MATHEMATICS AND ORIGAMI

Mathematics and Origami

One can understand why we said earlier that a flexagon is alike a Möbius band: looking
afterwards how it works, we ́ll see its topological connotations observing its endless conti-
nuity of surfaces.

Here it is the folding process. Fig. 1 starts with a triangled paper strip as in Solution 1 ̧
Point 8.2.3. Figs. 1 and 2 show the folds to get Fig. 3; this indicates how ∆Y has to be
folded and stuck over ∆X.
In Fig. 4 we number with 1 the six triangles of the obverse, and with 2 the six of the re-
verse side. Then we produce in the same Fig. 4 three mountain and three valley folds,
bearing in mind that the seen triangles that meet, respectively, in A,B,C show paper conti-
nuity whereas this is not the case with the others. That ́s why it is a must to perform the
folds as indicated.
So we reach Fig. 5 with the beginning of the operation that gets points ABC meeting at
the bottom vertex of Fig. 6. Once this operation is over, a new hexagon will appear, sur-
prisingly, with six unnumbered triangles. If we number them with a 3 (Fig. 7), we can
check that the reverse of Fig. 7 has numeration 1.

Fig. 8 is Fig. 7 after the triangles XY have been unglued and the rest developed. It
shows in its obverse and reverse the corresponding numbers in each triangle (six 1, six 2,
six 3), the X, Y triangles and vertices A, B C.
If we manipulate Fig. 7 as we did with Figs. 5, 6, we can see appear hexagons with
these combinations in obverse / reverse: 1 / 2; 1 / 3; 2 / 3.
Till now we have got a tri-hexa-flexagon: A hexagon based flexagon, with three faces
(marked with 1, with 2 and with 3). But we can introduce a certain number of variations.
For example, we can draw on each one of said three faces, three different figures having
internal symmetry within the hexagons. When manipulating that flexagon and taking into
account obverses and reverses we come across, as before, with two faces 1, two faces 2 and

X

Y

1

1

1

1

1

1

2

2

2

2

2

2

3 4

A

B

C

7

3

3

3

3

3

3

8

3

3

3

3

3

3

1

1

1

1

1

1

2

(^22)
2
2
2
Y
X
C
B
A
C ABC
A