MATHEMATICS AND ORIGAMI

Mathematics and Origami

cial graduated squares able to configure any type of angles. That craft has even assigned specific
names to some of these angles. Through a rather awkward application of those tools, the craftsmen can
cope with any wood construction such as slants, dovetails, etc.
At present, instead of appealing to the old techniques, I have preferred to use those taken as
modern nowadays. CAD, nevertheless is not a panacea. What I mean is that, for example, to figure out
the angles in the trapeziums of Fig. 2, we must know beforehand, which is the intersection line of two
planes in space. To solve problems like this, I have been forced to develop calculus programs such as
those that give the angle formed by two planes, the point of intersection of line and plane, the distance
to a plane from a point, etc.
We have to bear in mind that CAD works with points alike the 3D measuring machines; there-
fore a plane is defined by three of its points.

Fig. 3 is a dimensioned half-section of the small cut in Fig. 1. From that we can figure out the
length of the segments forming the broken line EDCBA.

ABsenα+BCsenβ=c −EDcosβ+DCcosγ=d−e

−ABcosα−BCcosβ=d−b EDsenβ+DCsenγ=c−a

()

()β α

α α

−

+ −

=

sen

ccos d bsen

BC

() ()

()β γ

β β

+

− + −

=

sen

d esen c a cos

DC

α

β

sen

c BCsen

AB

−

=

β

γ

sen

c a DCsen

ED

− −

=

3

c

a

e

b

d

a

C

D

E

A

B

a E

c

d

H

4

a

C

I

F G