers (PDP-11, Moto 68000, 8086, etc.) >>>>>> 32-bit computers (IBM 360, VAX, 80386, ARM, etc.) >>>>>> 64-bit computers (everyone!) >>>>>> >>>>>> It remains to be seen if I'll make it to 128 or 256... >>>>>> >>>>>> Son of a bits! >>>>>> --- >>>>>> >>>>> >> https://apple.slashdot.org/story/17/06/07/0538233/apple-to-phase-out-32-bit… >>>>>> >>>>>> Apple To Phase Out 32-Bit Mac Apps Starting In January 2018 >>>>>> >>>>>> Apple will be phasing out 32-bit apps with iOS 11, and soon the >> company >>>>> will make the same changes on its macOS operating system. During its >>>>> Platform State of the Union keynote at the Worldwide Developers >>>>> Conference, >>>>> Apple told developers that macOS High Sierra will be the "last macOS >>>>> release to support 32-bit apps without compromises." >>>> _______________________________________________ >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Fwd: Fibonacci divisibility
by Bill Gosper 07 Jun '17

07 Jun '17

Martin Gardner would have loved this. --rwg ---------- Forwarded message ---------- From: GEORGE E ANDREWS <gea1(a)psu.edu> Date: Wed, Jun 7, 2017 at 6:57 PM Subject: Re: Fibonacci divisibility To: billgosper(a)gmail.com Cc: gea1 <gea1(a)psu.edu> Dear Bill, Davis's Moriarty identities are, in fact, instances of the Chu-Vandermonde summation. Thus it appears Moriarty turned to crime even earlier than thought previously. Best wishes, George ------------------------------ *From: *"Bill Gosper" <billgosper(a)gmail.com> *To: *math-fun(a)mailman.xmission.com *Cc: *"Julian Ziegler Hunts" <julianj.zh(a)gmail.com>, "Zack Chroman" < zacchro(a)gmail.com>, "Rohan Ridenour" <catisfluffy(a)hladkyschool.org>, "Neil Bickford" <techie314(a)gmail.com>, "Nico Brown" <nicobrownmath(a)gmail.com> *Sent: *Wednesday, June 7, 2017 9:54:32 PM *Subject: *Fwd: Fibonacci divisibility Hilbert's 10th was: How do you find the integer solutions (= "Diophantine") to multivariate polynomials? Matiyasevic et al showed that, in general, you can't. The problem is Turing complete. For example, somebody constructed a polynomial in n, p, and a couple of dozen auxiliary variables that can only vanish when p is the nth prime! Matiyasevic once sent me a book. (Soviet era, incredibly cheaply made.) Insensitive to what must have been his austere economic situation, I sent him several packets of Pop Rocks. I addressed the envelope in Cyrillic, and John McCarthy impishly had me append "имени Ке́ренский" to the name of the research facility. Kerensky was much despised in Soviet propaganda. I think the book was one about hypergeometric identities. One such, written with binomial coefficients, mentioned "Moriarty" in cyrillic. I hit the ceiling, thinking that the Russians had fallen for some prank, based on the Sherlock Holmes quote, "At the age of twenty-one, he wrote a treatise upon the binomial theorem <https://en.wikipedia.org/wiki/Binomial_theorem>, which has had a European vogue." I sent a xerox to Martin Gardner, who sent it to the Baker Street Irregulars, who sent it to the CIA. But the hoax was on me. One Harold Davis discovered the theorem, and playfully named it Moriarty's. https://en.wikipedia.org/wiki/A_Treatise_on_the_Binomial_Theorem Had my Russian been better, I'd've noticed that "Дэвиса" was not a "device", but rather how they spell "Davis". --rwg ---------- Forwarded message ---------- From: Warren D Smith <warren.wds(a)gmail.com> Date: Wed, Jun 7, 2017 at 4:04 PM Subject: Fibonacci divisibility To: billgosper(a)gmail.com, marniekanarek(a)gmail.com The fact that GCD[Fibonacci[a], Fibonacci[b]] == Fibonacci[GCD[a, b]], is proven as lemma 4 on p.221 in John H. Halton: ON THE DIVISIBILITY PROPERTIES OF FIBONACCI NUMBERS (1966) http://www.fq.math.ca/Scanned/4-3/halton.pdf Halton also in lemma 5 says F_m is divisible by F_n , if and only if either m is divisible by n, or n = 2. Divisibility properties like these were a key ingredient of the famous proof by Julia Robinson et al, that established the undecidability of diophantine equation solving. Robinson et al had everything they needed except for this missing ingredient, which was found by Yuri Matiyasevich thus competing the proof in 1970. However, it would seem Robinson et al could have avoided the need for Matiyasevic if they had bothered to read this 1966 paper by Halton? See https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem

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[math-fun] Fwd: Fibonacci divisibility
by Bill Gosper 07 Jun '17

07 Jun '17

Hilbert's 10th was: How do you find the integer solutions (= "Diophantine") to multivariate polynomials. Matiyasevic et al showed that, in general, you can't. The problem is Turing complete. For example, somebody constructed a polynomial in n, p, and a couple of dozen auxiliary variables that can only vanish when p is the nth prime! Matiyasevic once sent me a book. (Soviet era, incredibly cheaply made.) Insensitive to what must have been his austere economic situation, I sent him several packets of Pop Rocks. I addressed the envelope in Cyrillic, and John McCarthy impishly had me append "имени Ке́ренский" to the name of the research facility. Kerensky was much despised in Soviet propaganda. I think the book was one about hypergeometric identities. One such, written with binomial coefficients, mentioned "Moriarty" in cyrillic. I hit the ceiling, thinking that the Russians had fallen for some prank, based on the Sherlock Holmes quote, "At the age of twenty-one, he wrote a treatise upon the binomial theorem <https://en.wikipedia.org/wiki/Binomial_theorem>, which has had a European vogue." I sent a xerox to Martin Gardner, who sent it to the Baker Street Irregulars, who sent it to the CIA. But the hoax was on me. One Harold Davis discovered the theorem, and playfully named it Moriarty's. https://en.wikipedia.org/wiki/A_Treatise_on_the_Binomial_Theorem Had my Russian been better, I'd've noticed that "Дэвиса" was not a "device", but rather how they spell "Davis". --rwg ---------- Forwarded message ---------- From: Warren D Smith <warren.wds(a)gmail.com> Date: Wed, Jun 7, 2017 at 4:04 PM Subject: Fibonacci divisibility To: billgosper(a)gmail.com, marniekanarek(a)gmail.com The fact that GCD[Fibonacci[a], Fibonacci[b]] == Fibonacci[GCD[a, b]], is proven as lemma 4 on p.221 in John H. Halton: ON THE DIVISIBILITY PROPERTIES OF FIBONACCI NUMBERS (1966) http://www.fq.math.ca/Scanned/4-3/halton.pdf Halton also in lemma 5 says F_m is divisible by F_n , if and only if either m is divisible by n, or n = 2. Divisibility properties like these were a key ingredient of the famous proof by Julia Robinson et al, that established the undecidability of diophantine equation solving. Robinson et al had everything they needed except for this missing ingredient, which was found by Yuri Matiyasevich thus competing the proof in 1970. However, it would seem Robinson et al could have avoided the need for Matiyasevic if they had bothered to read this 1966 paper by Halton? See https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem

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Re: [math-fun] When I'm 64
by Bill Gosper 07 Jun '17

07 Jun '17

The MIT Life hackers once got an absolutely surreal demo of the "PDP-3" in the basement of Searle Medidata, out on rte 128, by it's sole caretaker, builder(?), and programmer Charles Corderman. If memory serves, some salesman leaked that DEC was building a 36 bit challenge to IBM's mainframes, and Searle staked much on this belief. When it became clear that DEC was abandoning the project, Searle insisted on seeing the plans, and charged Corderman with completing them. The machine computed parity nearly everywhere. What a waste of bits. So Corderman changed it to a 37bit machine! He then built an *entirely* menu-driven OS, where the only pointing device was a light pen! To make things weirder, the Type 30 display had a purple phosphor. On the console was a Flexowriter, powered off. We expressed suspicion that he had programmed his (utterly peculiar) OS on it. "No, but it does work," he said, powering it on, and waving his light pen. The Flexowriter typed a lowercase "r", and he powered it off. (Kids: The only data from a light pen is a single bit, saying whether or not it saw the most recent point flashed on the (non rastered) screen. During one second, the machine was only fast enough to light a very few thousand of the 2^20 addressable points. The pen's field of view depended on how close it was to the screen, but it needed to be pretty close. How can you possibly do anything useful with such a thing?) This is the machine with which Corderman found his famous Switch Engines, http://www.conwaylife.com/w/index.php?title=Switch_engine , bizarre c/12 diagonal puffers, one of which shoots (Doppler shifted) gliders out the front! Or did I dream all this? --rwg On 2017-06-07 16:39, Fred Lunnon wrote: > Manchester Atlas I with 48-bit words. WFL > > On 6/8/17, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> wrote: >> My first computer was the IBM 650. It had 2000 10-decimal digit words >> stored on a rotating magnetic drum. Senior year in high school class in >> applied math in 1959, IBM gave us access to their computer. It was punch >> cards, card sorters, and line printers. >> >> -- Gene >> >> >> On Wednesday, June 7, 2017, 3:52:26 PM PDT, Victor Miller >> <victorsmiller(a)gmail.com> wrote: >> >> And 60 bit (cdc). I also programmed a univac 418 -- 18 bit 1's complement >> >> >> On Wed, Jun 7, 2017 at 18:40 Tom Knight <tk(a)mit.edu> wrote: >> >>> You left out the 40 bit machines… >>> >>> > On Jun 7, 2017, at 3:50 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>> > >>> > FYI -- >>> > >>> > "Will you still need me, will you still feed me >>> > When I'm 64?" >>> > >>> > I've now lived through: >>> > >>> > decimal computers (IBM 1620, IBM 1401) >>> > 36-bit/15-18bit-address computers (IBM 7040, IBM 7090, GE635, PDP-10, >>> DEC-20) >>> > 12-bit computers (PDP-8) >>> > Microcoded machines (Xerox Alto, Lisp Machines) >>> > 1-bit computers (ICL DAP 32x32x1 "square word", Connection Machine) >>> > 4-bit computers (Intel 4004) >>> > 8-bit computers (6502, Z-80, 8080, 8051 (still used inside "USB flash >>> drives"!), 6800, etc.) >>> > 16-bit computers (PDP-11, Moto 68000, 8086, etc.) >>> > 32-bit computers (IBM 360, VAX, 80386, ARM, etc.) >>> > 64-bit computers (everyone!) >>> > >>> > It remains to be seen if I'll make it to 128 or 256... >>> > >>> > Son of a bits! >>> > --- >>> > >>> https://apple.slashdot.org/story/17/06/07/0538233/apple-to-phase-out-32-bit… >>> > >>> > Apple To Phase Out 32-Bit Mac Apps Starting In January 2018 >>> > >>> > Apple will be phasing out 32-bit apps with iOS 11, and soon the company >>> will make the same changes on its macOS operating system. During its >>> Platform State of the Union keynote at the Worldwide Developers >>> Conference, >>> Apple told developers that macOS High Sierra will be the "last macOS >>> release to support 32-bit apps without compromises." >> _______________________________________________

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[math-fun] Minsky/Trinsky questions
by Bill Gosper 07 Jun '17

07 Jun '17

Date: 2017-06-05 17:48 From: Henry Baker <hbaker1(a)pipeline.com> To: math-fun(a)mailman.xmission.com Reply-To: math-fun <math-fun(a)mailman.xmission.com> >What is known about which orbit sizes are possible? By size, you must mean period, since x₀ and y₀ can be anything. I'm pretty sure we found many a small set of multipliers (δ, ε) whose periods grew periodically with (x₀, 0), say, covering the integers. If not, I'll bet Julian can construct such a covering fairly easily. Another approach: Removing the Floor functions yields an exactly solvable recurrence of any real "period", independent of (x₀, y₀). When these are large, restoring the Floor functions has only a small effect. I'm guessing you'd only need to try a few. Which isn't much help if you need exactly a googol and one. >What is known about which orbit sizes are impossible? The above, if correct, says no period is impossible. >How many bits of state are required for an orbit of size N ? (non-trivial max ?, non-trivial min ?) For fixed multipliers (δ, ε), the state is entirely contained in the current point (xn,yn). Usually, a very small shift in x₀ and y₀ will simply translate the whole orbit by that shift. Often, a small shift in (δ, ε) will not change the orbit at all. E.g., the large rectangles in the (mysteriously cropped) "Monster" plot on p12 of gosper.org/g12a.pdf (un-animated part of the talk). But see the discussion of accumulation points in the book. (Speaking of the animations, the one Rohan misprepared, "The worst circle I ever drew", should have drawn gosper.org/circle.png . Try to picture a Minsky making that!) >The video <https://youtu.be/lXsVWwPa7bc> talks about parallelizing; is there any kind of "speedup" -- e.g., repeated squaring of matrices or some such? No. NeilB figured out how to heuristically subdivide the space of possible (x,y) starting points with about a 12% chance of connecting up with the desired orbit, so breakeven needed about eight processors. Although TomR never managed to close the Minsky Stock Index, he did manage to close disjoint loops of record length, in the quadrillions. >Suppose that a Minsky orbit of size N is known; how hard is it to map the integers 1..N to the elements of the orbit, You just count. We don't have a way of jumping ahead a prescribed distance. > and how hard is it to map back? Or is a table lookup the fastest method in both directions? More generally, can I use Minsky orbits for addition mod N ? I don't know. Which is the answer to most questions about Minskyspace. >Are there any Minsky orbits which "decompose" into direct products of Minsky orbits? (I'm not entirely certain how such a direct product would work, but perhaps the peculiarities of particular numbers might allow the separation of a "high order" part and a "low order" part that happen not to interfere with one another.) Julian? Help! >Does Minsky/Trinsky work on any other algebraic objects other than standard integers -- e.g., polynomials over Q in a single variable ? Probably. Important: The algorithm described in the talk and the book is over the reals, not the integers. >Perhaps I didn't understand the video well enough, but are there elegant cellular automata that perform the Minsky/Trinsky computation? I'd be surprised. --Bill

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Re: [math-fun] math-fun Digest, Vol 171, Issue 35 re Bathroom square dance tilings
by Stuart Anderson 07 Jun '17

07 Jun '17

> But all that's completely irrelevant to my purpose, which is simply to > enquire if these tilings have an established name --- preferably one rather > more euphonious than my current nomenclature: The bathroom floor tiling! > > Fred Lunnon Wikipedia has a good article on these tilings; Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling) The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html Stuart Anderson

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