How to Draw a Straight Line
November 28, 2005 10:49 AM   Subscribe

How to Draw a Straight Line - Until 1873, virtually all mathemeticians and engineers agreed that it was impossible to build a linkage that could convert circular motion to perfectly straight motion. In that year, Lipmann Lipkin rediscovered the Peaucellier cell which had been quietly created a decade earlier. Although much simpler to build, it was predated by Pierre-Frederic Sarrus' non-planar solution. Nowadays, though, linkages can do some extremely complex things. (via)
posted by Plutor (25 comments total) 5 users marked this as a favorite

My sone and I go to the boston Museum of Science every November, where the Pierre-Frederic Sarrus' solution, pictured, is located (in the "Mathematica" exhibit). I'm always somehow strangely facinated watching it.
posted by ZenMasterThis at 11:08 AM on November 28, 2005

Ummmm... use a ruler?

Just kidding, this is a very cool post.
posted by anomie at 11:16 AM on November 28, 2005

Without the java applets or the lego model, I'd have no idea how this thing worked.
Great post, Plutor.
posted by rocket88 at 11:19 AM on November 28, 2005

Yes, fantastic post.
posted by dwordle at 11:21 AM on November 28, 2005

(this is good)
posted by Rothko at 11:24 AM on November 28, 2005

I can't believe I misspelled mathematicians.
posted by Plutor at 11:24 AM on November 28, 2005

Great! Thanks!
posted by vacapinta at 11:30 AM on November 28, 2005

Excellent post.
posted by loquacious at 11:35 AM on November 28, 2005

Beautiful. The main KMODDL site is a find. I loved kinematics as an engineering undergrad, but was exceptionally poor at visualisation.
posted by scruss at 11:44 AM on November 28, 2005

Incredible post. This is fascinating.
posted by [expletive deleted] at 11:52 AM on November 28, 2005

Four bar linkages!! We had a whole class on them in Mechanical Engineering. They can do some pretty freaky things, all determined simply by the length of the linkages and positions of the pivots. This was ten years ago, but I remember plotting out, on graph paper, the path that you want the linkage to trace, then iteratively crunch through equations that spit out the linkage lengths. Then we wrote a Fortan program to do the same thing. Good times.

Now, however, teh intarweb has made my education obsolete, dammit all..: Create your own!
posted by LordSludge at 11:53 AM on November 28, 2005

Being a model builder, I've often wondered just how the first straight line was drawn. Perhaps it was something like this? Perhaps the shadow of a rock with a 90-degree cleavage? In any case -- good post.
posted by benATthelocust at 12:02 PM on November 28, 2005

It's so simple! Amazing.
posted by ab'd al'Hazred at 12:06 PM on November 28, 2005

This post is friggin awesome. I was just looking for info on this stuff too - funny how that works.
posted by Astragalus at 12:24 PM on November 28, 2005

benATthelocust: Hang a weight from a pliable vine, then trace the shadow? I'm assuming the first straight line ever drawn was in someplace viney, and that drawn straight lines predate string.
posted by aaronetc at 1:02 PM on November 28, 2005

I just learned about linkages from a colleague last week, and meant to follow up, but forgot. This is awesome.

Historical detail: Kempe, the "barrister who pursued mathematics as a hobby" mentioned in the first link, is best-known (among mathematicians) for his false proof of the 4-Color Theorem. Cayley, who had been the first to propose the problem in print, encouraged Kempe both to work on it and to publish his proof in Sylvester's American Journal of Mathematics. The flaw in Kempe's argument wasn't found for 10+ years, and does in fact prove that any planar graph can be five-colored.
posted by gleuschk at 1:25 PM on November 28, 2005

This is really interesting. I'm reading Kempe's lecture on the topic. One wonder's how exactly the Peaucellier cell was invented/discovered, it doesn't seem particularly obvious how it works (although Kempe proves easily geometrically that it does draw a perfect line).

And why do only odd numbers of links give a continuous curve locus? (Even numbers apparently give only discrete solutions, which is certainly obvious for two links, but not - to me - for four).
posted by snoktruix at 3:11 PM on November 28, 2005

Even numbers apparently give only discrete solutions

Actually, that can't be right, because you could always fix a pair of adjacent links in a static orientation and call them a single link, reducing say 4 links to 3, which certainly has continuous motions. So I must have misinterpreted something.
posted by snoktruix at 3:32 PM on November 28, 2005

Dang, I can't figure out from that site if there's a way I can see this collection in person. The university has cool stuff like this hidden all over the place, and you never see a tenth of it.
posted by Eideteker at 6:32 PM on November 28, 2005

Restores my confidence in MeFi bringing me weird new worlds heretorfore unknown.

Thank you, Plutor.
posted by rleamon at 8:38 PM on November 28, 2005

Heretofore. Solly.
posted by rleamon at 8:39 PM on November 28, 2005

[this is excellent]

Incidentally, for those who hadn't noticed, the (via) links to the blog by the creator of Bittorrent.
posted by Aknaton at 8:48 PM on November 28, 2005

Thank you! I've long wondered how the first flat surface or straight edge was machined and this is the first reasonable clue yet.
posted by jewzilla at 9:14 PM on November 28, 2005

I'm glad everyone liked this. As Aknaton pointed out, I saw a few of these links on Bram Cohen's blog, and spent half an hour searching and reading and being totally entranced by something I had never even thought existed. I just hoped that a few people would be as interested. Apparently I was right.

As for jewzilla and benATthelocust, I don't think this linkage has any significant historical value. I'm sure there were straight edges and flat surfaces long before 1873. Unless we're talking "perfectly, mathematically straight", which is impossible to machine, no matter how accurate your linkage. James Watt's Lemniscoidal Linkage is remarkably straight; in fact it's accurate enough for pretty much any real-life application.
posted by Plutor at 6:33 AM on November 29, 2005

Thanks, really interesting.
posted by teleskiving at 10:00 AM on November 29, 2005

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