MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Fig. 2 shows, as said, how to pleat fold, at the discretion of the folder (but with a great

accuracy), the large sides of the original rectangle. Fig. 6 shows how to get the pleat angle γ:

19 , 471221 º

2

1

γ= 90 − 2 Arctg =

18.2.2.2 EQUILATERAL-TRIANGLE QUADRANGULAR PYRAMID

Fig. 1 is the folding diagram and Fig. 2 is the pyramid we get. In Fig. 3 we can calculate

the altitude h and dihedral angle α, the side of the equilateral triangle being the unity.

0 , 7071067

2

2

4

1

2

3

2

 − = =













h= ;

3

1

arccos

2

2 3

1

α=arccos = ; α = 54.735613º

(equal to the α angle in Point 18.2.1.2)

18.2.3 PENTAGONAL PYRAMID

This pyramid is also lacking its base. Its lateral faces are equilateral triangles of side l (same

as the side of the base pentagon). The apothem of said pentagon is calculated in Point 18.6.1 though

Fig. 3 makes evident its value.

Fig. 1 is the folding diagram which is worked out in two steps: in the first place we form a

hexagon; then, the upper trapezium that appears is rotated 60º around the center of the hexagon, as

shown. While performing this operation the figure is filled out to attain its pyramidal volume (Fig. 2).

2

2

108

ltg

a= ; 37 , 377368 º

2

3

:

2

tg 54

arccos =













=

l l

α

1

1