By Kazuo Haga
The artwork of origami, or paper folding, is performed utilizing a sq. piece of paper to procure appealing figures of animals, vegetation or different standard figures. possible see that origami has hyperlinks with geometry. Creases and edges characterize strains, intersecting creases and edges make angles, whereas the intersections themselves signify issues. due to its manipulative and experiential nature, origami may perhaps turn into a good context for the educational and educating of geometry.
In this distinctive and unique booklet, origami is an item of mathematical exploration. The actions during this e-book vary from usual origami in that no figures of items outcome. fairly, they lead the reader to check the consequences of the folding and search styles. The experimental process that characterizes a lot of technology job may be famous during the booklet, because the manipulative nature of origami permits a lot experimenting, evaluating, visualizing, getting to know and conjecturing.
The reader is inspired to fill in the entire proofs, for his/her personal delight and for the sake of mathematical completeness. therefore, this e-book presents an invaluable, substitute technique for enhancing and making use of the theorems of highschool arithmetic.
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Extra resources for Origamics: mathematical explorations through paper folding
Let HB = x and BG = y. By the Pythagorean Theorem on HBG, x2 + y 2 = (1 − y)2 . And therefore y= Since EAH and 1 − x2 . 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.
Yoko Takamoto, a teacher of the Senior High School attached to Toshimagaoka Women’s Educational Institution, while a trainee at Tsukuba University’s Extension course entitled “Origami and Education”. August 11, 2008 11:23 World Scientific Book - 9in x 6in 41 Origamics˙noline X-LINES WITH LOTS OF SURPRISES ☎ ✄ ✟ ✆ ✡ ☛ ☞ ✍ ✠ ✌ ✝ ✞ ✁ ✂ ✍ ✌ Fig. 9 The length FH (= IG) is an interesting fact. 9). These are sides of triangles FJH and IJG respectively. Since J lies on the midline then FJ = JG and HJ = JI.
1(b)). Press the whole crease repeatedly with the finger nail to make it clear and distinct. 1(c)). Make sure that the fold is distinct. Now place the other lower vertex on the selected point and 3 These surprises come from the mathematical reasoning arise through paper folding. Through those surprises, we can find the invariant and it initiates us to inquire much more paper folding as mathematical science activity. 1(d)). 1(e)). Do you see the two creases? ✂✁☎✄ ✝✆✞✄ ✌✎✍✑✏✓✒✔✍✑✕✖✍✘✗✝✙✎✚ ✛ ✗✜✍✑✗✣✢✥✤✎✦✧✚ ✕✧✛★✦✩✕✪✛ ✫✎✒ ✬ ✤✧✤✎✒✘✗✭✒✯✮✧✰✧✒✩✱ ✂✟☎✄ ✍✵✴✯✒✶✛ ✫✓✒ ✦✩✷✥✒✑✗ ✒✯✸✹✛✓✺✓✒✑✗✣✛✹✒✑✻ ✲☛ ✦✩✕✧ ✳ ✛✹✦✼✛ ✫✓✒✽✍✑✳ ✗✾✙✎✚ ✛ ✗✡✍✘✗✣✢✥ ✳ ✤✎✦✧✚ ✕✧✛✣✱ ✡✠☛✄ ✍✵✴✯✒✶✛ ✫✓✒ ✒✽✦❀✙❁✛✹✍✯✚ ✕✖✦✩✕✓✒ ✿ ✚ ✕✓✒☎✱ ✳ ✂☞☎✄ ✦✩✷✥✒✑✗❂✗✜✚ ✰✩✫✧✛✓✺✓✒✘✗✣✛✹✒✵✻ ✚ ✕✓✍ ✲☛ ✦✩✕✧ ✳ ✛✹✦✼✛ ✫✓✒❄❃✂✍✑ ✳ ❅❆✒✽✍✘✗✝✙✎✚ ✛ ✗✡✍✑✗✝✢✥✤✎✦✧✚ ✕✧✛✣✱ ✢✩❈✯✷✪✒✽✦❀✙❁✛✹✍✯✚ ✕✪✛ ✷✪✦✪✴✑✗✜✒✯✍✯❃❉✒✯❃ ❇✚ ✕✪✛ ✫✓✳ ✒❄ ✳ ✸✝✦✩✗✝❅❊✦❁✸✎❋●✱ Fig.