Mathematics and Origamiference.It is pertinent to recall Point 11 about stellate polygons and Point 9.1 on how to divide a
circle. Note that for a greater pitch we get a smaller circumference.
Process is as good for a given circumference cut out of a piece of paper as for other
drawn on it. Of course, first case is easier.13.2 ORIGAMI AND PLÜCKERIAN COORDINATES
One could be disappointed when a curve is defined as the envelope of its tangents: Such
may be the case of the circumference in former Point 13.1.2, of the parabola –Point 1.2.4-, or
some other cases we shall see here after. In those cases the curve does not appear in its points,
as it is usual in Cartesian representation.
Nevertheless we must admit that origami does not play with Cartesian co-ordinates. Its
very name gives us the clue: origami, paper folding, i.e., paper (the plane) on which folds are
produced (folds are but lines –mountain or valley-).
Then we can agree to the natural representation in origami as made out of straight lines
(Plückerian co-ordinates) rather than of points (Cartesian co-ordinates). In origami the point is
normally a by-product, i.e., the intersection of two straight lines (two creases) through which,
most likely, a new straight line will pass.
What happens in practice when we deal with both, maths and origami, is that we bear a
hand to certain instruments. We may recall, e.g. Point 7.11 in which we had to set the co-ord-
inates of two points in the plane.
Once we know that Cartesian (points x,y) and Plückerian (straight lines u,v) co-
ordinates coexist, it will be good to dig out the mutual correspondence that will enable us to do
some verifications.
Fig. 1 shows the straight line t on which all its points (x,y) lie. It has a as x intercept and
b as y intercept, being its equation:yb
a xa
=
−
that can be transformed to1 01 1
− − y+ =
bx
a(1)Fig. 2 is the same Fig. 1 to which has been added the normal OP with point N (u,v) such
that:n = 16 ; p = 3 n = 16 ; p = 5