Jesús de la Peña Hernández
10 REGULAR CONVEX POLYGONS WITH MORE THAN 4 SIDES
Our attention will focus on folded constructions leaving aside the classic that use rule
and compass.
It is impossible to present the whole variety of origami solutions because of lack of
space: as far as pentagon only is concerned, well overt 10 different solutions may be registered.
What we shall do is to discuss various solutions because not all of them are perfect un-
der the point of view of geometry. Origami can deal with most of them though, even with the
added handicap of folding difficulties and the accumulation of overlapped paper.
Nevertheless we are convinced that the capacity to digest errors inherent to origami
should not impair to distinguish exact from approximate solutions; besides, some of these are
more imperfect than others.10.1 PENTAGON
10.1.1 FROM ONE ARGENTIC RECTANGLE
Before continuing I shall indulge myself in a semantic digression. In British origami lit-
erature, the DIN A rectangle is called silver rectangle: of course I have nothing to object (see
D. Brill ́s BRILLIANT ORIGAMI).
Since DIN A rectangles are well enough defined as such, I do not think they need of any
added qualification despite of their 2 singularity.
On the contrary, I do fill it necessary to name argentic rectangle that special one whose
diagonal and small side form an angle half of that of a regular convex pentagon. I like to put
together, semantically speaking, those singular rectangles: argentic and auric.
To keep alive two synonyms like silver (taken as adjective) and argentic may lead us to
confusion though we are aware that synonyms may have important differentiating nuances, as
the present case manifests.
Former discussion would be useless not to be for the fact that in the above-mentioned
publication, both silver and what I call argentic rectangles are mixed up. In it, pentagonal and
pentagonododecahedric properties are associated to the first one, what is incorrect, but do cor-
respond exclusively to the second. This is so, though the inexactitude is admited by the author
of said publication. Let ́s go, anyway, to look for the solution we are interested in.
It is a perfect geometric solution produced from an argentic rectangle whose sides are
proportional to 5 + 2 5 (small side) and 2 + 5 (large side). See its construction in Point
6.6.7.
To give an idea of those proportions, we may say that they are equivalent to 210 (width
of DIN A4) × 289,04021.REMARKS TO FOLDING PROCESS.
1- BF = 1, therefore:
FD = 1.3763819 ; BD = 1.7013016 ; Ang. DBF = g = 54º ; Ang. BDF = p = 36º
2- Ang. DCB = 2g = 2 × 54 = 108º (angle of convex regular pentagon).
3- Isosceles ∆ DCB is produced: Ang. CDB = Ang. CBD = Ang. BDF = p
4- Fold D to lie on CB being HL parallel to BD. Folding operation has to be made by the rule
of thumb (it ́s easy to get a good result). Its analytical development is as follows: