Exploring the Limits of Ruler and Compass Constructions in Abstract Algebra

[mishima]

Hi, I know next to nothing about abstract algebra (had one intro-class years ago), so sorry if this is a dumb question. I was browsing through "A book on abstract algebra" by Pinter and had a thought. In the chapter called "ruler and compass" (chapter 30) he talks about how abstract algebra can prove the impossibility of certain constructions using ruler and compass: doubling the cube, trisecting any angle, and squaring the circle.

I was wondering if abstract algebra can also suggest new instruments which include those prohibited constructions as possibilities?

Like can it describe new tools which are able to perform those constructions which haven't been invented yet?

Or is it really saying that there can never be ANY tool, or combinations of ANY tools, which can perform those feats?

Thanks for any insight on this.

edit: Or maybe a better way to ask this would be to ask if it can suggest new "operations" which mimic the behavior of real-life tools.

 

[HallsofIvy]

Certainly there are ways of trisecting angles with other instruments. In fact, I notice you titled this "Ruler and compass". That's sufficient! The usual "instruments" are a straightedge (unmarked) and compass, not "ruler" and compass. In fact, allowing you to mark a single distance on your straight edge and "measure" with that is enough. There is also the "trisection tool" (four lines with hinged connectors), the "tomahawk", "carpenter's square", and "limacon", "MacLaurin's trisectrix", etc.. You can read about them at
http://www.jimloy.com/geometry/trisect.htm#tools

 

[mathwonk]

the straightedge and compass constructions suffice to solve quadratic equations. thus only lengths can be thus constructed that occur as solutions of repeated quaDRATIc EQUATIONS.

the problems you describe require either solutions of cubic equations, or in one case at least, a length that satisfies no algebraic equation.so your question is sort of like asking "if a number cannot be a solution of quadratic equations, can it still be found some other way?" often, yes.

 

[mishima]

Very cool.

So is there a minimum set of tools that could offer comprehensive constructions for all elementary algebra (polynomials < 5th degree)? Is that question answerable using abstract algebra?

Can the physical appearance of those "appropriate" tools be learned from abstract algebra alone? Like if the tomahawk was never invented, could abstract algebra have suggested its existence?

And I may as well ask this semi-related question, does anyone know any references that show actual Greek compasses and straightedges or describe how they were made in ancient times? Thanks again.

 

[SteveL27]

Very cool.

So is there a minimum set of tools that could offer comprehensive constructions for all elementary algebra (polynomials < 5th degree)? Is that question answerable using abstract algebra?

Can the physical appearance of those "appropriate" tools be learned from abstract algebra alone? Like if the tomahawk was never invented, could abstract algebra have suggested its existence?

And I may as well ask this semi-related question, does anyone know any references that show actual Greek compasses and straightedges or describe how they were made in ancient times? Thanks again.

You might be interested in taking a look at the basic Wiki page.

http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions

One factoid of interest is that if you are allowed to put a mark on your straightedge indicating a known distance, you can then solve problems you couldn't solve without the mark.

 

[mathwonk]

I have read that factoid, but when I also read the subsequent construction it seemed to em it involved a procedure that cannot be made precise. I.e. given a straightedge with two marks on it , one is required to position it so that one mark lies on a certain line and the other on another line and the ruler also passes through a given point off the lines, (see Hartshorne, Geometry, Euclid and beyond, page 260). I cannot see any precise way to actually do this even with a marked ruler. I.e. it is an "eyeball" procedure that has no algorithm for executing it. Perhaps I have missed the point.

You could do it if you had holes in your ruler at each marked point, and nails through the holes resting in tracks corresponding to the two lines. Then you could slide the mark on the ruler along one line and the track would force the mark to remain also on the other line as the ruler moved until it passed through the given point, but this requires a lot more than a mark on the ruler. So I consider this "construction" a bit forced, and not fully described by the usual language of just a marked ruler.