By Kazuo Haga

The artwork of origami, or paper folding, is conducted utilizing a sq. piece of paper to procure beautiful figures of animals, plant life or different primary figures. you can see that origami has hyperlinks with geometry. Creases and edges symbolize strains, intersecting creases and edges make angles, whereas the intersections themselves symbolize issues. as a result of its manipulative and experiential nature, origami may possibly turn into an efficient context for the training and educating of geometry.

In this distinctive and unique booklet, origami is an item of mathematical exploration. The actions during this publication range from traditional origami in that no figures of items outcome. relatively, they lead the reader to review the results of the folding and search styles. The experimental process that characterizes a lot of technological know-how job could be famous through the booklet, because the manipulative nature of origami permits a lot experimenting, evaluating, visualizing, researching and conjecturing.

The reader is inspired to fill in all of the proofs, for his/her personal pride and for the sake of mathematical completeness. hence, this booklet offers an invaluable, substitute process for boosting and using the theorems of highschool arithmetic.

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**Extra resources for Origamics: mathematical explorations through paper folding**

**Example text**

You cannot obtain these results as easily 5 by using compass and straightedge. Comment on teaching. As a classroom activity, how a teacher develops the topic depends on his/her style. However, before bringing up a mathematical proof it would be more concrete and interesting for the students if a folding procedure to support the above findings is brought up first. Following is a suggestion. The point F is first located by Haga’s Second Theorem Fold. The teacher then proposes that the position of F is as stated above.

2 (1) HBG are similar, HB : BG = AE : AH, or x : y = √ 1 : 2−x . 2 (2) Equations (1) and (2) together lead to the quadratic equation, √ √ 2 2 5 2 3x − 4 2x + 1 = 0. Therefore, x = . 19745 . . August 11, 2008 11:23 World Scientific Book - 9in x 6in Origamics˙noline 31 EXTENSION OF THE HAGA’S THEOREMS TO SILVER RATIO RECTANGLES Fig. 9 Haga’s Third Theorem fold with A4 paper in a vertical layout. ✠ ☎ ✝ ✄ ✟ ☛ ✡ ✏ ✞ ✌ ☞ ✍ ✁ ✂ ✆ ✎ Fig. 10 Mathematical principles of Haga’s Third Theorem fold with A4 paper in a vertical layout.

If the length of a side of the square is 1, then √ 5 , obtained by using the Pythagorean the length of the crease EC is 2 theorem on right EDC. 2. This is the new position of ED after folding. Suppose segment EF is extended to the left edge at point G. Where does this extended line reach on the 11 August 11, 2008 11:23 World Scientific Book - 9in x 6in Mathematical Principles Related to Haga’s Second and Third Theorems Origamics˙noline 12 ✂✁☎✄ ✝✆✞✄ Fig. 1 How to fold the upper edge through the midpoint?