(c) Copyright AZUMA Hideaki. All rights reserved.

(Last updated: February 17, 2008)

The basic self-similar (sub) structure of the iterative [almost self-] similar flatly foldable pattern appeared in the previous article is:

The above polygonal structure has eight edges (colored in red) as its boundary: these edges come from two - large and small - squares that are placed concentric at 45°[π/4 (rad)] each other.

The (homothetic) ratio of the large square to the small square is

(√2 + 1)/1 [= 1/(√2 - 1)].

Such closed and connected regions, each of these is homeomorphic to a 2-dimensional closed annulus [/compact cylindrical surface], will make "alternating homothetic" [this "expression" might not be correct in the strict sense of mathematics...] patterns over a square as follows:

- "Alternating homothetic" patterns (in 5 times iterations) -

- A self-similar unit adjusted in scale to compare -

- Corresponding colored diagram in the same scale -

Click here to see a larger image - in 1285 pxls by 1285 pxls - of the above diagram; and here to see its corresponding patterns by the regions over a square in the same scales.

The above polygonal structure has eight edges (colored in red) as its boundary: these edges come from two - large and small - squares that are placed concentric at 45°[π/4 (rad)] each other.

The (homothetic) ratio of the large square to the small square is

(√2 + 1)/1 [= 1/(√2 - 1)].

Such closed and connected regions, each of these is homeomorphic to a 2-dimensional closed annulus [/compact cylindrical surface], will make "alternating homothetic" [this "expression" might not be correct in the strict sense of mathematics...] patterns over a square as follows:

- "Alternating homothetic" patterns (in 5 times iterations) -

- A self-similar unit adjusted in scale to compare -

- Corresponding colored diagram in the same scale -

Click here to see a larger image - in 1285 pxls by 1285 pxls - of the above diagram; and here to see its corresponding patterns by the regions over a square in the same scales.

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2nd variation of foldable patterns in the preceding article and its reflection are composed of the basic 2 by 2 unit for tiling: it has been used as the background image of this site since its beginning [until the end of November 2005]:

A larger image of the above module is:

Same as mentioned at the article: On the background tiling patterns: in the companion site ManyFolds in Variety , above tiling module (as a diagram) may not be foldable in a practical manner.

A larger image of the above module is:

Same as mentioned at the article: On the background tiling patterns: in the companion site ManyFolds in Variety , above tiling module (as a diagram) may not be foldable in a practical manner.

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Iterative [almost self-] similar flatly foldable patterns can be designed on the cruciform shape that has appeared in the previous articles:

1st;2nd;3rd;

4th;5th.

- 6th, 7th, more and more can be designed [as mathematics] -

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

Some patterns above as diagrams are applied to his tiling works (tapestries of paper folding modules) that had been made in 1993: one was appeared in a photograph (by the side of him) at his article in the quarterly magazine ORU #6 (1994 Autumn), and modifications were also used in applications, such as floral models.

1st;2nd;3rd;

4th;5th.

- 6th, 7th, more and more can be designed [as mathematics] -

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

Some patterns above as diagrams are applied to his tiling works (tapestries of paper folding modules) that had been made in 1993: one was appeared in a photograph (by the side of him) at his article in the quarterly magazine ORU #6 (1994 Autumn), and modifications were also used in applications, such as floral models.

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4×4 cruciform shapes in a square: that is composed of 8 basic modules (in some modification) and 3 joining modules.

Material: Square origami, without using adhesives.

Date: From October to November 2005

Photo date: November 20, 2005

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The tiling structure in the previous article is also displayed as the pattern of folding modules in diagrams at corresponding positions, for example:

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