MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

18.2.5 RHOMBIC PYRAMID
The one to be studied now is an irregular pyramid whose base is a rhomb having its di-
agonals in the ratio 2 : 1. We obtain it from a DIN A rectangle with its small side equal to1
(Fig. 1).
Folding accordingly we get the mesh-like pyramid of Fig. 2. Its base is the rhomb
ABCD whose diagonals are:

AC = 1 ;

2

2

2

= =

HH

BD therefore = 2

BD

AC

Fig. 3 shows the right angle E: 90

2

1

arctg

2

2

1

. 180 arctg =
























AngE= − +

On the other hand we know (Point 9.8) that HE =

3

1

HF, hence

0 , 5773502

3

3

1 2

3

1

HE= + = = (Figs. 3 and 4)

Besides, folding to Fig. 1, points J and K will lie on O (Fig. 4).

Consequently

2

1

h=HJ=HK= are the altitude of the pyramid; (HD = HB) < (HA = HC)

In Fig. 4, h = HO. Likewise, Ang. HEO (formed by a triangular lateral face and the

rhombic base) measures:

arcsen 0 , 8660254 60 º

2 3

1 3

arcsen arcsen = =

×

= =

HE

HO

AngHEO

1

H A

J

O

B D

C

K

H

O

C

3

H A H

E

F