VIIIOrigami receives more sophisticated geometrical help to design folding bases. In passing, I
tackle this matter when dealing with the triangle ́s incenter and its related hyperbola. But one who
masters this subject is our ingenious creator Anibal Voyer. Not long ago, he lectured on that in the
conference held at the Spanish Institute of Engineering under the title of Engineering, origami and
creative design. Afterwards, and with the same materials, he wrote a comprehensive article in PA-
JARITA (No 68, October 1999; bulletin of the AEP –Spanish Association for paperfolding-), this
time entitled INTRODUCTION TO CREATION.
Many mathematical demonstrations can be fulfilled by means of origami. Nevertheless, to be
fair, both, maths and origami demonstrations should be performed in order to obviate the risk of
taking for exact a folded figure which is not such. Moreover, the best will be to add some CAD
(Computer Aided Design) evidences.
It should be recognised the ingeniousness that led origami to demonstrations such as the
limits of convergent series, the Poncelet ́s theorem on conics or problem solving of maxima and
minimums. The book will deal with all this and with some other simpler things developed under the
excuse of not having available neither a pencil nor a rule, a square or a set square.
In this respect, I wondered whether splitting out simple and no so simple matters. Eventually
I decide not to do so. I thought the entanglement produced would be greater as compared with the
advantages to be obtained.
I hope the reader will follow out each subject up to his mathematics limitations, ignoring the
points to be jumped: that will not impair him to proceed.
Another question I should like to point out is this: now and again I intend to prove the lack
of mathematical rigour shown in certain paperfoldings which, on the other hand are believed to en-
close perfection.
Anybody can object that my strict attitude does not make any sense because the inherent im-
perfection of folding (not being the less important that induced by the thickness of paper) hide, by
far, the supposed geometrical imperfection.
I am prepared to agree with the objectors. Not only that: I should like to render here my most
sincere homage to those who had the intuition to almost reach perfection in folding geometrical
figures.
But that will not weaken my purpose to discern perfect –there are many of them- and imper-
fect constructions, from the point of view of pure geometry. It should be added that the contrary is
equally true: a faultless design, mathematically speaking, will end up in an imperfect construction
by the reasons already mentioned. What matters is to know the cause of imperfection.
Finally I should like to assert my limitations. I am not an origami creator. To me, origami, as
well as mathematics are a source of recreation, of personal fun.
Much of the content of this book was already widespread throughout countless publications.
My main work has consisted in adding coherence and math demonstrations whenever needed. The
reader will be able to judge how much in it is due to my own creativity or to my profile of recreator.