MATHEMATICS AND ORIGAMI

Mathematics and Origami

1.3 RELATION WITH SURFACES

1.3.1 EULER CHARACTERISTIC APPLIED TO THE PLANE

If both figures are obtained by folding a rectangle, the above condition can be checked:

1) Faces = 12 Vertices = 20 Sides = 31 ........ 12 + 20 = 31 + 1

2) Faces = 11 Vertices = 16 Sides = 26 ........ 11 + 16 = 26 + 1

Total of faces by left sides: 7+1+1+1+1= 11 = total amount of faces = C (2)

Adding up (1) and (2) we ́ll have:

Total sides A = V – 1 + C ; C + V = A + 1

A

B

A ́

B ́

A

B

a

(^12)
3
I
I
I
I
I
I I
II
III
IV
V
We shall show it from figures 2 and 3.
The heavy broken line is such that it passes just once through all the
vertices. As any side has two vertices, it follows that the broken line is
made out of so many sides as vertices in it, minus 1.
15 A = 16 V – 1 ; Sides in broken line = Vertices – 1 (1)
There are left 26 – 15 Sides to be determined. Let ́s associate these
11 Sides left, to the amount of faces.

• Faces to which, from broken line, one only side is left (I): 7.


• After the former operation: faces to which one only side is left
  (II): 1


• Idem (III): 1


• Idem (IV): 1


• Idem (V): 1

Faces (C) + Vertices (V) = Sides (A) + 1

V A

1

C

C

V

A C

2

A