ManyFolds in Variety
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Welcome to the exhibition of Origami/paper folding and related works by H. AZUMA.
(c) Copyright AZUMA Hideaki. All rights reserved.
(Last updated: February 17, 2008)
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2006/7/21
[Some further variations of the fish base: these contain the cases appeared in the preceding article: Generalized fish bases defined on excircles of a triangle as special ones.]
For an arbitrary triangle, there exist (many) 'ex-ellipses': each of which contacts an edge of the triangle and two others at these extended parts. The following example shows isogonal conjugate relations with respect to a triangle and one of those ex-ellipses [(Apollonius)]:
Applying this fact to a given triangle at each of its vertices, then the following diagram/graphic image will be realized:
3
For an arbitrary triangle, there exist (many) 'ex-ellipses': each of which contacts an edge of the triangle and two others at these extended parts. The following example shows isogonal conjugate relations with respect to a triangle and one of those ex-ellipses [(Apollonius)]:
Applying this fact to a given triangle at each of its vertices, then the following diagram/graphic image will be realized:
3
2006/7/17
[Some 'eccentric' corollary to the results on incircular conditions of a triangle appeared in the previous article: Inellipses; isogonal conjugates; fish bases...]
For an arbitrary triangle, there exist three excircles: each of those contacts an edge of the triangle and two others at these extended parts. Moreover, it is possible to draw rays those start from the center of an excircle and to be composed into a flatly foldable structure at that point. The following example shows such angular relations at a (local) flatly foldable structure:
Thus, three flatly foldable structures at respective centers of excircles around a given triangle will be illustrated by:
[Let the above diagram print on a sheet of paper and cut its triangular region (colored in gray) away, then the paper with a (triangular) hole will be flatly folded with its creases of printed segments - as we have done it -]
A partial colored version of the above picture to show that half of three generalized fish bases are included in the diagram:
The three flatly foldable structures of generalized fish bases will be composed into the following respective diagrams and graphics:
1
For an arbitrary triangle, there exist three excircles: each of those contacts an edge of the triangle and two others at these extended parts. Moreover, it is possible to draw rays those start from the center of an excircle and to be composed into a flatly foldable structure at that point. The following example shows such angular relations at a (local) flatly foldable structure:
Thus, three flatly foldable structures at respective centers of excircles around a given triangle will be illustrated by:
[Let the above diagram print on a sheet of paper and cut its triangular region (colored in gray) away, then the paper with a (triangular) hole will be flatly folded with its creases of printed segments - as we have done it -]
A partial colored version of the above picture to show that half of three generalized fish bases are included in the diagram:
The three flatly foldable structures of generalized fish bases will be composed into the following respective diagrams and graphics:
1
2006/7/11
Another case in graphics for supplementary to the article: With isogonal conjugate points and an inellipse of a triangle. The result is given by the almost 'parallel' argument about the structures in the preceding article: Inellipses; isogonal conjugates; fish bases.
Part of a parallel strip and inner segments those are determined by the foci of a fixed inellipse and parallel rays for a flatly foldable structure, such as:
That will be composed into the following diagrams and a graphic image:
0
Part of a parallel strip and inner segments those are determined by the foci of a fixed inellipse and parallel rays for a flatly foldable structure, such as:
That will be composed into the following diagrams and a graphic image:
0
2006/7/8
Supplementary graphics to the article: With isogonal conjugate points and an inellipse of a triangle.
A triangle with segments that are determined by the foci of a fixed inellipse:
That will be composed into three pairs of modified fish base diagrams and graphic images as follows:
0
A triangle with segments that are determined by the foci of a fixed inellipse:
That will be composed into three pairs of modified fish base diagrams and graphic images as follows:
0
2006/7/1
>> Click the image to see in larger scales <<
Corresponding diagrams to the image above are as follows:
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