MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

1.3.2 INTERACTION OF STRAIGHT LINES AND SURFACES.

We shall call paper strip to a rectangle such that its length is much greater than its width.

1. If two paper strips of the same width are superimposed, the common surface is a rhomb (one
   square if the strips lay orthogonally). That common surface is, obviously, a parallelogram: con-
   gruent acute angles in A, B, C (vertical and alternate / interior). ∆ADE = ∆CDF as straight an-
   gled with ang.A = ang.C and ED = EF (same strip width). The parallelogram with two adjacent
   congruent sides is a rhomb or a square.


2. If both strips have different width, the common surface is a rhomboid (a rectangle in case of
   orthogonality): Parallel lines cut by those other parallel produce a parallelogram with not con-
   gruent adjacent sides.


3. A strip of paper folded over itself in any way, produces an isosceles triangle as common surface.
   If fold and paper edge form a 45º angle, the triangle is a straight angled one; it ́s equilateral if
   that angle is 60º. By means of this one can make very useful bevel squares.
   The angles marked in A or B are equal, now because of symmetry now for being alternate inte-
   rior angles; so AB is the base of an isosceles triangle. Moreover, sides OA and OB of that isos-
   celes triangle are equal as the fold p makes evident: O → O ; B → A. Therefore, ∆OAB is
   isosceles.

1.3.3 THE RIGHT ANGLE

Let a strip having A in its upper edge and make the fold AD according to α.

Then produce fold AF to carry AC over AE.

Resulting angle FAD is a right one:

Symmetry in last figure makes equal the pair of α angles and the pair of β, respectively.

Straight angle in A gives:

α+α+β+β= 180 ; α+β= 90

C A B

D

(^12)
A
E B F
C
D
A
B
3
O
p