MATHEMATICS AND ORIGAMI

Mathematics and Origami

18.2 PYRAMIDS

We are going to study various regular pyramids and one irregular, using when possible,

for their construction, the Solution 1 of Point 8.2.3 to generate equilateral triangles.

18.2.1 TRIANGULAR PYRAMID

18.2.1.1 TETRAHEDRIC
Fig. 1 is the folding scheme and Fig. 2 is the obtained solid. It is evident that the result-
ing pyramid is a tetrahedron since its three lateral faces are equal and also equal to the base.

To draw Fig. 2 by means of CAD (Point 18.1 showed to which extent CAD is an origami tool
and not a mere ornament) we must know the value of the α angle in the tetrahedron (Fig. 3), for CAD
usually plays with plane revolving.
The same requirement will be put forward with other polyhedra and it will have to be satisfied
in each occasion.
In the tetrahedron of Fig. 3 we have:

• side = l


• altitude of one of its faces
  2

l 3

h=

• distance AB between two opposite sides:

 =











 −













=

(^22)
2 2
l 3 l
AB 0 , 7071067
2
2
l =l

• altitude of tetrahedron: distance from the pyramid ́s vertex to the center of its base = H


• dihedral angle of two faces: angle formed by two segments h meeting on the same side = α.


• angle formed by two segments l and h meeting on the same vertex = β.

3

2

3

2

2

H l^2 h =l











= −

arctg 2 2 70 , 528779 º

3

=arctg = =

h

H

α

arctg 2 54 , 73561 º

3

2

=arctg = =

h

H

β

1