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.)