Go to Fun_Math Content Table Three Famous Problems Origami Solutions

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".

But first let us go back to Plato's Delian solution.

The basis of this construction is that trianlges OAM, OMN, and ONB are similar. (angles AMN, and MNB are right angle.) Therefore if ON = a .OB Then OM = a |

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Draw x ,y coordinate axis through "O". Let OP = 1 (unit length), OQ = a (arbitrary length) At Q , draw a line perpendicular to PQ. This line intersects x-axis at R. At R , draw a line perpendicular to QR. This line intersects y-axis at S. |

Draw a horizontal line DD' through S. Draw line FF' at the distance a |

You can see the process in animation.

************ Abe_Delian_model.dwg** *********

**To create this drawing and animation: **
** Load Abe_Delian_model.lsp (load "Abe_Delian_model")**

Then from command line, type **Abe_Delian_model **

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

(2) Line AF, DG, BJ & HK are creases 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) 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.

Note:The extension of MN looks like intersecting AE at H

, but it is not the case.

You can see the process in animation.

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

**To create this drawing and animation: **
** Load Abe_Delian.lsp (load "Abe_Delian")**

Then from command line, type **Abe_Delian **

As usual, for a quick look , run test & test_2.

What is required is a square or rectangle paper.

But here a rectangle graph paper is used .

In this example: OP = 2 inches, OS = 1 inch

Check the result against the exact values.

OR = 2^{1/3} = 1.2599.., and OQ = 2^{2/3} = 1.5874..

OR = between 1.25 ~ 1.3 and OQ = between 1.55 ~ 1.6

So in spite of a simple folding operation, precision is very good.

*************** Abe_Delian_test.dwg** ************

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

Go to Fun_Math Content Table Three Famous Problems Origami Solutions

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved .