MATHEMATICS AND ORIGAMI

Mathematics and Origami

Fold A over D to get C ́:

()

()

4 4

1

2

1

4

1

2

1

2

2

1

1

2

1

2

́ ́

x x

x

AD BD

AC CD = − +

−

= −

−

−

=

−

= = =

Fold B over C ́ to get D ́:

8 8

1

4

1

2

1

2

4 4

1

2

1

1

2

1 ́

2

́

́ ́ ́

x

x

BC AC

BD CD = − + −











− − +

=

−

= = =

Fold A over D ́ to get C ́ ́:

16 16

1

8

1

4

1

2

1

2

8 8

1

4

1

2

1

1

2

1 ́

2

́

́ ́ ́ ́ ́

x

x

AD BD

AC C D = − + − +













− − + −

=

−

= = =

Fold B over C ́ ́ to get D ́ ́:

=













− − + − +

=

−

= = =

2

16 16

1

8

1

4

1

2

1

1

2

1 ́ ́

2

́ ́

́ ́ ́ ́ ́ ́

x

BC AC

BD C D

32 32

1

16

1

8

1

4

1

2

1 x

− + − + −

and so on. The following step (7th fold) will give:

AC ́ ́ ́= − + − + − + −.....=

2 2

1

2

1

2

1

2

1

2

1

2

1

1 2 3 4 5 6 6

x

Lim x

− + − + − + −.....=
2 2

1

2

1

2

1

2

1

2

1

2

1

2 3 4 5 6 7 7

x

2

1

Lim x

It may be noted that the limit of x is made up of the algebraic sum of certain powers of 2, like

2 (for j values from 1 to n).−j

Adding up the former two expressions, we ́ll have:











 +

2

1

1 Lim 6 7 7

2 2

1

2 2

1 x x

x= + − +

2

3

Lim x 7 6 7 x 7 7 x

2

3

2

1

2

1

2

1

2

1

2

1

2

1

 = − +









= − + +

(^3) A C C ́ D B
(^4) A C C ́ D D ́ B
(^5) A C C ́ C ́ ́ D D ́ B
(^6) A
C C ́ C ́ ́ D D ́ D ́ ́ B