Conference rusure::math

    Trisecting any angle in general is no big sweat.  However, trisecting
    an arbitrary angle with only a ruler and a compass with finite steps
    has been proved to be impossible.  The proof involves fields extention 
    (There I go again :-). 
    The idea is to find out the set of real number constructable with
    a ruler and a compass (R&C).  What this means is as follow:
    First you use the ruler to draw a line on the plane, and call the 
    length of the line 1 (unit length).  Now you try to find out the 
    kind of line you can construct with R&C.  Well you can construct 
    sqrt(2) by constructing a square (the diagnol is of length sqrt(2)).
    Now the theorem:
    Any constructable number c is algebraic; moreover c is of degree
    a power of two over teh field of rationals.
    What this means is that the only numbers you can construct are the
    rationals and the square roots (Note: this means sqrt(2 + sqrt(3))
    is constructable, but 2^(1/3) is not).
    Now the above theorem not only says that trisecting an arbitrary
    angle is impossible with R&C but also gives an easy criterion to
    determine whether a specific angle is trisectable with R&C (Unlike
    computing the Galois group, fortunately).  Basically,
    you divide the angle by three and see if the resulting angle is
    constructable or not.
    Eugene     
    

In the general case, trisecting an angle can be restated as the same 
problem as solving a cubic equation, e.g. ax^3+bx^2+cx+d=0. The solution
of this equation is known in general, and it includes expressions involving 
cube roots. Now the ruler-and-compass constructions permitted in Euclidian 
plane geometry permit the construction of numbers that are the *square* 
roots of other numbers, but not *cube* roots. So the solution set of the 
trisection problem lives on a different 'floor' in the hotel of irrational 
numbers than the ruler and compass can visit. By using other tools than 
the ruler and compass one can venture onto that 'floor', but that's
regarded as cheating from the point of view of Euclideans. 

    Re .2
    
    That's very interesting. Does that mean that if you had an instrument
    that "drew" 3-D spherical shells and you used a 2-D plane "ruler"
    you could make a construction that could trisect angles? Does the
    process extend to higher order spaces, from prime number sub-divisions
    of angles, such a 5,7,11 etc? Does the lack of closed form solutions
    for polynomials of higher order than four stop the process? Do I
    ask too many questions?

     I don't know about 3-D constructions, but there is a plane
     construction using non-standard tools for trisecting an
     arbitrary angle.  It uses a straight-edge with a length
     marked on it.  I vaguely recall that you slide the ruler
     between a line and a circle until each endpoint of the
     marked off length falls on one of them; then a line is drawn
     there.  I may have the whole "construction" at home.
     
     Dan

>    That's very interesting. Does that mean that if you had an instrument
>    that "drew" 3-D spherical shells and you used a 2-D plane "ruler"
>    you could make a construction that could trisect angles? 

No. Adding independent variables to the equation without changing the 
degree of the equation leaves you with the same solution set in each 
variable. Spheres and planes are still exponent=2 surfaces.

>    Does the process extend to higher order spaces, from prime number
>    sub-divisions of angles, such a 5,7,11 etc? 

Yes. The solutions of these problems are on still different floors of the 
hotel.

>    Does the lack of closed form solutions for polynomials of higher order
>    than four stop the process? 

The closed form does not matter - it's the difference in degree that is
relevant.

>    Do I ask too many questions?

Definitely not!

    Here is a method I developed to trisect an angle in an infinite number
    of steps.

    First, is it correct?

    Second, has anyone seen this method before?

    I've constructed the following series which converges to 1/3.

                infinity
                ----
	 1      \     n     1
	--- =    >  -1 * -------  =  1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 . . .
	 3      /	     n+1
	        ---- 	    2
		n=0

    To apply this series with a ruler and compass simply bisect an angle
    and then alternately bisect the resultant angle above and below each
    new bisection line.

    Martin H.

>>    First, is it correct?

	The series you gave is absolutely convergent and its
	partial sums do "correctly" converge to 1/3.

	But what do you mean by "is it correct?"?  I take it
	that your "it" refers to "method" in your previous
	sentence.  What do you mean by your method being
	correct?  Does it construct a specific point such that
	the line through it and the vertex trisects the angle?
	No.  Can it be used to construct a sequence of points,
	alternating between greater than and less than 1/3 the
	given angle, which converge to what would be a point
	on the trisector if a trisector existed?  Yes.

	Dan

RE: .7 (Dan)

    Hold on now.  The trisector *exists*.  It just cannot be "constructed"
    using a finite sequence of specific operations which have proven a
    fruitful set of operations for certain purposes.  Other sets of
    operations (such as those using a "tomahawk trisector") allow the
    angle trisector to be constructed.  It exists in any case as a well
    defined locus of points, i.e., those whose perpendicular distances from
    one side of the angle is � their perpendicular distances from the other
    side.

    Martin has pointed out that if we change our arbitrary rules for what
    constitutes a "construction" by leaving out the restriction that the
    sequence of operations must be finite in length, then a construction
    is possible.  He proves his point by presenting a rather elegant, and
    I believe correct, such infinite construction.  This is no more
    stranger than the infinite sequences of (less restrictive) operations
    used to construct fractals.

    Whether or not a theory of "infinite ruler and straightedge"
    constructions is a fruitful area for mathematical investigation remains
    to be seen, but it certainly has some potential.  For example, it might
    tie into the current theory of "numeric complexity" which relates to
    how hard it is to specify infinite algorithms which converge to a
    specific irrational number.

				    Topher

RE: .6 (Martin H.)

    Following in the same line here is a procedure which can be used to
    construct any angle "n-sector".  Call the initial angle A, with sides
    s1 and s2.

    1) Construct an approximation to the n-sector, i.e., a line which
    approximately n-sects angle A.  We will specify what constitutes
    a valid approximation later.  Call this approximate n-section a.
    Construct it off of side s1, and call the resulting line, sa.

    2) Duplicate "a" n times (including sa) from s1 in the direction of s2,
    to produce side "sn".  If sn=s2, quit.

    3) Call the angle between s2 and sn, angle B.  Use the same approximate
    angle n-sector as in 1, to n-sect angle B.  If B<A "add" the result
    to "a", otherwise "subtract" it.  Call this the new "a".  Repeat step
    2.

    Any angle division such that n*a<2*A, will work as an approximate
    n-sector with this method.  The "better" the approximation the faster
    the convergence, but any angle construction which meets this criterion
    will converge to the n-sector.  Repeated bisectors can always be used,
    for example, by using enough of them.

    For trisection and using a single bisection as an approximation, the
    procedure goes through the same series of approximate trisectors as
    Martin's procedure, though more clumsily.  For trisection and double
    bisection (4-section or ?quadsection?) as an approximation it goes
    though every other of Martin's approximate trisections (i.e.,
    1/4 + 1/16 + 1/64 + ... = (1/2 - 1/4) + (1/8 - 1/16) + (1/32 -
    1/64)...)

				    Topher

    Re: .8 & .9 (Topher Cooper)
    
    Thanks, that answers my questions.
    
    Martin H.

	re .8

>> RE: .7 (Dan)
>>
>>     Hold on now.  The trisector *exists*.  It just cannot be "constructed"
>>     using a finite sequence of specific operations which have proven a
>>     fruitful set of operations for certain purposes.  Other sets of
>>     operations (such as those using a "tomahawk trisector") allow the
>>     angle trisector to be constructed.  It exists in any case as a well
>>     defined locus of points, i.e., those whose perpendicular distances from
>>     one side of the angle is � their perpendicular distances from the other
>>     side.

	If you think of your geometry as being embedded in an
	algebraically complete field, then a/the trisector does
	exist.  But if you work in the extension of the rationals
	by all algebraic numbers of degree a power of two, then
	only constructible points exist.

	Dan

RE: .11 (Dan)

    In other words, if you assume that it doesn't exist unless you can
    construct it, it doesn't exist if you can't construct it.

    But this is *not* the standard treatment of geometry.  The *points*
    exist *before* construction.  The construction finds them. Sets of
    points can be classified into the categories of "constructable" or not.

    Do you really claim that there is no such thing as a regular 7-gon in
    Euclidean geometry?  Or an ellipse (we can find arbitrary individual
    points on an ellipse using c&se, but it would take an infinite number
    of steps -- an *uncountable* infinite number of steps, what is more --
    to construct the ellipse)?  Am I disallowed from saying as part of a
    hypothesis, "assume that I have an arbitrary angle and another angle
    which trisects it..." (presumably I could clean this up by describing
    the smaller angle as arbitrary and the larger one as derived, but it
    would seem from your viewpoint that the first statement is "outside the
    system" and until I convert it to the allowed form. There is a really
    beautiful theorm dealing with the intersection of the angle trisectors
    of the vertexes of an arbitrary triangle which would loose all its
    elegance if restated this way)?  And I don't see any way to allow
    "assume that I have a 7-gon...".

    I'm sure that you could develop a meaningful geometry where lines and
    curves exist only if you can construct them, but I do not believe that
    that is the standard approach.  But you have to avoid all curves except
    straight lines, circles, some circular arcs, and some finite
    collections of the above.  And think of the circumlocutions necessary
    to state theorems about the non-existence (non-constructability) of
    things since you can only describe them directly by describing their
    construction.

					Topher		    

	re .12

>>    Do you really claim that there is no such thing as a regular 7-gon in
>>    Euclidean geometry?

	No.  I didn't say anything like that.  I just carefully
	stated .7 so that I didn't need to assume completeness.
	I don't see that as much different than wording a proof
	so as to avoid gratuitous use of the axiom of choice.
	(Though perhaps your complaint was more as if I was
	avoiding gratuitous use of commutivity of addition.) :-)
	When you said it "*exists*" then I merely explicitly
	named the assumption being implicitly used.

	Dan

.12>    ...						There is a really
.12>    beautiful theorm dealing with the intersection of the angle trisectors
.12>    of the vertexes of an arbitrary triangle which would loose all its
.12>    elegance if restated this way)?
    
    Not to digress too much, but in fact you _couldn't_ state the theorem
    at all.  You have to start with angle theta and construct angle 3�theta
    for one vertex.  Do the same for another vertex, but then you end up 
    with a third vertex forced upon you and you'd have to materialize the
    third set of trisectors from a given angle -- which is what you were
    trying to avoid in the first place.
    
      John

	re .-1,

	Actually, you can trisect the third angle of a Euclidean
	triangle given trisections of the other two angles.  Just
	subtract one-third of each other angle from pi/3 (60 degrees).

	Dan