I guess this thread is about the physical inexactness of compass and straightedge constructions versus their logical exactness. When I first started using CAD tools (like AutoCad) I expected the tools to be based on inputing exact coordinates of the vertices. Instead, I found that they are based on the same techniques used in drafting, compass and straightedge. This intrigued me because I thought that this would produce inexact constructions with gaps showing up between the final two points. I even tried to prove this to myself by constructing polygons with many sides to see the final side not exactly connect to the first. There were no gaps. Even when I zoomed in. I realized that the way these tools get around this problem is that internally they use numbers with the full precision of the computer (say 15 digits) but for display and output they use only the first 10 digits of precision. This way, unless you cause a gap that is more than 5 digits in magnitude, you will never! see it. At 10 digits of precision (which is itself still very precise) everything appears exact.
Another example of this has to do with older calculators versus newer ones. I mentioned before how as a kid I would take a number and divide it by 3 repeatedly till I had almost nothing left, and then reverse the process by multiplying by 3 again, till I got back to my original number. I would also do this with square root and log. Obviously, you don't get back to the original number, just close to it, depending on how far you went with the first operation. This is due to the successive errors that are introduced in each calculation because the computer is limited to finite precision. On newer calculators this trick doesn't work as well as it used to and for the same reason as with the CAD software above. Newer calculators employ the same trick of storing the results internally at a higher precision than what they display. Thus, 1 divided by 3 is 0.33333... and when you multiply that by 3 you get back to 1, rather than 0.99999... like you did before. Internally, the result i! s actually 0.99999... but when it is rounded to the display precision it is exactly 1.
This trick (maintaining an internal representation that is more precise than the displayed precision) works very well at overcoming the problems with finite precision and computation.
On Aug 23, 2013, at 11:09 AM, Joe Niederberger <email@example.com> wrote:
> J Crabtree says: > 3) Explain the proofs and the mathematics are fine, yet it is inevitable that errors will accumulate for you (the student/reader) just as they do for me (the expert/author). > > If you want to explain some mathematics, why not give us your analysis of what errors occur, their relative magnitudes and distributions, the overall accumulation of such errors for each of 5 or so different pentagon constructions? > > Then you can set "acceptable" tolerances on the resulting figures drawn according to the rules. > > Further explain what actions might be taken outside the regimen of strict straightedge and compass construction to improve upon the error limits explored above. > > We could then separate those "improvements" into two categories - those that are so subtle they may go unnoticed (cheating) and those that are quite explicit. > > Armed with your knowledge, go and promote a "Pentagon Drawing Contest" TV show - where the winner takes home $1,000,000 (Ausie). Have it become the #1 watched show in Australia, and then move into the Merican market. > > Create a huge scandal when the cheaters are exposed... > > Make a movie about the scandal... > > And so on.. > > > Cheers, > Joe N