History   Help on query formulation   Algebra and geometry by means of paper folding. (Algebra og geometri ved hjelp av papirbretting.) (Norwegian)
Normat 46, No. 4, 170-185 (1998).
If two points \$A\$, \$A’\$ in the plane are symmetrical with respect to the line \$c\$ we say that, by folding the paper along the crease \$c\$, the point \$A\$ is sent to the point \$A’\$. “Admissible” paper-folding procedures are listed as game rules: starting with an initial set of points one must reach new points by applying a number of admissible steps. The most far-reaching procedure in the list is the following: given two points \$A\$, \$B\$, and two lines \$a\$, \$b\$, find a crease which simultaneously sends \$A\$ to a point \$A’\$ on \$a\$ and \$B\$ to a point \$B’\$ on \$b\$. The reader is invited to experiment with thin semi-transparent paper in order to see that up to three such creases exist. The analytical approach leads to the surprising conclusion that this game includes and surpasses ruler-and-compass constructions, by solving all geometrical problems of degree \$\le\$ 3, e.g. the duplication of a cube, the trisection of an angle and the construction of regular \$n\$-gons, \$n=2^h3^kq_1q_2\dots q_m\$ for distinct primes, of the form \$q_i=1+2^{u+1}3^v\$. (orig.)
Classification: G40