MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

18.2.1.2 OF TRI-RIGHT-ANGLED VERTEX

From a square of side l (Fig. 1), we get a virtual pyramid (I name so, in this case, the

pyramid lacking its base) with these characteristics (Figs. 2, 3):

• Its base is an equilateral triangle of side l and altitude
  2

l 3

h=

• Its three equal lateral faces are isosceles right triangles. The vertices of their right angles
  coincide with the pyramid ́s vertex; their legs are the pyramid ́s lateral sides and measure

2

l

CALCULATION OF DIHEDRAL ANGLE α

Altitude of pyramid 0 , 4082483

2

3

3

2

2

2 2

 = ×













 −











= l l

l

H

tg 2 54 , 735613 º

3

= tg =Arc =

h

H

α Arc

We should note that this angle α is equal to β in Fig. 3, Point 18.2.1.1.

The folds in lower triangle of Fig. 1 allow pyramid interlocking.

18.2.2 QUADRANGULAR PYRAMID

18.2.2.1 VIRTUAL QUADRANGULAR PYRAMID

It is quite defined by its vertex, the four base ́s vertices, two full lateral faces and

the other two semi-full ones; it is lacking the base.

The starting rectangle, according to Fig. 1 is a DIN A4 with sides 1 (the small)

and 2 (the large). Fig. 1 shows the folds previous to final folding performed to Fig. 2:

pleat its large sides in such a way that the distance between its endpoints will be 1.

Thus we obtain the complete folding diagram of Fig. 3 and hence the pyramid of

Fig. 4. The construction requires that both pleats in the semi-full faces, will be fixed.

The final pyramid has these characteristics:

• The side of the square of its base is 1


• The diagonal of this square is 2

1