[FOM] Compass and straight edge constructions

Joe Shipman joeshipman at aol.com
Tue Sep 27 15:37:24 EDT 2016

With a marked ruler you can solve all cubics (and quartics), which gets you all polygons whose number of sides is a power of 2 times a power of 3 times distinct primes of the form (2^a)(3^b)+1.

With a marked ruler AND compass you can solve some quintics and equations of higher degrees, but it's not clear you can quinquesect angles, I'm actively researching that.

With standard origami folds you can solve cubics and quartics the same as with a marked ruler, but if you allow rolling of cones or cylinders you can n-sect angles for any n, and this get all algebraic numbers with abelian galois group by the  Kronecker-Weber theorem.

The next step would be an "nth rooter" tool which would combine with the previous to give you all algebraic numbers with solvable Galois group.

With linkages (jointed rulers) you can get all algebraic numbers.

-- JS

Sent from my iPhone

> On Sep 26, 2016, at 1:55 PM, Steve Stevenson <steve at clemson.edu> wrote:
> Some students I'm mentoring asked me the other day about constructions.
> We all know that we can't do everything with straight edge and compass
> alone. My question is, what is the minimum extension needed to do, say,
> all polygons?What might be a good reference along this line?
> ​Thanks,
> steve​
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