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 1997