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September 16th, 2015, 12:25 PM  #1 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  What does a geometrical construction mean today?
The word construction in geometry derives from the ancient Greek word, but even in their day there were differences as to wht it meant. Some incorporated the use of a straight edge, some led by Plato allowed only the use of a pair of compasses. Neither allowed measurement of linear distance by ruler. Of course the Greeks had more limited technology. They did not know how to manufacture a straight edge to any desired accuracy and their straightness technology was substantially inferior to their technology for drawing circles. Further they directed their mathematics with religous motivations, believing in absolute perfection. Anyway this all resulted in an elaborate and inmany ways admirable edifice of what could be drawn using only the allowed 'construction techniques' to be developed up as far as the late renaissance, a period of about 1700 years. The Victorian scientific explosion led to the use of geometrical diagrams for many new purposes, not covered or possible under the either of the old greek rules. One of these was the introduction of signed lengths. Another was the introduction of a method of solving arbitrary polynomial equations by Lill (1867) by means of drawing a series of signed lines of measured length on a rectangular grid. This introduced the T square and set square and as well as the ruler into 'construction'. The protractor was soon added. These additional tools proved so useful that by the early twentieth century their use was taught in the scool curriculum in geometry classes for construction (Durrell 1927), Latimer and Smith (1937) Mobius introduced the drawing of reciprocal figures well before the end of the 19 century. All this was brought together in books by Luigi Cremona Graphical Calculus & Reciprocal Figures and Ewart Andrews Elements of Graphic Dynamics So today we have added many draftman's tools to the armoury of geometrical construction, even without the computer. This thread was started as a discussion of the question where are we today? 
September 18th, 2015, 04:42 PM  #2 
Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: noneuclidean geometry 
Isn’t geometric construction accomplished by whatever rules we set up, using whatever tools we allow? My understanding of this question is different from the way you are asking it. Shouldn’t the question be about whether or not the geometry itself has ever been truly understood? The way that you approach this question makes a lot of sense, and it’s actually a process of thinking that mimics the order in which our understanding of these things evolved. But something else has also been happening during the last hundred years or so, where our abstractions have become much more abstract and they are used more and more to describe things that don’t really exist. (Might have something to do with Tim Leary and the 60's, don't know, just hypothesizing.) Which takes us right back to your description of a construction with a compass. Does a circle really exist? Some of the most interesting things (to me at least) seem to occur whenever some abstract thing that is believed not to exist is shown to actually exist. I’ve felt for a long time that this notion of orienting three orthogonal planes in space is an extremely poor abstraction of what three dimensions really are. If plane geometry can be so rich, as it is, with just one plane, shouldn’t threedimensional geometry be even more so? We’re adding another dimension; we’re not just adding two more planes. That should be a very easy proposition to accept. The way that I view things nowadays, the word geometry itself is sort of a miracle. Its meaning is much more descriptive to me now than it ever has been before. In threedimensional geometry, every point seems to have the same value, but it also has its very own geometry. By this I mean that it has a special view (“spherical” sort of describes it, but “geometric” nails it) of all the points around it. Direction has a whole new relationship in three dimensions. I’m still working on an animation, to show this, but until that effort is completed we’ll have to make do with a lot of frantic hand gestures. I hope that I haven’t wandered off topic here, with this. I might ask, can an animation be a geometric construction? 
September 19th, 2015, 03:59 AM  #3 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
Thank you for replying, Steve. I was difficult to know where to place this thread since there is not a higher level geometry section in this forum. It was also difficult to know how to phrase my thoughts as a question, so fear not all comments are welcome and not off topic as I don't know exactly what the topic is. However I agree with you that geometry seems to be a poor relation these days, and that any consistent set of rules should be OK so long as they are made clear at the outset. Clearly, however some sets of rules are more useful than others so perhaps that is the purpose of this thread. To find the best set of reules, if there is one. FYI I started this thread as promised to CR GReathouse in posts 15 and 16 here, after much pondering the above points. Error in pi number 
September 19th, 2015, 09:47 PM  #4 
Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: noneuclidean geometry 
The rules do seem to matter a lot. The compass and straight edge is probably still the basic definition of geometric construction. If I were to respond directly to the question raised about whether or not an arc can represent pi, I would say no, because pi represents a length and length is the distance between two points. But I could easily be wrong. I know that you pointed out to me the other day that, right or wrong (not you, of course you are correct, but rather the folks who make the rules), it has been decided that the center of a sphere is not part of the sphere. Makes absolutely no sense at all to me. 
September 19th, 2015, 10:03 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
There are lots of ways to do geometric constructions! The rulerandstraightedge is of course wellknown. The equivalent (!) methods of marked straightedge and compass or origami have greater constructive power. (I believe mechanical linkages give the same ability as well.) I recently read an article about constructions with compass and straightedge with *two* marks and that yields another type inequivalent to the others. Of course allowing a ruler with all lengths marked gives something like Cartesian geometry. If you know about field and Galois theory, each gives rise to a field closed under certain operations. For example, the compassandstraightedge lets you construct lines with length (relative to some given unit) which is in the quadratic closure of the rational numbers, and circles with radii in same. 
September 23rd, 2015, 05:26 AM  #6 
Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: noneuclidean geometry 
Galois theory is something I'd never even heard of. It does seem to cover much of what was raised by the OP. After pondering the basic claim made in the ancestral thread to this one, I can’t see any error in it. Absent some identifiable error, doesn’t the theory have to stand? I’m not saying that there is no error; I’m saying that I don’t know where it is. Too bad the discussion was shut completely down before any error could be identified. Afaict, the only push back against the proposition of pi being wrong was a link to some blog where someone supposedly debunked the theory. I am not at all swayed by the debunking effort, which amounts to nothing more substantial than hurling slurs and insults, and contains no mathematical arguments of any at all which contradict the poposition. The debunker seems to get so caught up in all the woo surrounding the question (stories about NASA and such) that they fail to even try to find the error in the basic proposition. My guess is that the red circle in the “geometric construction” used to explain the theory doesn’t really fit neatly into a unit square the way that it is drawn. But I haven’t been able to figure out a way to show that these two things are not equal. So it’s merely a guess, and that’s all it is. I will withhold my disbelief or ridicule until a proof is shown. It’s easy for me to accept that pi is something other than what we believe it to be. It’s based on a complete abstraction that may or may not exist in the physical world. There may be a difference between what we believe about pi and what is real. I guess I just don’t know enough math to allow me to rule out the possibility. I’m not suggesting that pi is not accurately defined mathematically; I do believe that it is very accurately defined. I just think that we may have defined something other than what we believe we have defined. It is fundamentally an abstraction. "A subtle thought that is in error may yet give rise to fruitful inquiry that can establish truths of great value."  Isaac Asimov 
September 23rd, 2015, 05:45 AM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
You should read through some of the basic proofs, then. Once you inform yourself you won't need to rely on trusting authorities (or crackpots, for that matter).  
September 23rd, 2015, 07:19 AM  #8  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
Just that its originator has abandoned it since he was rumbled. In that ancestral thread I gave an exact value for pi. (which was not the value offered in the OP) There may have been discussion about drawing methods and measuring methods, but it remains exact, by definition. The purpose of this thread is to discuss drawing methods in the modern context since it is irrelevant to the exact value of pi or any other number.  

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