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2005/12/16
「Fish base in modification with isogonal conjugates」 study in math
There exist some capacities for uploading. gifs, so that we will introduce some generalized mathematical relation in paper folding...
Isogonal conjugate points P and Q of a given triangle ABC and the angular relations are illustrated as follows:
[Details about the mathematical notion: isogonal conjugate will be found at some articles in the site: MathWorld , for instance.]
For any given triangle, there exists a kite shaped quadrilateral that consists of two copies of the triangle to be joined in a symmetric arrangement.
[Notice that such quadrilateral may not be a convex shape, if the given triangle has an obtuse angle; if the given triangle has a right angle, then a convex quadrilateral may degenerate into triangle, since it has twice of right angle as its angle at a vertex.]
The kite is to be folded into a modified fish base, and the relations of isogonal conjugacy will be found in such folding operation:
In the above image, segments as creases in red are mountain-creases, and those in black are valley-creases (and those will be in vice versa, i.e., creases in red will be valley-creases, and those in black will be mountain-creases, if the other orientation is given to that compact surface).
The angles illustrated as above can be varied with keeping those isogonal conjugate relations in each admissible range; namely, a fish base diagram on such kite can be continuously modified.
This fact gives some generalization of the results about configuration/modification of a bird base diagram on a square, given by Prof. K. Husimi (Fusimi) and Mr. J. Maekawa: the bird base is considered as a diagram which consists of two fish base those are joined at two segments of each [the shapes of fish bases can be different in spite of the segments to be joined], so it can be continuously varied on both (partial) fish bases with keeping those respective isogonal conjugate relations. Frog base, or other similar diagrams that consist of fish bases, also admit such modification.
[The original illustrations were made at November 1, 2002]
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Isogonal conjugate points P and Q of a given triangle ABC and the angular relations are illustrated as follows:
[Details about the mathematical notion: isogonal conjugate will be found at some articles in the site: MathWorld , for instance.]
For any given triangle, there exists a kite shaped quadrilateral that consists of two copies of the triangle to be joined in a symmetric arrangement.
[Notice that such quadrilateral may not be a convex shape, if the given triangle has an obtuse angle; if the given triangle has a right angle, then a convex quadrilateral may degenerate into triangle, since it has twice of right angle as its angle at a vertex.]
The kite is to be folded into a modified fish base, and the relations of isogonal conjugacy will be found in such folding operation:
In the above image, segments as creases in red are mountain-creases, and those in black are valley-creases (and those will be in vice versa, i.e., creases in red will be valley-creases, and those in black will be mountain-creases, if the other orientation is given to that compact surface).
The angles illustrated as above can be varied with keeping those isogonal conjugate relations in each admissible range; namely, a fish base diagram on such kite can be continuously modified.
This fact gives some generalization of the results about configuration/modification of a bird base diagram on a square, given by Prof. K. Husimi (Fusimi) and Mr. J. Maekawa: the bird base is considered as a diagram which consists of two fish base those are joined at two segments of each [the shapes of fish bases can be different in spite of the segments to be joined], so it can be continuously varied on both (partial) fish bases with keeping those respective isogonal conjugate relations. Frog base, or other similar diagrams that consist of fish bases, also admit such modification.
[The original illustrations were made at November 1, 2002]
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投稿者: kaishonashi
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