We all "know" that it is impossible to trisect an arbitrary
angle with traditional compass and straightedge, so I was surprised
to see, while browsing Wikipedia, a trisection method, discovered by
Archimedes no less!
Obviously the method goes beyond Plato's restrictions.
Some might want to discover such a method for themselves:
I think it would deserve a very big pat-on-the-back.
I guess the "traditional" way to go beyond Plato's restrictions is
to allow the constructor to make marks on the straight-edge.
Archimedes' method does that, but, it seems to me, makes
a more dubious assumption: in Archimedes' trisection
method one needs to align the marks on circle and line,
even though neither has a marked point-- in other words
the anchor is yet a 3rd point. This gives the manipulation
2 degrees of freedom instead of 1. (With 1 d.o.f. one moves
the straightedge continuously and stops when appropriate;
with 2 d.o.f.'s one seems to need an iterative approximative
search.)
Is this correct? Is there a way to trisect an arbitrary angle
without such a dubious manipulation?
James Dow Allen
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