As you could tell from my previous question: ~ sci.math: How could you multiply and/or divide using straightedge and compass only? ~ I am interested in the details of why have we Mathematicians used certain means and ways both logical and mechanical to operate and express ourselves ~ Using a string would have let them produce not only circumferences, but also ellipses, so they wouldn't need conic sections ~ "Pythagorean" triplets were known and used to survey land using strings possibly way back before Pythagoras himself knew of them, so, I guess they certainly knew they could use a string. Why didn't they? ~ Is it possible that they found conics as a natural part of an extension of 2d geometry into 3d and they could produce ellipses from them and/or is it a consequence of the huge influence that regular polygons and in general Platonic philosophy had in those times? ~ Thanks lbrtchx
> As you could tell from my previous question: > ~ > sci.math: How could you multiply and/or divide using straightedge and > compass only? > ~ > I am interested in the details of why have we Mathematicians used > certain means and ways both logical and mechanical to operate and > express ourselves > ~ > Using a string would have let them produce not only circumferences, > but also ellipses, so they wouldn't need conic sections > ~ > "Pythagorean" triplets were known and used to survey land using > strings possibly way back before Pythagoras himself knew of them, so, > I guess they certainly knew they could use a string. Why didn't they? > ~ > Is it possible that they found conics as a natural part of an > extension of 2d geometry into 3d and they could produce ellipses from > them and/or is it a consequence of the huge influence that regular > polygons and in general Platonic philosophy had in those times? > ~ > Thanks > lbrtchx
Geometry was the issue.
A compass allows an angle to form the part. A single inference of parts then causes.
A string as a part must be relatable to allow a like inference. What placment of length would allow another parts relative size.
A true relative angle to part as a science of geometry defines the science.
A like analogy could be constructed using a string?
How?
Place the string. What relation would be used? I can think of NO possible relation. A mere string as the part itself was to be caused in exact inference.
A compass was examined and found a unique part effector.
It is for this reason geometry IS NOT mathematical.
> On Dec 9, 3:48 am, lbrt...@gmail.com wrote: >> As you could tell from my previous question: >> I am interested in the details of why have we Mathematicians used >> certain means and ways both logical and mechanical to operate and >> express ourselves >> ~ >> Using a string would have let them produce not only circumferences, >> but also ellipses, so they wouldn't need conic sections
I'm not sure the study of conic sections by the ancient was motivated by "extending the compass and straightedge".
> Geometry was the issue. > ... > It is for this reason geometry IS NOT mathematical.
Humm. I leave this opinion to YOU.
However a string (apart from the added ellipses(*)) allows to construct a lot of things too. Of course straight lines (tightening the rope, that's the way gardners and other house builders draw straight lines, with a rope soaked with chalk). Of course a circle. So all what can be drawn with compass and straight edge can be drawn with rope alone. But as you said, it allows even more, as does just paper folding for instance (allows trisection of angles, and construction of 7 sides regular polygons).
However for the history of maths, I think nobody knows *exactly* what was in the mind of ancien geometers.
I've been told that the discovery of irrationnality (sqrt(2)) gave them a big shock ! Hence they tried to prevent the occurrence of such further shocks by not allowing too powerfull tools, and limiting to just compass and straightedge. The Idea is as is.
In that time, geometry was quite "contemplative", many proofs being "just look" from a well drawn figure. Hence the search of exact constructions in that time. Then much effort has been made to make the geometry more rigourous. First one being Euclides, then more recently Hilbert etc. Then the geometry today is much more algebraic, considering sets and metrics and manifolds etc. (but it IS mathematical)
Considering geometry constructions as just a game, we need however to fix the rules. Changing the rules changes the results (what can and can't be constructed).
As well known, the use of rope (the 12 knots rope) to construct right angles was a current practice. Why didn't they choose the 'rope only' constructions ?
To mention also that even the ancient tried to overcome these limits, by using specific curves (trisectrix, quadratrix etc) instead of just circles and straight lines, also "Neusis constructions" like the well known angle trisection by Archimedes (humm, not sure he was the first) with a marked straightedge etc.
As there are mechanical tools to draw continuously conic sections, you could use "construct by conic sections" instead of just compass and straightedge (straight lines and circles being degenerate cases of conic sections, we call that just "conic sections").
But these are quite "recent" discoveries.
A big problem also is that a straightedge is quite hard to make. (How do you ensure it is perfectly straight ?) But a compass is much more robust. Just suffice it is enough rigid.
Hence many geometers tried to discard the straightedge. I mentioned Mascheroni in another post. The study of inversion also gave a mean to *construct* a straight line from scratch (from a circle).
The world of construction rules, and geometry fundaments is quite wide...
(*) draw an ellipse with a rope. Proove that it is a true ellipse, considering the diameter of the pencil etc... The right method : use a LOOP of rope, going around 3 poles of same non null diameter : two fixed poles, one moving (the pencil). It is not if the 3 diameters are unequal.
>>> Using a string would have let them produce not only circumferences, >>> but also ellipses, so they wouldn't need conic sections ~ >I'm not sure the study of conic sections by the ancient was motivated >by "extending the compass and straightedge".
~ I was speculating because I am trying hard to reason around this to me seemingly unexplainable fact ~ To me geometry definitely is Math, which is part of I am researching on right now. Trying to demonstrate or elucidate to a certain extent that Geometry and Arithmetic are to kinds of different Math languages and that Geometric proofs may be/are as valid as verbal axiomatic ones ~
> I've been told that the discovery of irrationnality (sqrt(2)) gave them a big shock!
~ Of, yeah! The Pythagorean school thought natural numbers to be the language of nature ~
> However for the history of maths, I think nobody knows *exactly* what was in the mind of ancien geometers.
~ Well, I am not really trying to get to their minds, but there has to be some explicitly enough and clear indications of why they preferred some tools and to other ones, given the options they had and very well knew about ~ So far I haven't found a totally clear explanation as to why it happened to be this way. Maybe they just considered a compass to be their latest toy that looks like some conic, who knows ~ lbrtchx
"Philippe 92" <nos...@free.invalid> writes: > However for the history of maths, I think nobody knows *exactly* > what was in the mind of ancien geometers.
Obviously. For that matter, nobody knows *exactly* what is in the mind of modern geometers either. I don't even know what is in my own mind.
> I've been told that the discovery of irrationnality (sqrt(2)) gave > them a big shock !
"Ioannis" <morph...@olympus.mons> writes: > Robert Israel wrote: > > "Philippe 92" <nos...@free.invalid> writes:
> >> However for the history of maths, I think nobody knows *exactly* > >> what was in the mind of ancien geometers.
> > Obviously. For that matter, nobody knows *exactly* what is in the > > mind of modern geometers either. I don't even know what is in my > > own mind.
> I *always* know *exactly* what's in my own mind: A big juicy Ponderosa > T-bone > steak with fries!
Ah, but if you always know exactly what's in your mind, then you must also know the following facts:
1) A big juicy Ponderosa T-bone steak with fries is in my mind 2) "A big juicy Ponderosa T-bone steak with fries is in my mind" is in my mind 3) Fact #2 is in my mind 4) Fact #3 is in my mind ...
Do you have an infinite mind? -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
Robert Israel wrote: > "Ioannis" <morph...@olympus.mons> writes:
>> Robert Israel wrote: >>> "Philippe 92" <nos...@free.invalid> writes:
>>>> However for the history of maths, I think nobody knows *exactly* >>>> what was in the mind of ancien geometers.
>>> Obviously. For that matter, nobody knows *exactly* what is in the >>> mind of modern geometers either. I don't even know what is in my >>> own mind.
>> I *always* know *exactly* what's in my own mind: A big juicy Ponderosa >> T-bone >> steak with fries!
> Ah, but if you always know exactly what's in your mind, then you must also > know the following facts:
> 1) A big juicy Ponderosa T-bone steak with fries is in my mind > 2) "A big juicy Ponderosa T-bone steak with fries is in my mind" is in my mind > 3) Fact #2 is in my mind > 4) Fact #3 is in my mind > ...
> Do you have an infinite mind?
I think anyone who understands large numbers and the classic definition of infinity in analysis, does. For example:
On your favorite night before going to sleep, close your eyes in the dark and think of the largest possible mental space that can be perceived by your mind, outwards.
You will easily discover that it is a void black sphere of radius R, with your consciousness at the very center. Think of any radius R you want. 25 light years, say. You can make your mind's radius larger than that: You can visit, say, Vega, mentally. Think of another radius: 10^10^10^10 light years. You can again make your mind's mental sphere radius greater than that, provided you can approximately perceive the immense size of this number, relative to unity.
For _any_ radius R I give you, you can expand your mind to a mental sphere of radius larger than R, provided you have sufficient understanding of the size of R.
Doesn't that imply that the mind is infinite? -- Ioannis
Robert, ~ on your investigative work, ~ http://www.math.ufl.edu/~rcrew/texts/pythagoras.html ~ IMO (H or not you may find or care) I think you should research more the issues relating to Hippasus of Metapontum. As I understand/have extensively read, he was creating great disruptional problems in the Pythagorean school, which was more of a society or even a sect, to the point that it split in Akusmatikers, those following P absolute believe about natural numbers being some sort of "nature's language" (to a large extent motivated by the correlation of frequencies in the musical scale as it nature was speaking for itself ...), and Matematikers, who sought the truth of matters in demonstrations ~ The school after their master's dead moved to Metapontum and Hippasus, whose work on Acoustics has reached our times, proved that even the proportion of the side to the diagonal on the emblem they used (a pentagon) was not rational. ~ The most succinct historic account I have found on this in in: ~ // __ http://www.amazon.de/dp/3322004767/ ~ Mathematik im Reich der Töne (Broschiert) von Eberhard Schröder (Autor) ~ # Broschiert: 111 Seiten # Verlag: Teubner Verlag; Auflage: 4. A. (März 1997) # ISBN-10: 3322004767 # ISBN-13: 978-3322004765 ~ You specially want to read chapter 9: "Glanz and Verfall des Weltbildes der Pythagoreer". Since you are interested in this info and I know this book is very hard to get I took pictures of these 5 pages, mrt_pg_[58-63].jpg, you can get from: ~ http://www.geocities.com/tekmonk2005/pub/Math_Reich_Toene/ ~ The book was written in German ~ Even Plato, someone who would have been more inclined to accept old- school thinking, expressed his very harsh opinion on what happened (whatever it was, even if not a holiwood-like act, it seemed to be downright drastic) ~ Hey Ioannis, as Philippe 92 told you, even thought it became kind of a modern thing after David Hilbert's work, I think your stat about Geometry not being part of Math is more than half way off. Also, kneading my balls has a pacifying effect on me and I have heard some other people suck their thumbs, caress their noses or curl their hair, but I don't try to see it as a Mathematical thing at least not primarily, let along a transcendent one, that other Mathematicians would care about. It is hard to tell if you are just joking, but in this case, keep trying making better jokes or relevant ideas of the relation between Math and your autogenic feelings. ~ Again, we are talking here about why we mathematicians help ourselves in certain ways and not other ~ lbrtchx
Philippe 92 hasn't "told me" anything. I don't see any responses to _me_ by him. I only had a short, joking exchange with Robert.
> even thought it became kind of > a modern thing after David Hilbert's work, I think your stat about > Geometry not being part of Math is more than half way off.
"My stat"? I NEVER claimed that Geometry is not part of math. How could I? I was taught Geometry when I was 13. In fact, Geometry was what got me into math. But I guess you are too clueless to even KNOW that Geometry is obligatory in Greek education, of all things.
> Also, > kneading my balls has a pacifying effect on me and I have heard some > other people suck their thumbs, caress their noses or curl their hair, > but I don't try to see it as a Mathematical thing at least not > primarily, let along a transcendent one, that other Mathematicians > would care about.
Whatever the above may mean, anyway... Feel free to elaborate. We are all VERY interested in your psychological meanderings.
> It is hard to tell if you are just joking,
Is it THAT hard? Robert is saying something totally hillarious and funny ("I don't even know what's in my own mind...") and I thought I might just add a little humorous spice in reciprocation with my T-bone steak (which, btw, is abolutely true. Honest...)
Gosh. I guess it was REALLY hard to figure out if I was joking.
> but in > this case, keep trying making better jokes or relevant ideas of the > relation between Math and your autogenic feelings.
Hunger is not an "autogenic feeling", you clueless maroon. It's a primary human DRIVE.
Well, I guess there's nothing wrong with being just another humor impaired blockhead, but what's wrong with your REAL name? Why aren't you using it?
> lbrtchx
^^^^^^^ "Liberty Checks"? That's a heck of a name you've chosen for yourself, absent your REAL name. -- Ioannis
On Dec 10, 2:18 pm, "Ioannis" <morph...@olympus.mons> wrote:
> ...
~ Sorry I meant eaglesondouglas I was just typing fast and generally, also as anyone could see I am not trying to be offensive in fact I am making myself part of the joke ~ lbrtchx
That's not my investigative work, it's Richard Crew's.
BTW, Crew seems to have missed one of the references to Hippasus in Iamblichus: De vita pythagorica 246-247
They say that the man who first divulged the nature of commensurability and incommensurability to men who were not worthy of being made part of this knowledge, became so much hated by the other Pythagoreans, that not only they cast him out of the community; they built a shrine for him as if he were dead, he who had once been their friend. Others add that even the god became angry with him who had divulged Pythagoras' doctrine; that he who showed how the icosagon (that is the dodecahedron, one of the five solid figures) can be inscribed within a sphere, died at sea like an evil man. Others still say that the same misfortune happened on him who spoke to others of irrational numbers and incommensurability.
> IMO (H or not you may find or care) I think you should research more > the issues relating to Hippasus of Metapontum. As I understand/have > extensively read, he was creating great disruptional problems in the > Pythagorean school, which was more of a society or even a sect, to the > point that it split in Akusmatikers, those following P absolute > believe about natural numbers being some sort of "nature's > language" (to a large extent motivated by the correlation of > frequencies in the musical scale as it nature was speaking for > itself ...), and Matematikers, who sought the truth of matters in > demonstrations > ~ > The school after their master's dead moved to Metapontum and > Hippasus, whose work on Acoustics has reached our times, proved that > even the proportion of the side to the diagonal on the emblem they > used (a pentagon) was not rational. > ~ > The most succinct historic account I have found on this in in: > ~ > // __ http://www.amazon.de/dp/3322004767/ > ~ > Mathematik im Reich der T=F6ne (Broschiert) > von Eberhard Schr=F6der (Autor) > ~ > # Broschiert: 111 Seiten > # Verlag: Teubner Verlag; Auflage: 4. A. (M=E4rz 1997) > # ISBN-10: 3322004767 > # ISBN-13: 978-3322004765 > ~ > You specially want to read chapter 9: "Glanz and Verfall des > Weltbildes der Pythagoreer". Since you are interested in this info and > I know this book is very hard to get I took pictures of these 5 pages, > mrt_pg_[58-63].jpg, you can get from: > ~ > http://www.geocities.com/tekmonk2005/pub/Math_Reich_Toene/ > ~ > The book was written in German
And it was written recently. What sources does Schroeder base his conclusions on? Or is he just making it all up?
As far as I know, there are very few references to the discovery of incommensurability in the ancient literature, and nothing to indicate that this caused a crisis in the Pythagoreans' thinking. What they were upset about was the fact that Hippasus disclosed their secrets to people outside the Pythagorean brotherhood.
> Even Plato, someone who would have been more inclined to accept old- > school thinking, expressed his very harsh opinion on what happened > (whatever it was, even if not a holiwood-like act, it seemed to be > downright drastic)
And where does Plato say this? -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
Discussion subject changed to "Why didn't ancient Greek Mathematicians use a string instead of a compass for their constructions?" by lbrt...@gmail.com
> That's not my investigative work, it's Richard Crew's. > BTW, Crew seems to have missed one of the references to Hippasus in Iamblichus: De vita pythagorica 246-247
~ Well, I found a little strange that you or anyone that should know better had written this, because I have read about this from many reliable sources. ~ Also I noticed a weirdly persuasive style of writing that is not common in scientific papers. What could conspiracy theories about Kennedy's assassination possibly prove or disprove in relation to whatever happened to Hippasus of Metapontum? If you want to hear my personal opinion in this full-of-weird-off-topic_themes-and-jokes thread: "I find downright insultingly stupid that some people try to make other people believe that Kennedy was killed by Marinus van der Lubbe single shot from the sixth floor of the Texas School Book Depository, ... and (among many other things) that same bullet kept bouncing around till it fatally reached his brother Bob who was investigating what had happened ... ~ We Scientists/Mathematicians do that kind of stuff too. We don't stop being human. Einstein was ostracized by his own friends at der Preußischen Akademie der Wissenschaften in Berlin, all of which to me are like semi-gods (the ones who put an end to the mechanic view of physics/nature), who would not even sit next to him and David Hilbert ran that French Mathematician (No, not Poincaré' ;-)) from the Math society he was chairing basically for disagreeing with him ~ If we, people that should seek objectivity foremost, do that, what could we expect from politicians ... no wonder they invade, occupy and totally destroy a country in an outright act of abusive state terrorism, based on WMD stories (Do you remember that preposterous sh! t?!?) and as if it were not enough, that happens in our times in the "war against (not made in the USA) terrorism", because "we should defend our freedoms", "God blessed us", "America ueber alles", ... and all that crap ~
>> The book was written in German > And it was written recently. What sources does Schroeder base his conclusions on? Or is he just making it all up?
~ I would doubt to a very large extent that Herr Schroeder, who was one of my dear teachers, made all of this up. I had to sit his classes about the history of Physics/Math twice, because when I was young I was very opinionated/problematic and refused to sign the attendance sheets ;-), but his Vorlessungen were a gem anyway ~ This is just a short chapter of some 110-page brochure he wrote about the Mathematical foundation in the evolution of the musical scale, but anyway; this is what he wrote on the book and my fast and even if true, far from unqualified translation to (my not native) English ~ ~ ~ ~ ~ ~ ~ ~ ~ GERMAN ~ ~ ~ ~ ~ ~ ~ ~ ~ "Ich have ja wohl auch selbst erst recht spaet etwas davon vernommen and musste mich ueber diesen Uebelstand bei uns hoechlich wundern. Es kam mir vor, als waere das gar nicht bei Menschen moeglich, sondern nur etwa bei Schweinevieh. Und da schaemte Ich mich, nicht nur fuer mich selbst, sondern auch for alle Helenen." ~ ~ ~ ~ ~ ~ ~ ~ ~ ENGLISH ~ ~ ~ ~ ~ ~ ~ ~ ~ Sure, I heard well myself late about it and I was quite surprised about such grievance. Such acts seemed to me to be absolutely impossible by us human beings. They would be expected only from a pig cattle. And I was ashamed, not only for me, but also for all Hellenistic people(?) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Now, if you have read Plato, you would know that he didn't tend to engage in this kind of -Socratic- way to use language, specially calling the Pythagorean school a -pig cattle- ("pig" among middle eastern, Mediterranean/Greek people had/has a very derogatory charge) and being ashamed not only "for himself, but also for all Hellenes (?)" ... ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ It truly amazes me that so little is being said about this great Mathematician ~ http://en.wikipedia.org/wiki/Hippasus ~ http://el.wikipedia.org/wiki/%CE%8A%CF%80%CF%80%CE%B1%CF%83%CE%BF%CF%82 ~ This guy, short of being canonized as some soft of "Universal Saint" patron of "The Pursuit Truth", deserves a Milos Forman rendition of his life as a movie!!! (Please, Hollywood don't trash this poor guy even more!) ~
>> Even Plato, someone who would have been more inclined to accept old- >> school thinking, expressed his very harsh opinion on what happened >> (whatever it was, even if not a Hollywood-like act, it seemed to be >> downright drastic) > And where does Plato say this?
~ You are right I couldn't find the exact reference to it in the book, but, again, this book was primarily about Musik. I did post two more pictures with the two bibliographic pages about it ~ However, I found here such a reference: ~ http://www.greektexts.com/library/Plato/laws_(books_7_-_12)/eng/19.html ~ LAWS, by Plato (translated by Benjamin Jowett) ~ My trans from German wasn't that bad after all ;-) ~ lbrtchx
well, if the Pythagoreans merely cast him out, *and* erected a shrine to him as though dead, that's dramatic.
I thought that I was original in using "icosagon" for the dodecahedron, although Bucky did it with a neologism, *-vertexion*; actually, since I was wont to reserve the "-gon" for flat figures, I proposed using *-asteron*.
> BTW, Crew seems to have missed one of the references to Hippasus in > Iamblichus: De vita pythagorica 246-247
thus: Mascheroni is cool; I'll have to look at that, again.
it seems to me that the wookypoopieya thing is just a trivision of a segment, which seems to be realted to an equilateral trigon; I was referring to n-section.
lbrt...@gmail.com wrote: > On Dec 10, 2:18 pm, "Ioannis" <morph...@olympus.mons> wrote: >> ... > ~ > Sorry I meant eaglesondouglas
Well, then it's not my fault that you got the stück.
I think it helps to be careful WHO you reply to, and HOW you reply to them. Consider sci.math to be a newsgroup where "surgical precision" matters the most (whether in arguments or in blows).
> I was just typing fast and generally, > also as anyone could see I am not trying to be offensive in fact I am > making myself part of the joke
> > On Dec 9, 3:48 am, lbrt...@gmail.com wrote: > >> As you could tell from my previous question: > >> I am interested in the details of why have we Mathematicians used > >> certain means and ways both logical and mechanical to operate and > >> express ourselves > >> ~ > >> Using a string would have let them produce not only circumferences, > >> but also ellipses, so they wouldn't need conic sections
> I'm not sure the study of conic sections by the ancient was motivated > by "extending the compass and straightedge".
> > Geometry was the issue. > > ... > > It is for this reason geometry IS NOT mathematical.
> Humm. I leave this opinion to YOU.
> However a string (apart from the added ellipses(*)) allows to construct > a lot of things too. Of course straight lines (tightening the rope, > that's the way gardners and other house builders draw straight lines, > with a rope soaked with chalk). Of course a circle. So all what can > be drawn with compass and straight edge can be drawn with rope alone. > But as you said, it allows even more, as does just paper folding > for instance (allows trisection of angles, and construction of 7 > sides regular polygons).
> However for the history of maths, I think nobody knows *exactly* > what was in the mind of ancien geometers.
> I've been told that the discovery of irrationnality (sqrt(2)) gave > them a big shock ! > Hence they tried to prevent the occurrence of such further shocks by > not allowing too powerfull tools, and limiting to just compass and > straightedge. The Idea is as is.
> In that time, geometry was quite "contemplative", many proofs being > "just look" from a well drawn figure. Hence the search of exact > constructions in that time. > Then much effort has been made to make the geometry more rigourous. > First one being Euclides, then more recently Hilbert etc. > Then the geometry today is much more algebraic, considering sets and > metrics and manifolds etc. (but it IS mathematical)
> Considering geometry constructions as just a game, we need however to > fix the rules. Changing the rules changes the results (what can and > can't be constructed).
> As well known, the use of rope (the 12 knots rope) to construct > right angles was a current practice. Why didn't they choose the > 'rope only' constructions ?
> To mention also that even the ancient tried to overcome these limits, > by using specific curves (trisectrix, quadratrix etc) instead of just > circles and straight lines, also "Neusis constructions" like the > well known angle trisection by Archimedes (humm, not sure he was the > first) with a marked straightedge etc.
> As there are mechanical tools to draw continuously conic sections, > you could use "construct by conic sections" instead of just compass > and straightedge (straight lines and circles being degenerate cases > of conic sections, we call that just "conic sections").
> But these are quite "recent" discoveries.
> A big problem also is that a straightedge is quite hard to make. > (How do you ensure it is perfectly straight ?) > But a compass is much more robust. Just suffice it is enough rigid.
> Hence many geometers tried to discard the straightedge. > I mentioned Mascheroni in another post. > The study of inversion also gave a mean to *construct* a straight line > from scratch (from a circle).
> The world of construction rules, and geometry fundaments is quite > wide...
> (*) draw an ellipse with a rope. Proove that it is a true ellipse, > considering the diameter of the pencil etc... > The right method : use a LOOP of rope, going around 3 poles of same > non null diameter : two fixed poles, one moving (the pencil). > It is not if the 3 diameters are unequal.
very pedantic, in a sort of Korbzyski Pidgen E' sort of way; ete-vu mon General Bourbaki?... the new math is dead -- long live the new math!
anyway, I forgot to state the real genesis of my reply. it seems obvious that making constructions with a catenary is going to be problematic -- and it's a Heck of a problem in musical theory, two!
if whoever stated the "theorem" about the diameter of the poles is correct, vis-a-vu the *hypodenaptae* (sp.?), it certainly wil be a practical matter for any constructions on the desktop, if it was ever formally done with a particualr set-up. clearly, linkages are very adequate, although I never figured the old-style "pair of compasses."
> Just be reminded that functions are not just mathematical. Functions > preceed mathematical sets as a mapping of set element to ANY other > set.
I recalled, the Latinization of the Greek term, Harpadenaptum. so, you have to account for teh tension in the catenaries, and a 3-4-5 trigon, knotted on a 12-unit loops, gives you a nice sequence of superparticular ratios, plus an odd one of 5/3.
