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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)

The following two-toned diagram of tiling consists of locally flat foldable patterns, but it is not to be folded like 'Hira-Ori':

>> Click the image to see the larger picture <<

Notice that all 2-cells in the structure above [considered as an Eulerian finite 2-dimensional connected graph on a flat torus T^{2}] are triangles of similar shapes (these are isosceles right triangles, precisely) in two sizes and every vertex is valued at six of its (graph theoretical) degree.

[Refer to the previous article:*"Some mathematical observation on geometric structures in origami"* about mathematical terms.]

>> Click the image to see the larger picture <<

Notice that all 2-cells in the structure above [considered as an Eulerian finite 2-dimensional connected graph on a flat torus T

[Refer to the previous article:

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An applied result of the previous article: "Twisted folding around a triangle #3", that is also homeomorphic to the not foldable tiling.

It consists of the patterns that are based on the twisted folding around a regular triangle [at its Brocard angle, that is evaluated at π/6(rad) (30°) in this case]. If every edge of the triangles colored in light bamboo-green is set into a mountain-crease, while every edge of the other triangles - colored in light yellow-green - is set into a valley-crease, then the rest of edges in the diagram will be uniquely defined into mountain/valley- creases because of the [flatly] folding conditions.

Then the diagram will be folded into:

It consists of the patterns that are based on the twisted folding around a regular triangle [at its Brocard angle, that is evaluated at π/6(rad) (30°) in this case]. If every edge of the triangles colored in light bamboo-green is set into a mountain-crease, while every edge of the other triangles - colored in light yellow-green - is set into a valley-crease, then the rest of edges in the diagram will be uniquely defined into mountain/valley- creases because of the [flatly] folding conditions.

Then the diagram will be folded into:

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Here is a flatly foldable tiling that is homeomorphic [topologically equivalent] to the preceding *not* foldable tiling:

>>Click the image to see the larger picture<<

[This two-toned tiling is based on a diagram that appears in the previously mentioned book (in Japanese):*"Souzou suru Origami Asobi heno Shoutai (Invitation to creative playing in Origami)"* by S. Fujimoto and M. Nishiwaki.]

Supplements:

>>Click the image to see the larger picture<<

[This two-toned tiling is based on a diagram that appears in the previously mentioned book (in Japanese):

Supplements:

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A not foldable (crystallographic) tiling over the plane:

>>Click the image to see the larger picture<<

The significance of the example above is:

the diagram consists of foldable structures around each vertex [namely, it is generated by 'locally' foldable structures], but it is not composed into foldable tiling as a whole.

In fact, it is satisfied [necessary and sufficient] flatly foldable conditions at each vertex represented in the angular relations [given by T. Kawasaki in his papers up to general cases of geometry] as follows:

π/3 + π/2 + 2π/3 + π/2 = 2π (rad) [60°+ 90°+ 120°+ 90°= 360°];

π/3 - π/2 + 2π/3 - π/2 = 0 (rad) __[60°- 90°+ 120°- 90°= 0°].

On the other hand, it is not flatly foldable because of the combinatorial folding conditions with respect to the relations between mountain-/valley- creases and values of angles. [Try to fold it: then you will realize it is not foldable.]

Around 1993, the author found a monotone drawing of that tiling - like this - in the figures on Archimedean solids and semiregular tessellations, those appeared in the Japanese translation of the book by L. Fejes Toth:*"Lagerungen in der Ebene auf der Kugel und im Raum" (2nd ed.), Springer, Berlin, 1972* - although there was no discussion about 'foldability' of such (planar) tilings/tessellations -.

[And he informed several paper-folding artists about this diagram and the book: within his remembrance...]

>>Click the image to see the larger picture<<

The significance of the example above is:

the diagram consists of foldable structures around each vertex [namely, it is generated by 'locally' foldable structures], but it is not composed into foldable tiling as a whole.

In fact, it is satisfied [necessary and sufficient] flatly foldable conditions at each vertex represented in the angular relations [given by T. Kawasaki in his papers up to general cases of geometry] as follows:

π/3 + π/2 + 2π/3 + π/2 = 2π (rad) [60°+ 90°+ 120°+ 90°= 360°];

π/3 - π/2 + 2π/3 - π/2 = 0 (rad) __[60°- 90°+ 120°- 90°= 0°].

On the other hand, it is not flatly foldable because of the combinatorial folding conditions with respect to the relations between mountain-/valley- creases and values of angles. [Try to fold it: then you will realize it is not foldable.]

Around 1993, the author found a monotone drawing of that tiling - like this - in the figures on Archimedean solids and semiregular tessellations, those appeared in the Japanese translation of the book by L. Fejes Toth:

[And he informed several paper-folding artists about this diagram and the book: within his remembrance...]

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The following is the result of an attempt to make some 'latticework', made of modules as appeared in the preceding article and sub-modules with another shape for linking adjacent modules together.

It has been displayed in the previous articles already: "A modified modular planar folding" and "Larger modules with light blue links".

Here is a partial diagram of those modular folding structures:

Folded shape of six modules in 2 by 3 viewed from front/back sides:

>> Click each image to see the larger picture <<

It has been displayed in the previous articles already: "A modified modular planar folding" and "Larger modules with light blue links".

Here is a partial diagram of those modular folding structures:

Folded shape of six modules in 2 by 3 viewed from front/back sides:

>> Click each image to see the larger picture <<

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