ManyFolds in Variety
% The preparation is (still) in progress, however [forever?], %
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)
Categories
Calendar
2022年
July 
July 
| Sun | Mon | Tue | Wed | Thu | Fri | Sat |
|---|---|---|---|---|---|---|
| 1 | 2 | |||||
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| 17 | 18 | 19 | 20 | 21 | 22 | 23 |
| 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 |
Archives
Search This Site
Recent Articles
Latest Revisions
Edited top page.
(February 17, 2008)
Edited "Some directions for articles with photos"
(July 10, 2007)
Edited top page.
(July 4, 2007)
QR Code
このブログを
ブログサービス
Powered by
« Foldable tiling: homeomorphic to the preceding structure | Main | Twisted folding around a triangle #10 »
2005/7/12
「Not foldable tiling in two-toned」 foldable tilings
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...]
0
>>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...]
0
投稿者: kaishonashi
トラックバック(0)