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Sheila King writes in a message dated 12-06-92:

SK>Next year we plan to incorporate a great deal of use of Miras and also
SK>paper folding (we call it using Patty Papers...)
SK>In any case, it is very easy to perform ALL of the classic constructions
SK>done with compass and straightedge with the Mira and Paper folding.
SK>My question is this: If it is possible to perform all of the
SK>constructions with the Mira and paper folding, is there any reason that
SK>we would need to teach our students more traditional construction
SK>methods? Would anything be lost by not covering compass and straightedge
SK>constructions? (And if so, what?)

I think some familiarity with classical construction tools is important,
not only from a historical perspective, but mathematically because of
two more advanced theorems of geometry and the nice connections between
Euclidean contructions and transformations, especially inversion.  The
two theorems are the Mohr-Mascheroni Construction Theoerem and the
Poncelet-Steiner Construction Theorem.  The first proves that the
straight-edge is redundant, i.e. insofar as the given and required
elements of a construction are points they can be constructed with
Euclidean compass alone.  The second shows that all Euclidean
constructions are possible with a straight-edge and a given circle and
its center.  Other nice results are attributed to Adler which state that
all constructions are possible with a double-edged ruler (parallel or
intersecting at a given angle).  I mention these not to suggest that
they are part of a High School Geometry course but to argue that *some*
familiarity with the classic tools are useful if for no other reason
than that students may encounter their use in an abstract form at a
later date and that construction proofs are always given in terms of
circles and lines (i.e. straight edge & compass).  It would be useful
and beneficial to students to explore the equivalence of constructions
they perform with "patty papers" and the same constructions with compass
and straight-edge.  For example, constructing a perpendicular bisector.
If you don't want to spend time with this, I think two areas that need
further enhancement in *any* geometry course are transformational
approaches and non-Euclidean geometries.  Anyway, enough of this.
Please keep us informed, Sheila, as you struggle with revising and
reforming your geometry curriculum.  I'd certainly like to hear how the
students react to it.
Howard

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