MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

2

1

2

= = = =

AB

AC CD DA

Thus we have settled ∆CDI which is homothetic to ∆EFG. Let ́s find now the center of

homothety P.

As FE= 2 and the angle in A of a regular octagon measures 135º, we can write:

0 , 7653668

2

135

sen

2

2

AE= =

The ratio of homothety being expressed by:

CP

EP

CD

FE

= ;

CP

EC+CP

2 2 = ;

2 2 − 1

=

EC

CP

EC= AE^2 −AC^2 = 0 , 5794708 ; hence CP = 0,3169231

With the information gathered till now we can draw Fig. 3 that is a face of the trapezo-

hedron: we know its two diagonals and the intercepts of both of them. The sides of the trape-

zohedron are AE (already known) and AP whose value is:

AP= AC^2 +CP^2 = 0 , 5919799

Fig. 4 is the folding diagram of one of the 8 sections that make up the whole trapezohe-

dron (recall Fig. 2).

We must say that the trapezoid of Fig. 3 does not enjoy the auric proportion seen in the

Penrose tesserae (case 1B, Point 12).

3 4

E

E

A

P

B

C

A

F

G

B

B

P

Interlude