Doubling the Cube

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### Origami(Paper Folding) Solution

#### 1. Origami Solution by Hisashi Abe (ref.1)

Hisashi Abe published a very unique Delian solution in a Japanese mathematical magazine in 1980.
The method is now described in ref.1, published in 2003.

He uses a traditional Japanese paper folding technique , called "Origami".

(1) Prepare a rectangular paper. All the corner angles must be exactly 90 degrees.

(2) Line AF, DG, BJ & HK are crease lines after folding along the lines.

(3) Distance between horizonral lines is 1 unit length.

(4) Distance between vertical lines is twice the unit value.

(5) With point H as a pivot, fold paper so that point A

falls on line HK and point B ,onto line DG.

Then distance ON is the cube root of 2.

You can see the process in animation.

For detail, go to the section Abe Hisashi's method.

************ Origami_Delian_desc.dwg** *********

#### 2. Origami Solution by Kazuo Haga (ref.2)

Ref.2 shows how to make a side length (15329/20000) by folding a square paper.

ABCD is a sqaure paper.

Step 1: move side CB so that C matches point E.Side CD cuts line AB at point f.
Find point F to bisect Af.

Step 2: move corner C to point F. Folding line cuts CB at point g.
Extend Bg by 50 %, to define point G.

Step 3: move corner D to point G. Foling line cuts CD at point H.

Length AH = 1.25993873 ,is a very good approximation of 2^{1/3} = 1.259921...

You can see the process in animation.

For detail, go to the section Haga Kazuo's method.

************ Origami_Delian_2_desc.dwg** *********

#### References

1. Abe, Hisashi: "Amazing Origami" , (in Japanese), 2003, ISBN 4-535-78409-4

2. Haga, Kazuo: "Origamics part - I" , (in Japanese), 1999, ISBN 4-535-78293-8

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Last Updated Nov 22, 2006

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