MATHEMATICS AND ORIGAMI

Mathematics and Origami

A variant of Fig. 4 is Fig. 8 that in turn produces Fig. 9.

It is not docile but it is collapsible to a virtual right quadrangular parallelepiped (instead

of the cube of Fig. 2). The result is a prismoid with altitude h= a^2 − 1 = 0 , 66 (a = 1,2). It has

its four valley-fold diagonals in contact.

After all seen till now, one could think that the condition to fold-flat a quadrangular

prismoid is that the angle lg = 45º. That is not true, though.

Let ́s look over Fig. 10 that exhibits these conditions:

l = 1 ; lg = 40º ; lp = 95º

hence:

sen 135

1

sen 40

=

p

; p = 0,909

a = p sen 95 = 0,9056

The passage from Fig. 10 to 11 is not docile but accepts collapsing to fold-flat. If we

analyse the paper arrangement around vertex A (Fig. 11), we have:

lg – gp + pl – ll = 0

Being ll = 90 and lg + gp +pl = 180 (the sum of the angles of triangle lpg), we end up

with:

lg + lp = 135

What means that the condition to fold-flat is this: the sum of the angles adjacent to l

must be 135º. Fig. 10 accomplishes that condition. Besides angle lp has to be greater than 90º.

Fig. 10 is lacking the bases to allow observing the small interior square appearing in

Fig. 11. This square may vary in size according to the chosen combination of angles.

18.5.3 PENTAGONAL PRISMOID

Up to here, the prismoids we have considered were lacking, in general, their bases and

the necessary elements (flaps, interlocks, etc.) to conform them tight. In the present occasion,

the folding diagram of Fig. 1 is an example of how to close the lateral surface of a prismoid by

pocketing (though bases are not shown either).

1

8

1,2

90º 50,2º

10 11

a

95º 40º

p

g

l A