MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

We have seen up to now that CAD allows us to draw Fig. 13 as well as Fig. 11. Then

we ́ll see the analytical procedure in case CAD would not be available.

Here are the data:

• The side of the square = a.


• The distance between squares = h.


• The angle turned around each time = ω.


• Amount of turns and consequent upward moves given to the square = n. Obviously n
  may be as great as we wish if the paper size can cope with it.

Let ́s look again to Fig. 10 in order to get the co-ordinates of points A, B, C, D

A ≡ (0,0,0) ; B ≡ (a,0,0)

Ang. FAD =

2

45

2

180

180 45

π π

= +









 −

− −

2

2 sen

2

2 sen

2

2

π π

a

a

AD= =













= = +

2

cos 45

2

cos 2 sen

π π

FA AD FAD a













= +

2

sen 45

2

2 sen

π π

DF a

∆DFA = ∆BEC after transformation by the rotation π.

BE = FA ; CE = DF













= − +

2

sen 45

2

2 sen

π π

xC a a

























= − +

2

sen 45

2

1 2 sen

π π

xC a













=− =− +

2

cos 45

2

2 sen

π π

yC BE a

zC=h

xD=yC













= +

2

sen 45

2

2 sen

π π

yD a

zD=h

Its three sides define ∆ABD. Besides, calling p to its half-perimeter, its area will be

S= p()( )( )p−a p−BD p−AD and therefore:

a

S

h

2

́=

Once the solid form of Fig. 11 is physically completed, we can check that the shrinkage

produced by the twist is equal to n()h ́.−h

We should recall that h ́ is the distance between horizontals in Fig. 13, and h is the same

distance taken in the space (sides DC and AB of Fig. 12, which cross each other): h ́ > h.