Jesús de la Peña Hernández
18.5.1 TRIANGULAR PRISMOID
Fig. 1 is a right prism whose bases are equilateral triangles. Fig. 2 is its folding scheme.
All five figures are not to the same scale.
Twisting 40º clockwise the upper base of Fig.1 with respect to its lower base, the alti-
tude of the prism is reduced to 70 %. By so doing (CAD) we get Fig. 3.To draw the folding diagram of Fig.4, the only thing needed is to measure by means of
CAD the sides of the eight triangles of Fig. 3.
We must pay attention to the fact that the altitude of prismoid 3 is not available in Fig.
4, but in Fig. 3: it is the distance between the centers of its bases.
In Fig. 5 we can see, overlapped, both Figs.1 and 3 in order to clarify the process.
We may observe that the process above can be fulfilled regardless of the bases of the
prism. Actually it is better to have them to afford a more consistent solid.Let ́s consider now the problem of twist to flattening both bases on the same plane.
It is not possible to twist flat the prismoid of Fig. 3 because of the interference of its
three valley-fold diagonals. It is required that the prismoid ́s altitude fulfils certain conditions,
as we shall see here after.
Let ́s observe what happens in Fig. 7 that is a flattened prismoid without bases. Fig. 8 is
a meshlike version of Fig. 7 showing the twist angle to make clear the process. The vertices of
the bases are, respectively, ABC, A ́B ́C ́.
In a folding diagram like that in Fig. 4 we see the six lateral triangles whose sides will
be named in this manner: l (side of the base ́s triangle); p (the small side); g (the great side). We
can see that one of the angles adjacent to the base l is acute and the other is obtuse. The same
configuration have the triangles in Fig. 8 (for example BCC ́).
Let ́s see now how the paper is arranged around vertex C. Each angle will be named af-
ter the letters of the sides including the angle. Therefore:
lp – pg + gl = 60º
lp + pg + gl = 180º (the three angles of a triangle, together)Adding up we have: lp + lg = 120º24