ManyFolds in Variety

Supplements:

Statement of proposition at the counter-twisting direction:

How to draw Brocard point(s) and Brocard angle(s):

How to draw those point(s) and angle(s) above mentioned
in the counter-twisting direction:

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The following series of manuscripts in Japanese [from #1 to #10] were written in January 1993 to shape the author's study about some foldable structures during 1992 into an article. That was mostly motivated by Dr. T. Kawasaki's achievements in the mathematical theory of origami/paper folding.

Although the flat foldable condition around a vertex with radial rays [as creases] in a 2-dimensional Euclidean plane had been given/defined by Kawasaki [and partially by some people], it seemed that there were few approaches on the foldable condition with respect to several - more than two - vertices or faces [(convex) polygons].

The proposition stated below was at first given by the author himself; however, he supposed that it might be described in some books of trigonometry since it was not so difficult. It turned out that a geometrically equivalent proposition had been described as an exercise in a Japanese dictionary of trigonometry for high school students -- he found it at the public library of his prefecture. So it should say that this is rediscovering of the proposition to be the foundation of composing (2-dimensional) foldable diagrams/patterns.

Note that the dictionary above mentioned did not refer to the name and works of H. Brocard, that is the reason why the following manuscripts do not refer to the notions or theorems named after this French army officer in the 19th century [the author knew his name and results from another series of encyclopedia of (Euclidean) geometry in Japanese later].

Statement of basic proposition:
Proof of the proposition:
Twisted folding structure around an arbitrary triangle:
[Later, the author has found that the statement above can be generalized into the case of twisted folding structure around a convex polygon/polygonal domain.]

Brocard angle of a triangle gives the critical folding conditions:
[When all three edges of the given triangle as creases are set at the same folding 'direction' - i.e., all the edges are folded into mountain/valley creases -, then the [obtainable] value of twisting angle must be upper bounded by the value of Brocard angle of this triangle.]

References:
[The manuscripts that appear in the articles from #1 to #10 were displayed or provided to some paper folding 'artists' before 1995. (Some models that seemed to be motivated/derived from those have been published without his recognizing...)]