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Re: BUSTED! Re: Mythbusting pentagons by compass & straight edge.
Posted:
Aug 20, 2013 8:27 PM


Re: >> The "myth" has always been in your brain not in the mathematics. In 6th grade it was obvious to me that constructing a regular hexagon with a compass was a triviality (and understood the trivial underlying mathematics) but had already observed that, if I really wanted a regular hexagon, not a cute construction were the final arc never hits the starting point, I should use a protractor. It's the proof that matters and, even more so, that the regular septagon can NOT be so constructed, not some Easter Bunny in your head. Why persist in embarrassing yourself?
Wayne
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Hello Wayne (and future readers of this discussion)
The word 'embarrassed' never would have occurred to me. On several occasions I stated both the mathematics and the proofs work. My only surprise was to discover that the 'regular' pentagons carefully drawn with compass and straight edge for publication and labelled as 'regular' have been so irregular.
I enjoy exploring math and math history and I am always open to learning something new. For me, discovery is addictive. I enjoy the selfdiscovery part much more than simply being told what to think and how I should feel.
Today I will be enjoying an original edition of Euclid's Elements printed in 1482. I will be turning the very same pages in the 21st century others did in the 15th century.
I might feel a little embarrassed to have to wear white cotton gloves, yet I'm sure I'll get over it!
I'm now proud to declare I have failed to draw a regular pentagon with compass and straight edge just as others have failed.
Failure is feedback and I have been reminded all diagrams may now be 'wonky' as long as the math, proofs and ideas behind them are pure and true.
Like you, I also admire Gauss's proof that a regular septagon cannot 'in theory or practice' be inscribed on a circle with compass and straight edge.
Wayne, you wrote earlier, "I really like compass and straightedge construction and one of my bedtime curses is that our generation has been somehow able to kill it."
Maybe your generation has killed the joy of compass and straightedge construction. So perhaps two lessons to come out of this for future students might be:
1) Explain that it is REALLY HARD to draw accurate constructions with straight edge and compass
2) Do NOT label pentagons as 'regular' when they are not!
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).
Otherwise students and readers may be misled by such published diagrams as the following...
http://jonathancrabtree.com/private/Pentagon_Construction.jpg http://jonathancrabtree.com/pentagonsbycompassandstraightedge/WonkyPentagonDrawnFollowingEuclid_BookIV_Proposition11.jpg http://jonathancrabtree.com/pentagonsbycompassandstraightedge/Beautiful_Pentagon_Is_Wonky.jpg
Thank you for your interest and contributions to this discussion, for which I am most grateful.
Best wishes Jonathan P.S. Yes, as I enjoy the Latin, I'll be sure to run a ruler and protractor over the 1482 pentagon in Euclid's Book IV Proposition 11 just for fun!



