96m:51024 51M15
   Geretschläger, Robert
   Euclidean constructions and the geometry of origami. (English)
   Math. Mag. 68 (1995), no. 5, 357--371.
   
   In this entertaining paper the author shows that Euclidean ruler and
   compass constructions can be equally well accomplished by folding
   (admittedly infinite) sheets of paper. Conversely, almost every
   folding (or origami) construction can be also constructed by ruler and
   compass, with the exception of finding the common tangent of two
   parabolas given by their foci and directrices. This last corresponds
   to solving a cubic equation, and the author shows how the usual
   unsolvable (in the Euclidean framework) problems of doubling the cube
   and trisecting the angle can be accomplished by judicious
   paperfolding.
   
   The paper is accessible to high-school students, and can be an
   excellent source of challenging geometry problems.
             Reviewed by Igor Rivin
                 

   © Copyright American Mathematical Society 1996, 1997
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   97c:51011 51M15 52A37
   Hull, Thomas(1-RI)
   A note on "impossible" paper folding. (English)
   Amer. Math. Monthly 103 (1996), no. 3, 240--241.

   In this paper the author presents a paper-folding method for
   trisecting angles that was invented by H. Abe. He also proposes the
   addition of a sixth axiom to the five origami axioms from the paper by
   D. R. Auckly and J. Cleveland [Amer. Math. Monthly 102 (1995), no. 3,
   215--226; MR 95m:12001]. The author proposes determining the limits of
   origami construction as an open problem. There is quite a bit of
   undocumented mathematics related to geometric constructions. Given a
   set of construction axioms, it is usually an exercise in elementary
   abstract algebra to determine the set of constructible points. For
   example, with the five axioms from the above-cited paper plus the
   sixth axiom, proposed by Hull, a point is constructible if and only if
   the degree of each coordinate of the point is $2\sp n3\sp m$. The
   paper ends with a list of eleven papers on mathematics and origami.
             Reviewed by Dave Auckly

   © Copyright American Mathematical Society 199