id: 02327603
dt: j
an: 1999c.01797
au: Scimeni, Benedetto
ti: Algebra and geometry by means of paper folding. (Algebra og geometri ved
hjelp av papirbretting.)
so: Normat 46, No. 4, 170-185 (1998).
py: 1998
pu: Nationellt Centrum för Matematikutbildning (NCM), Göteborgs Universitet,
Göteborg
la: NO
cc: G40
ut: ruler and compass constructions; duplication of a cube; trisection of an
angle; construction of regular n-gons
ci:
li:
ab: 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.)
rv: