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Gears, Computers And Number Theory
UncleJosh writes: "The latest issue of
American Scientist
has an interesting article
"On the Teeth of Wheels" about the relationship between gears, computers and number theory." It's as much a well-meshed detective story (hunting for books and obscure references) as a historical and mathematical introduction to the science of gears. Prime numbers, watchmakers and the Fibonacci sequence all play a part.
There are certain problems which are considerably easier using analog (though not necessarily mechanical) computers than to use a digital computer. The one that leaps to mind is a particular minimization problem-- given three cities, let's say, at the vertices of an equilateral triangle, what is the shortest amount of road needed to connect them? The answer is to do this: make a new vertex at the centroid of the triangle and connect roads from each of the three cities to that new vertex.
This uses considerably less road than, say, connecting city A to city B and then city B to city C. The problem gets considerably harder if you are not using a regular figure or if you are using more than three points. It's really a calculus of variations problem (I think) -- you're minimizing over an infinite number of paths so this makes it a pretty tough problem for a computer.
This problem is a piece of cake, however, if you use a very special analog computer -- namely soap bubble solution. If I make two plexiglas plates and put pins between them representing the cities I am trying to connect, then dip the construction into bubble solution, a soap film forms connecting the pins. As long as the soap film doesn't close on itself (that is, as long as it doesn't make a 'real' bubble) you will get a solution to the problem. The number of solutions to a given problem grows depending on the number of vertices, but suffice it to say it's a lot quicker to check all the solutions using soap bubbles than it is to use a computer. Also, each solution is quite close to the absolute minimal solution, so within certain parameters it may not be necessary to check every solution. My old differential geometry teacher tells me that they actually use the soap bubble method to do minimization problems for such things as highway planning.
There's a lot more interesting stuff about this problem which I shan't go into, partly because I don't remember and partly because it's not very relevant to the discussion at hand. In any case, this is a good example of an analog 'computer' being demonstrably faster than a digital one.
Is this refutable?
Refutable? Not without a strict definition of great. But you hit the nail on the head when you said, "snotty".
I would say the scientific method, and empiricism itself, qualify as great contributions. Hell, deductive reasoning. And personally, I'd be kind of proud if I'd discovered evolution, or DNA, or atoms.
Anyway, who exactly does he mean by, "the West", and to whom is he contrasting them? I seem to recall the Chinese were developing calculus about the same time as Liebnitz and Newton... so maybe the west has had only one great contribution. :-)
One must wonder what contribution Mr. Berlinski has made to science, to so judge the contributions of others. I guess /. doesn't have a monopoly on technological ingrates!
"The best we can hope for concerning the people at large is that they be properly armed." - Alexander Hamilton
The Swedish Navy developed a programming language in the 1950's called Kvikkalkul, in honour of Plankalkul. The language was entered on 5bit Baudot paper tape, and lacked any alphabetic characters. It used solely punctuation and numerals.
,0 :3 .8 -/- ,0005 ,49975 -) :2 .9 -/- ,0005 ,49975 -) :1 :1
The code looks like this:
1030:
.8 (-
1040:
-)
1050:
.8 (-
.8 (
.9 (-
.9 (
:1 -) 666
-)
It actually had some halfway decent flow control, and even pointers.
What is ultimately very scary is that the Swedish Navy was still using this language as late as 1991. And, code written in the 50's still compiles on the "modern" compilers. Now they have OO and some GUI functions in it, but still no alphabetic characters.
I went through most of high school with a slide rule; it was a *very* big deal when I bought my first calculator, for hundreds of dollars, from Sears...
And yet, before that, a whole lot of computational stuff we now take for granted was all done mechanically.
Check out TIDE-PREDICTING MACHINE No. 2
"This machine was designed by Rollin A. Harris and E.G. Fischer and constructed in the instrument shop of the U.S. Coast and Geodetic Survey.
It was completed in 1910 and replaced the Ferrel Tide-Predicting Machine in 1912."
"The machine summed 37 constituents and was capable of tracing a curve graphically depicting the results."
Whoa! So you don't have to write down the output! Now that's a feature! But don't laugh! It was necessary to write down the output on previous models.
"It is about 11 feet long, 2 feet wide, and 6 feet high, and weighs approximately 2,500 pounds."
And this was state-of-the-art, at the time!
t_t_b
--
I'm on PJ's "enemies" list! Are you?
This reminds me of an "extra points" question on a final exam in a course on operating systems taught by Vint Cerf at Stanford many years ago. Similar setup except the bar was some number of light years long and was rotated at a rate that would make the tip move faster than the speed of light. In this case what happens is that the "bar" (no matter how rigid) wraps up in a spiral around the axis of rotation, with the result that the tip never exceeds (or for real a physical bar, even approaches) the speed of light. I put this answer (that the bar winds up around the axis of rotation) on the exam and concluded with "Rigid rods are illegal in physics, Freud notwithstanding" and got 15 out of 10 points extra credit: I think Vint (or perhaps the grader) got a laugh out of my answer :-)
It takes me back to the earlier days of /., before the days of the Four-Letter Crusades (MPAA, RIAA, DMCA). Back when you could still find articles on science and technology instead of <contempt>legal depositions</contempt>.
(I gotta admit, tho, Valenti's depo was kinda funny, up until I got sick.)
"Genius may have its limitations, but stupidity is not thus handicapped." --Elbert Hubbard (1856-1915)
So he went into the auditorium and sat down. The lecturer began: "The theory of gears with a finite number of teeth is well-understood. However..."
      I was surfing around a little while back when I decided to see if I could find a sliderule for sale somewhere. (yes..I AM a nerd :) During my search I ran across refrences to a mechanical calculator called the Curta. :) thanks to Bruce Flamm at the first link for some of the info.
      According to this web page, the Curta was designed and built by a gentleman named Curt Herzstark of Austria. Although several prototypes were made, the first production began in April, 1947. The last Curta was made in November, 1970 but they were still sold until early 1973. Over the course of about 20 years approximately 80,000 of the Curta I and 60,000 of the Curta II were constructed.
      Additional links, articles, and pictures of this awesome little device can be found here and at Curta.org
I gotta say..The Curta is one sexy little calculator
(And there's more of them to look at than number-calculation devices
And for an orrery, virtually every gear-ratio is an approximation of a non-factorable ratio, so I found the article of particular interest because I'm currently working on one at home. (Though it's a desktop sort of thing, I aspire to eventually do something along the lines of Aughra's awesome device <grin>)
The only real link I've got on hand is this one: Brian Greig's Orrery Page :-)
(He makes orreries for museums, collectors etc, and some of them are pretty cool
BTW, for those that haven't seen much of these things, an orrery (named after the Earl of Orrery, who commissioned one of the first built) is a device that shows the motion of the planets to scale (but not the size of the planets to scale...). And like the calculation engines, orreries today are done through software.
If you know a bit about the complexities of planetary motion (eg non-circular orbits, inclined orbits in which the plane of inclination drifts or rotates), seeing the various means of incorporating these aberrations into a clockwork model is quite fascinating.
One particularly nagging thing about the article was the assumption that the problem is finding the best gear ratio. Ha! The best ratio might be 103:17 but have you ever tried to find a gearcutter? The last one I saw was in a museum (I must have been a pathetic sight - pressed up against the glass like a kid outside the candy store...), which means I have to buy manufactured gears. Which means finding the best gear ratio out of the gears available to me. Sure, it cuts down on the computation, but you need to make a longer gear train to get even remotely close :-(
Ah well.
It seems a shame that the skills and tools of so many of these crafts are dying or dead (if only because they could make amazing things that modern manufacturing methods are currently simple incapable of producing).
Take an infinitely rigid bar (or near-infinitely rigid) which is about four light-years long, and weighs almost nothing (a few pounds, maybe a few tons) and put it between Sol and Alpha Centauri. Wiggle it back and forth, and you have instantaneous morse code across four light-years.
It has to be nearly infinite in its rigidity or it will bend and the girl on the other side won't see movement, and it has to be light-weight or you will have trouble keeping it straight (a few pounds with a 2 light-year moment arm will be a pain to keep straight).
This is the best example I know of where mechanics wins over electronics/optics/sub-space/new-tech.
Louis Wu
Thinking is one of hardest types of work.
If you were able to do an infinitely rigid bar you might get instantenous communication (infinite speed of sound), but the trouble is, there is no such thing as an infinitely rigid bar...
Say no to software patents.
The History of Mechanical Computers
Early Calculators
Zuse
"I believe that a scientist looking at nonscientific problems is just as dumb as the next guy." -Richard Feynman
In certain applications, like raising an arbitrarily high number to a very high non-integer power, if one were to create a very specialized machine, would it be possible to create a machine that could give the high accuracy in less time? Of course, you'd probably need a machine that could spin at a very high speed in order to get the necessary clocks, but if one were to use numbers that could be expressed rationally, but are very large, it at least in my semi-limited knowledge, could be a savings of time.
Marxism is the opiate of dumbasses
Keep away from the wheels! They have TEETH. We ask also that you keep any small children away from wheels, especially while in motion. Even though the wheels MAY look harmless enough, they have consumed a fair amount of mathmaticians ('s time?).
Also, please don't feed the wheels: they become enourmously fat and refuse to reproduce. We have been issued various legal threats from auto manufacturers about this. You WILL BE FINED.
Thank you. (forgive my spellign!)
After reading through the Scientific American article, I suddenly found I wanted to re-read The Story of Mel again, the tale of a programmer's programmer from an era gone by. Our old-timers often lament the extinction of code laboriously hand-tuned to run tight and fast on elegant machines from days gone by -- and those days have been gone only a few decades. The gear makers worked their craft a century or more ago.
Today, sometimes I wonder, what was the point? Why not just shovel in and ship out the first thing that works? A year and a half from now, the hardware will be twice as fast, and probably cost half as much. The software we wrote will be obsolete, as will be the hardware it ran on.
But maybe it does matter. It would be a terrible thing if our decendants did not surpass us. But even as they gaze back upon us from those lofty, distant heights, maybe we can give them a reason to listen to how it was done in the Good Old Days.
"Lest a whole new generation of programmers
grow up in ignorance of this glorious past,
I feel duty-bound to describe,
as best I can through the generation gap,
how a Real Programmer wrote code..."
I'll second that -- Konrad Zuse's work on electromechanical computers is fascinating. He developed the first stored-program computer in Germany during WWII, with bombs falling and severe shortages of materials. (At one point he "liberated" copper from a public power line for his machine, later realizing that the police/soldiers would have killed him on the spot if they'd caught him.)
I recommend reading his autobiography The Computer - My Life . It's entertaining and informative (unlike many autobiographies).
-- ;-)
Kuro5hin.org: where the good times never end.
After read about mechanical computers, I recalled having learned about asynchronous computers. Have asynchronous computers been built that run a lot faster than conventional clock-controlled computers? As events in an asynchronous computer are triggered by a previous event, and not a central clock, it would seem to me that they could really scream, if timing issues could be worked out.
---------------
After reading this article, I went to look things up at Eric Weisstein's World of Mathematics (courtesy of the makers of the great but ridiculously overpriced Mathematica).
Anyway, here are some quotes from articles in mathworld related to the original article:
Stern-Brocot tree. "A special type of binary tree obtained by starting with the fractions 0/1 and 1/0 and iteratively inserting
(m+m')/(n+n') between each two adjacent fractions m/n and m'/n'. The result can be arranged in tree form as illustrated above. The Farey sequence Fn defines a subtree of the Stern-Brocot tree obtained by pruning off unwanted branches (Vardi 1991, Graham et al. 1994)."
Gear curve. "A curve resembling a gear with teeth given by the parametric equations x = r cos t, y = r sin t, where r = a + 1/b tanh [b sin (n t)]."
Phi, the golden ratio. "A number often encountered when taking the ratios of distances in simple geometric figures such as the pentagram, decagon and dodecagon. It is denoted [phi], or sometimes [tau] (which is an abbreviation of the Greek ``tome,'' meaning ``to cut''). [phi] is also known as the divine proportion, golden mean, and golden section and is a Pisot-Vijayaraghavan constant. It has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers."
(Note: the above quotes from mathworld fall under the definition of "fair use". Please don't sue me!)
To the editors: your English is as bad as your Perl. Please go back to grade school.
These Brocoult trees are also known as Farey sequences by the mathematicians.
The farey sequence is known to naturaly enumerate the buds of the mandelbrot set. Julie Tolmie at math anu edu au has just finished a PhD thesis in this area.
Check it out; lot's of great pictures.
!#
As long as each successive gear chain "link" reduces the final ratio further, it should be infinite. Rather, it *could* be infinite, as in, an infinite number of gears.
This would make an interesting physics problem. I'm sure it's already been done somewhere.
You'd be adding up a series of terms based on the drive ratio of each pair of gears (where each driven gear has a pinion attached that drives the next driven gear), figuring out the speeds and thus the amount of power required to turn each gear which depends on the speed each gear is turning.
So, as the number of gears approaches infinity, what does the function for the required power input look like? Assuming it's a simple scenario where all of the driven/pinion gears have the same ratio.
Now if you're increasing the speed with each gear, the power required will skyrocket, friction will take over and you'll break your legos. Fortunately, theoretical physicists and mathematicians only have to deal with massless, unbreakable gears with precisely known friction functions...
-CausticPuppy "Of all the people I know, you're certainly one of them." -Somebody I don't know
Many years ago, I remember having a set of Lego Technics which included a bunch of gears. I had a good time building gear trains, but only up to a certain size -- ultimately, it became too hard to move the gear train.
What's considered a practical limit on the length of gear trains used for watches, mechanical computers, etc? At some point, the amount of force needed to overcome the static friction of the shafts would break the teeth off the gears first. Even before that, the force required to keep everything turning might be too high for a spring/watch battery/other power source to produce for any useful length of time.
If trains of four, five, six, etc gears are mechanically practical, the computational problem of choosing their ratios still seems interesting.
In the 1940s, Zuse also designed "Plankakül", which is widely considered to have been the first algorithmic language. It has some of the features characteristic of today's high-level languages.
The paper The "Plankalkül" of Konrad Zuse: A Forerunner of Today's Programming Languages [Bauer and Wössner, Mathematisches Institut der Technischen Universität München] is available in HTML form at Eric Raymond's Retrocomputing Museum. It describes Plankakül in excruciating detail... it's a very fun read (if you're into ancient and bizarre programming languages, that is).
To the editors: your English is as bad as your Perl. Please go back to grade school.
Eh, I don't know where you went to school, but the math in this article doesn't go beyond high school levels. At least not on where I went to high school.
I'm not sure what you mean by the average slashdot geek. Just because it doesn't mention Linux or free software doesn't mean it ain't interesting.
-- Abigail