17 ms·
The Patek Philippe Caliber 89 and Horology’s Easter Problem
- noonespecial 9y agoMechanical computing is awesome. Miniaturized mechanical computing doubly so. I'm glad it's alive and well in the world of watches. I especially love this sort of thing because there's not really a practical reason for this to exist. Its pure art. Technical art of amazing effort and skill. The best part is that people appreciate it and pay big money for it (lots of people). People pay for good art! That's all kinds of awesome.
- raverbashing 9y agoIt seems the website is already gone Edit: it's back up
- joncrocks 9y agoAs the developer said to the QA tester: "It works on my machine."
- cafebabbe 9y agoAh, love for needlessly complicated things. If it's code, it's an engineering flaw, but if it's mechanical, it's a marvel.
- Cookingboy 9y agoWhy do you say it's needlessly complicated? If the design parameter is no use of electricity and the requirements are those features, can you come up with simpler and more reliable solutions?
- minikites 9y agoIt's trivial to write complicated code (meaning code that is itself complicated, not code that handles complicated tasks) and difficult to create a complicated physical mechanism.
- cafebabbe 9y agoIs it though? Is it trivial to write, and maintain, over-engineered code that actually accomplish its business goals ?
- minikites 9y agoYou don't think it's trivial to write GOTOs everywhere and paint yourself into a corner? Of course it's difficult to maintain, that's why it's considered "an engineering flaw" like you said.
- cr0sh 9y agoCompared to a physical device, I would surmise it is. Consider this - you can write code in any manner you want, starting anywhere you want. You can build it from the inside out, or top-down, or bottom-up. You can make it monolithic, or modularize it. You can do all of these things at once, if you want! There was a commercial more than a few years back that illustrated something about "how clients want things done" vs "how things are done" - I think it was for insurance, IIRC. Basically, they showed a "modern construction of a pyramid" in some "city". It was a complicated construct; but apparently the client wanted it constructed from the top down! So there was this huge gantry crane like system, hanging the apex of the pyramid, and other stuff going on with workers and machines gradually building downward to the base... You can't do this physically, of course (though I am aware of people who try to get engineers to do this - it's actually a common thing in custom home building for a client to change their mind in the middle of a project and want a window or door moved a few feet over - and trying to get them to understand what that would change in the already-in-place infrastructure of wiring, plumbing, etc - not to mention demolition, re-engineering, permits, inspections, etc - but some clients have enough money and clout and hubris to insist on it). But you can do this with software. When it's done in the middle of a project (because of design changes, or to fix issues), we call it refactoring. It is this malleability of software that makes it easier to work with to express complicated constructs; there isn't any comparison to physical mechanical systems. Such systems, as exemplified by horological mechanisms, require careful planning of where parts are put for their relative functions, and in relationship to the rest of the device, and how those parts are put together, what order, layers, etc. Not to mention the various calculations and allowances needed for tolerances, forces, friction, lubrication, as well as simply the fasteners and framework holding the whole thing together! We haven't even considered taking into account external forces, and how they relate to the accuracy and/or longevity of the system. All of these things, and many more, apply to all mechanical systems, and require careful planning and execution. Fortunately (especially for many of these complex timepieces of today, as I am led to believe by the reading of a few of these kinds of articles), master watchmakers like these have at their disposal modern CAD/CAM systems to aid them in planning and designing these complex works of art, as well as to guide them in assembly of them. Even with such help, though, I imagine that the process is still a very consuming and maddening one (this is evidenced by the fact of how long it takes for these craftspeople to build only a few of these timepieces). Now imagine having to do this without such help! Not as complex (but still pretty insane given the time period) timepieces (and other mechanical clockwork devices and automata) were planned, design, calculated, drawn, and constructed by hand in the past, long before computers were available - some of which contained thousands of carefully crafted and assembled parts; many of these devices still exist and operate today. Interestingly - very little of our software that we construct today is likely to still be operating 100 or more years from now (though there are a few examples that are working their way to that point)...
- nf05papsjfVbc 9y agoThe demoscene is (rightly) appreciated in a similar manner. They do a lot in a small bundle of code. It's likewise with these watches. I don't own such a watch but I do admire them.
- ericfrederich 9y agoIn fairness, with timepieces "complications" are just what they call features. A second hand, the date, the day of the week, etc... all complications. I'm not saying some don't go overboard for the sake of going overboard, but complications are just features.
- pavel_lishin 9y agoThis is actually a pretty elegant solution to a difficult problem. Difficult calculation? Pre-compute it!
- encoderer 9y agoOn a tangent, my favorite example of how php is a giant bag of rando functions is easter_date(). This ships with the language core.
- quirkot 9y agoAbsolute best example of "why the heck should I calculate that myself"
- mrrsm 9y agoI wonder if they will deprecate the function after 2037?
- deleted 9y ago[deleted]
- Chyzwar 9y agonah, they will just add real_easter_date().
- CompanionCuube 9y agoWouldn't that be easter_real_date()?
- mrrsm 9y agoThey will probably go with one for a single release before leaving it deprecated for years and switching to the other but never remove it from the core.
- kalleboo 9y agoThere's already an easter_days(), and comments complaining that it doesn't revert to the Gregorian calendar before the 1700's... http://php.net/manual/en/function.easter-days.php http://php.net/manual/en/function.easter-days.php
- tangue 9y ago
- ce4 9y agoI think it's one of the lamer complications. The date-of-easter complication consists of a notched program wheel - practically a look up table. Due to the limited LUT-size of 28 (1989-2017) this program wheel needs replacement.
- smiley325 9y agoThe Patek Phillipe that the article opens with yes, but they go on later to talk about the /computus/ which is a purely mechanical Easter calculation accurate w/o modification to year 10,000 AD.
- lisper 9y agoAlso, don't neglect to follow the link to the Vacheron 57260: https://www.hodinkee.com/articles/the-vacheron-constantin-reference-57260 https://www.hodinkee.com/articles/the-vacheron-constantin-re... That will surely sate your lust for mechanical audacity.
- eternalban 9y agoAlso don't neglect the ad for auction of "one of only four" going on sale soon. This read as a very elaborate advertisement for that auction.
- erik-g 9y agoFor something that's expected to sell for over $10m, I doubt they figured one internet article would surely help sell it. I'm fairly confident anyone who seriously wants to buy one of these already knows it's up for auction.
- garethsprice 9y agoI think it's more advertising for Patek in general. The handful of people who have both the means and desire to purchase such an object likely got informed by their personal networks long ago. But Patek wants to sell $20k Calatravas to merely affluent people who want "the best watch" and this piece helps position the brand. I'd not be surprised if there was significant PR effort from Patek behind the scenes on this one.
- ericfrederich 9y agohttps://github.com/dateutil/dateutil/blob/master/dateutil/easter.py https://github.com/dateutil/dateutil/blob/master/dateutil/ea...
- wyager 9y agoWhat's the embedding for division and modulation into systems of gears? I imagine it's somewhat expensive.
- a_c 9y agosome miniature version of this perhaps https://www.youtube.com/watch?v=wwh0KH-ICCw https://www.youtube.com/watch?v=wwh0KH-ICCw
- Someone 9y agoMultiplying angular velocity by a constant is easier than adding a constant angular velocity. To multiply by m, use two gears with diameters in ratio m:1. To divide, switch the gears. I wouldn't even know how to subtract a constant angular velocity. I don't think you need that, though; if you know at what speed all your gears will rotate, a constant addition/change can be replaced by a constant multiplication/division: a+b = a * (a+b)/a
- pavel_lishin 9y agoIf the angular velocity doesn't have to be constant, could you skip gears?
- thechao 9y agoDifferentials have the property of being universal angular velocity adder/subtractors at fixed ratios. output := K * (input0 + input1) / 2 Spur-gear differentials are suitable for watches.
- a_c 9y agoWow! I didn't know if not (1 <= method <= 3) is supported syntax! I would always use method >= 1 and method <= 3
- criddell 9y agoThe idea of this watch is confounding. Presumably, a person that is interested in having their watch tell them what day Easter falls on would be a follower of and believer in Jesus. From what I know about Jesus, he probably wouldn't be impressed by somebody spending millions of dollars on this. What watch would Jesus wear? Or maybe I'm looking at it wrong. Maybe it's just a great piece of art in the form of a clock.
- nailer 9y agoChristians generally don't follow Jesus' teachings literally. Selflessness is an ideal rather than a way of life.
- garethsprice 9y agoIt's a show-off piece by master craftsmen at the pinnacle of engineering. Patek does not make these pieces to make a profit, but to position themselves at the top of the pile when it comes to engineering as their customers want "the best", and are willing to pay a huge markup for it over "the next best". It sounds crass to call such a beautiful thing "marketing", but it is - these show-off pieces are what sells the rest of the line (cheap at $20k starting price).
- dghughes 9y agoAnd Patek-Philippe won't even sell their million dollar watches to just anyone you have to own previous Patek-Philippe watches. It makes me laugh it sounds so silly. I want to buy that Apple computer? Do you own an Apple computer? No. Sorry I can only sell you an Apple product if you already own one.
- handedness 9y agoYou're comparing something with unit sales in the millions to something more in the range of several-to-hundreds. While Patek is probably among the worst offenders in that regard, a very reasonable consideration is that the stratospherically high-end watches are supply-constrained (partly due to there being so few people in the world capable of making them–I'm not taking about your $5K Rolex or even your $25K Patek, but the most complicated and finely-finished ones that can take a watchmaker months to years to produce just one example), and it's reasonable to sell them to known individuals who aren't just buying to immediately flip them at auction. It also makes sense to reward those most loyal to the brand rather than allow something to be bought by the nouveau riche who will quickly move on to the next flavor of the month. That's how you alienate lifelong customers.
- garethsprice 9y agoBeautifully written article, important to remember as developers that we'll only ever be able to capture the chaos and complexity of the real world as a rough model no matter how intricate the engineering.
- pavel_lishin 9y agoA good reminder to managers that you gotta keep developers on staff, too - can't just write a "fire and forget" app for a lot of cases.
- OliverJones 9y agoHeh heh. Churches have been squabbling about the date of Easter since there was Easter. It seems this particular resolution of the problem relates to the western church's choices for Easter. Orthodox people still, if I understand it right, use the Julian calendar. This stuff is hard to get right, even if you're a bishop with legions of scholars and theologians at your disposal. And, just for grins, look up how the modern state of Israel decided when daylight saving time begins and ends up until 2012. (They rationalized it in 2012.)
- asmithmd1 9y agoThis is why there is an official priest in the Catholic church who is an astronomer. Churches had lines inscribed on the floor showing where a spot of sunlight would be on noon of the Spring equinox: http://www.atlasobscura.com/articles/catholics-built-secret-astronomical-features-into-churches-to-help-save-souls http://www.atlasobscura.com/articles/catholics-built-secret-...
- LeifCarrotson 9y agohttps://en.wikipedia.org/wiki/Israel_Summer_Time#2005.E2.80.932012 https://en.wikipedia.org/wiki/Israel_Summer_Time#2005.E2.80.... > Until 2005, the start and end of IDT each year was established in an ad hoc fashion as the result of haggling between political parties representing various sectors of Israeli society. Parties representing religious groups wanted the start delayed till after Passover and the end to precede Yom Kippur, the Day of Atonement, while the secular parties would argue for starting it earlier and ending it later. Sounds like business as usual for a modern 2-party state. What was more interesting to me, as a programmer, was the next line: > In the past, the unpredictability of IDT in Israel became frustrating enough that Microsoft Windows stopped trying to track changes and just made Israeli time be Greenwich Mean Time plus two hours (GMT+2) (and disabled the daylight saving option). This has led to various ad hoc solutions to the problem in Windows systems and other Microsoft software (e.g. Outlook calendar entries are often off by an hour when shared, due to the lack of IDT support). On November 17, 2009, Microsoft released an update that has daylight saving time enabled for Israel. However, the date for transition back to Standard Time is set as the Second Sunday of September, regardless of the Hebrew Calendar date.[4] Windows 7 does contain correct IDT times up to 2023, but not all software makes use of this extra information. OK, I know that I'm supposed to just import my system date/time library for timekeeping functions. Because it's too complicated for mortals like me, while Microsoft, Apple, Google, and other major institutions can handle all the inanities. But Microsoft gave up on this one? Ouch.
- f_allwein 9y ago"the basic rule for Easter is that it falls on the first Sunday after the first full moon of Spring (that is, the first full moon after the Spring Equinox)" Given that the watch knows all these parameters (i.e. day of the week, moon phase and date (Mar 21 = Spring equinox), would it be possible to construct a mechanical complication that calculates the Easter date?
- scblock 9y agoThe article goes into that as it discusses the mechancial computus in the clock at the cathedral at Strasbourg, in Alsace, France.
- deleted 9y ago[deleted]
- stephen_g 9y agoWouldn't that only mean you'd be able to work out if it is Easter Sunday? So you could show an indicator on the day but it's a much harder problem to have it work out the right date for the next occurance from the start of the year (or from the last Easter Sunday).
- f_allwein 9y agoNo - there's a finite amount of possible dates (between March 22 and April 25), so every year could have a numeric value for the Easter date that the pointer could point to.
- CamperBob2 9y agoI don't understand this, myself. Why isn't this calculation implemented as a mechanical state machine? It's impossible to implement a 5,700,000-year cycle with an individual program wheel, but three or four smaller, simpler ones should be able to do the job. Hard to see why this is considered a difficult problem at this point, 2000+ years after the Antikythera Mechanism.
- ZoeZoeBee 9y ago
- sillypuddy 9y agoTLDR: Dates and times are hard
- danielam 9y agoInteresting article. One note... "[...] the whole structure of astronomical mechanical complications – whether in the Strasbourg cathedral clock, or in watches like Caliber 89 – is a manifestation of a world view." The idea of a "clockwork universe" is a mechanistic, modern, Enlightenment-era idea. Astronomical clocks predate that worldview by quite some time.
- pavel_lishin 9y agoBut people have been watching the skies for a long time, and have been noting regularities and using them for millennia - at least since the development of agriculture. The idea of Celestial Spheres alone dates from greek antiquity.
- lisper 9y agoYes, but until Newton people thought that gods were involved in moving them around.
- eru 9y agoNewton was a supreme weirdo who believed in all kinds of mystical things. If anything the best Greek scientists like Archimedes were probably more rational and less inclined to believe in Gods running things. If you changed you sentence to "but until Leibniz" I'd be more willing to by it. Even though it's the same time.
- gcb0 9y agothere are dozen of places where one can link to patents. and even if not, they have convenient index numbers. why not link/mention the number if you are going to mention the patent on the article some 20 times?
- peterkshultz 9y agoAstronomical Algorithms by Jean Meeus is a great read for anyone interested in these types of problems.
- coldpie 9y agoTangential to this, if you're interested in how the calculation of Easter led to using cathedrals as astronomical instruments, check out The Sun in the Church by J.L. Heilbron. Frankly, it's a slog to read at times, but there's some very fascinating stuff in there, including some impressive engineering feats.
- noir-york 9y agoA great read! Horology, Ecclesiastical history and lattice structures in one article. Thanks for posting.
- jacquesm 9y agohttp://www.paulgraham.com/submarine.html http://www.paulgraham.com/submarine.html
- te_platt 9y agoI'm always curious when Gauss turns up in the solution to a math problem. Wikipedia has a section on the algorithm here: https://en.wikipedia.org/wiki/Computus#Gauss_algorithm https://en.wikipedia.org/wiki/Computus#Gauss_algorithm The same article also explains why it is a hard problem: "Easter calculations are based on agreements and conventions, not on the actual celestial movements nor on indisputable facts of history." Sounds a lot like many user requirements I have had to code for.
- omoikane 9y agoGauss's algorithm was also used in an IOCCC entry: http://ioccc.org/years.html#1996_schweikh1 http://ioccc.org/years.html#1996_schweikh1
- DonaldFisk 9y agoAstronomically derived Easter is the Sunday after the full moon after the vernal equinox, and that's quite easy to calculate. I thought of implementing Computus, but gave up as it's so arbitrary that nothing worthwhile would be learned by implementing it. I implemented astronomical Easter instead, knowing that it's occasionally wrong: http://web.onetel.com/~hibou/fmj/tutorials/GregorianCalendar.html http://web.onetel.com/~hibou/fmj/tutorials/GregorianCalendar... - scroll down to Bell, Book, and Candle.
- jazzdev 9y agoThis was my first programming assignment (7th grade math class, 1975). It was a great introduction to computers and algorithms. I think I still have that BASIC program on a paper tape in my basement.
- microtherion 9y agoReminds me of the comment in a time zone file: From: Jonathan Leffler # [British Summer Time] is fixed annually by Act of Parliament. # If you can predict what Parliament will do, you should be in # politics making a fortune, not computing. https://github.com/lattera/glibc/blob/master/timezone/europe https://github.com/lattera/glibc/blob/master/timezone/europe
- jameshart 9y ago"As you can probably imagine, a program disk for the full cycle of Easter dates would be a wildly impractical thing as well; it would have to have 5,700,000 steps in order to encode the full cycle of Easter dates." That assumes that you want to encode the program as a single disc encoding the repeating cycle. But surely you could use a series of program discs to perform successive lookups and offsets and reduce it down to several more manageable discs? e.g. you have one disc that encodes the cycle of offsets of the spring equinox by year, then use that to offset the rotation of the wheel that encodes the lunar cycle...
- lmm 9y agoAt that point you're not using a single program disk, you're encoding the computation, which they also cover the practicalities of.
- jameshart 9y agoI'm still suggesting lookup tables based on year, not attempting calculations.
- grp 9y agoMaybe you will be interested by the perpetual calendar patents of Ludwig Oeschlin for Ulysse Nardin. (Note that perpetual calendars are relatively more simpler complications to design and make). So it is possible, but hard to fit into a watchmaking piece!
- TheOtherHobbes 9y agoThe problem isn't the size of the data, it's the limits of the mechanical technology used. I could be very wrong about this, but I suspect it might be possible to encode the data optically with much finer resolution and then contrive some kind of daylight driven optical -> mechanical transducer to extract it.
- 9y ago
- pjungwir 9y agoOnce upon a time I wrote a function for remind(1) to compute the date of Pascha (Easter in the eastern churches), which uses different rules. It uses the Julian calendar (mostly). Here it is on github: https://github.com/pjungwir/remind-pascha https://github.com/pjungwir/remind-pascha It lets you choose between 3 algorithms, one of which is Gauss's.
- cat199 9y agoNifty - cheers - will be INCLUDE'ing this at once :)
- tzs 9y ago"Easter is a feast, not a planet." --Johannes Kepler A good reference for anyone who has to write calendar related code is the book "Calendrical Calculations" by Dershowitz and Reingold [1]. [1] https://www.amazon.com/Calendrical-Calculations-Nachum-Dershowitz/dp/0521702380 https://www.amazon.com/Calendrical-Calculations-Nachum-Dersh...
- iaw 9y ago> the year indication goes to 9,999 and Schwilgué is supposed to have helpfully suggested that in 10,000, someone might paint in a "1" to the left of the year window I hope that's a true story, it's such a practical solution.
- tantalor 9y agoUntil you get to 20,000.
- pcurve 9y agoAn in depth, partially subbed video on Caliber 89 https://www.youtube.com/watch?v=CPWCl2FDKe8 https://www.youtube.com/watch?v=CPWCl2FDKe8
- bkeroack 9y agoOnly tangentially related, but if you ever get the chance to check out the Patek Philippe Museum in Geneva you should do so. I'm not really a watch geek so before going there I was kind of "meh", but that place is incredible. They have thousands of timepieces, some dating back to 1530. To see what people have been able to accomplish with nothing but springs and tiny gears is nothing short of mind-blowing. As a technologist it's pretty humbling, considering all the advantages we have today and yet ancient watchmakers could do this work literally with nothing but hand tools.
- ansgri 9y agoThere's also Uhrenmuseum Glashütte in Germany near Dresden, where A. Lange & Söhne and Glashütte Original factories are based. They have various vintage watchmaking tools on display, along with various assignments made by watchmaking students. Well worth stopping there if you're nearby.
- huhtenberg 9y ago+1 It's a fantastic museum. Several floors jam-packed with incredible watches. Really helps you understand where modern Patek's price tags come from :)
- mathattack 9y agoThis is also what makes Steampunk so fascinating.
- ChuckMcM 9y agoI always imagine that in the steampunk universe is 'the number of gears in a mechanism doubles every 2 years' and then extrapolate where we would be today :-)
- mml 9y agothere's a book along those themes. a corporation's power was measured in "cam feet", and the main plot revolved around the theft of a stack of punchcards with some important algorithm or other on them.
- cestith 9y agoI suddenly have The Buggles in my head singing "Software Killed the Hardware Star". The article's subtitle mentions the watch "needs a service". Well, that service could be CalDav if it was a software timepiece on a digital platform, and no skilled tradesman with tweezers would be involved. Yes, a Patek Phillippe is a thing of physical beauty. However, for functionality like this software wins the day.
- ksherlock 9y agoHow many Apple Watches will still be functional in 28 years?
- TheOtherHobbes 9y agoHow many people will have been keeping the same personal calendar data in some version of CalDav across a selection of different hardware endpoints and servers?
- cestith 9y agoMore than there will be of the 89, even given a 100x failure rate comparatively.
- gertef 9y agoCan you use a neural network to discover a formula for Easter date?
- gaius 9y agoNo
- ryanmarsh 9y agoSincerely thank you for posting this. I had no idea of these timepieces and I am in awe of their beauty and sophistication. I think I have a new obsession.
- Animats 9y agoWhat a kludge. They encoded a table in a variable-depth wheel, like the snail used in striking clocks, but not so regular. The table only has 29 entries. For a status-symbol complication watch, that's tacky. What does it do if not serviced? Display "needs service", or wrap around and show wrong values?
- Teknoman117 9y agoDoes anyone know of a video of the easter mechanism of the strasbourg astronomical clock running on the new year? I couldn't seem to track one down...
- lunaru 9y agoI'm big into mechanical watches and I find it surprising that more software developer types are not into watches. They're the mechanical parallel that completes my otherwise purely digital world. Also, everything about them (when done well) sounds like a top developer's wet dream, from the design, development, and down to the extremely rigorous QA processes around building these mechanical marvels. In particular, this "inside rolex" piece from the Hodinkee website is a must-read, and I believe was also posted to HN a few months ago: https://www.hodinkee.com/articles/inside-rolex https://www.hodinkee.com/articles/inside-rolex
- runeks 9y ago> I'm big into mechanical watches and I find it surprising that more software developer types are not into watches. They're the mechanical parallel that completes my otherwise purely digital world. The lack of interest from my side is due to the inferiority of mechanical watches in accomplishing their main purpose: keeping time. I know my smartphone always has the right time, because it's internet-connected and uses NTP. For a mechanical watch, that trust just isn't there: either it has the right time, or it's run out of power. Surely, I can appreciate the craftsmanship, but when it comes to knowing what time it is, this is not relevant for me at all. When I look at a clock, I really appreciate knowing that if it has run out of power it doesn't display the wrong time, as opposed to a mechanical watch.
- richardw 9y agoSame. I seem to consider them a Rube Goldberg machine that people use to indicate status. I was partly reading the comments here to see what I'm missing. I'm sure there's a lot. My wife loves watches (for the design). I'd rather have a nice bike.
- eo3x0 9y agoAnyone who buys a watch for the status is going to be disappointed. A $200,000 car will get you laid quick. A $200,000 watch wouldn't get noticed except by geeks you wouldn't want to bed down anyway. Fortunately, in 10 years, at least the latter will still be worth roughly just as much if not more while the former would be depreciated to less than half the value if that.
- gavinpc 9y ago> a true date-of-Easter complication is probably the single most difficult complication in horology Funny story. I was making the "final" commit before shipping a desktop application, and I wanted to make an Easter egg. But what should it be? It should be Easter-related, I supposed. I made a pastel color theme for the main screen, that should appear only on Easter. This product is used in DoD, and I doubted that anyone would ever be using it on a Sunday. The product had a design flaw, a kind of time bomb. I was a greenhorn working on it when it first shipped in 1994, and I thought nothing of the fact that its rate table was arranged horizontally, like so: id 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 1 4.49 4.59 4.79 4.99 5.16 5.26 5.68 5.78 5.99 6.44 2 ... etc Despite being a Navy budget forecaster by trade, the Captain who created the program lacked the hubris to worry about needing rates for the next millennium. This set of dates was hardcoded all over the the codebase. The Captain, a poor typist, kept resetting the bomb by adding more years. Fast forward to 2012. The product has been acquired, and I was brought in to port it for 64-bit machines. And the only task remaining—revision #1400—was to compute the date of Easter. It took me about 2 minutes to conclude that this wasn't going to happen (although I didn't know it was that hard). So although it lacks the elegance that you'd like in your Easter egg, this one hardcodes the dates of Easter Sunday... through 2020. I mean, it's going to be on the web by then, right? Yeah, it's still shipping. http://www.budgetbuilder.com/ http://www.budgetbuilder.com/
- BobMackay 9y agoJust for completeness, the 1990 paper referred to was by my father, Professor Alan Mackay, and was published in Modern Physics Letters B, Vol 4, No 15. I have put up a roughly scanned copy at http://bobmackay.com/Alan/AlanCV120.pdf http://bobmackay.com/Alan/AlanCV120.pdf in case anyone is interested.