Mathematics and OrigamiThis means that to fold flat an equilateral-triangle prismoid it is a must that the angles
formed by a base ́s side and the valley and mountain folds adjacent to it will add up to 120º.
There follows that values lg / lp must be between the limits 30 / 90 and 0 / 120. Under
those conditions the intersection of sides g / p will determine the prismoid ́s altitude that guar-
anties the folding flat of the solid.
The foresaid permits to draw the folding diagram 9 that in turn may produce the solid of
Fig. 6. In this we can measure the prismoid ́s altitude and the initial torsion before folding flat.
In Fig. 7 we can measure the twist to flatten.
We should note that the various combinations of angles lg / lp convey to different forms
for flat Figs. 7 and 8: there is not a unique solution.
Hence, the process will be as follows:- To fix the base ́s triangle with one unit as side.
- To choose the pair of angles, e.g.: lg = 25º; lp = 95º
- To get the development to Fig. 9
- To draw Figs. 7 and 8 to measure the torsion angle to folding flat (in this case it
happens to be 50º). - Fold to 9 to get the solid of Fig. 6.
- Refer solid 6 to a co-ordinate trihedron.
- Measure in it the lower base ́s co-ordinates and those of one of its upper base ́s ver-
tices. - With those co-ordinates, draw Fig. 6. In it we can measure the initial altitude of the
prismoid (it comes out to be 0,48) and the initial twist angle with respect to the right
prism (equal to 7,8º).
18.5.2 QUADRANGULAR PRISMOID
Let ́s start with a quadrangular right parallelepiped (Fig. 1) having A4 rectangles as lat-
eral faces. Fig.3 is its folding diagram. Fixing tight its lower base, we subject the upper one to a
clockwise twist of 90º. The result is the quadrangular prismoid of Fig. 2 with all the vertices in
a cube.
The valley sides of the prismoid meet in the center of the cube. If AB = 1 is the side of
the base ́s square, we ́ll have:
OA = 1 ; AR = 2 ; RB = 3BAC
B ́A ́
7 C ́A ́B
B ́8
ACC ́9