Mathematics and OrigamiThe kites are symmetric with reference to EC in that Fig. 8, and the darts do as well with reference to
GL in this Fig. 1.
Their peculiar configuration is a consequence of the inherent singularity of the argentic rectan-
gle as can be seen comparing that Fig. 8 and this Fig. 1:
- The small side of both tesserae (this Fig. 1) is the side of the pentagon in that Fig. 8. Likewise,
tesserae ́s big side is the diagonal of said pentagon. - As studied in Point 10.1.1 (AN INTERESTING VERIFICATION), the side of a regular convex
pentagon divides its diagonal in mean and extreme ratio, i.e., big and small sides of these teserae
keep an auric proportion - In point 10.1.1 we also saw the value of the angles in the argentic rectangle: diagonal and sides 54º
and 36º respectively. In consequence, the three acute angles of the kite measure 72º = 180 - 2×54 =
36 ×2 and the obtuse angle measures 144º = 360 - 3×72 (like the angle of the convex regular deca-
gon).
The obtuse angle of the dart measures 216º, its opposite 72º and the other two, 36º.
To get the folded tesserae out of an argentic rectangle it is advisable to aim at the largest possi-
ble ones to avoid, as much as we can, paper gathering and hiding.
This Fig. 2, derived from that Fig. 8, has the folds for the kite, and this Fig. 4, derived from that
Fig. 3 shows the fold lines for the dart (mountain, continuos; valley, dashed).
Figs. 3 and 5 are, to one half scale, the obverse and reverse of both tesserae.
Fig. 6 shows the seven possible combinations forming a perigon under the condition that the
adjacent sides of two distinct tesserae have to be always of equal length.
In Fig. 7 there are two tessellations: one of them is rather simple whereas the other renders it-
self complex as it grows up. In both, some virtual tesserae can be seen. One thing is clear about
these tessellations: they give the artists the chance to develop their own ingeniousness and to the
mathematicians that of disclosing the laws of formation.HGMK DLB F1
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