Mathematics and OrigamiIn Fig. 18 one can see how the polyhedron angle O (DBAE) can be revolved about its
base DOE from the situation in Fig. 16 till the coincidence of B and E. Note that this can be
done freely because the affected figures are triangles instead of the quadrilaterals of Fig. 14.
Another question to add is that Figs. 12 and 15 which, of course, lead to 3D figures, also
yield subfigures 2D about the hinges with 180º value dihedral angles.Let ́s dig out now in the process 2D → 3D → 2D already seen in case 2A, Point 12.
There we played with an octagon; here we ́ll do with scalene triangles. Each triad of figures(e.g. 19,20,21) keeps that order 2D, 3D and 2D (a flat configuration).We can note that the four figures at the beginning (19, 22, 25 and 28) are geometrically
identical: the central triangle is always the same, and it is subjected to a twist of 40º (α = 20º).The only difference is the nature (mountain or valley) of folding lines.Fig. 31 is a portion of Fig. 28 with some addenda. In it we can see that node A fulfils
the flattening condition of alternate angles adding up to 180º. This is so because of the configu-= 20 º19= 20 º21= 20 º2224= 20 º= 20 º25= 20 º28
= 20 º3027= 20 º