MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

10.2 HEXAGON

10.2.1 WITH A PREVIOUS FOLDING

The folding process that follows leads to a perfect convex regular hexagon that is also

the largest obtainable from the starting square.

In the last figure we can see the perfection of the hexagon produced (l = MN).

In ∆EBL:

Ang. BEF = 60 º

2

1

cos

2

cos =Arc =

l

l

Arc ; Ang. CEF = 30º

In ∆CFE:

sen 30 sen() 180 − 45 − 30

=

CF l

; CF = NM = =

2 sen 75

l

side of hexagon

l CF

CF

Arc

l CF

CF

Arc

−

=

















−

=

2

tg

2 2

2

α tg^2

30

2 2 sen 75 1

1

tg

2 sen 75

2 sen 75 2

tg =

−

=













−

= Arc

l

l

l

α Arc º

From what was shown in corner F and because of folding symmetries, we can deduce

the perfection for the rest of the hexagon.

10.2.2 KNOT TYPE HEXAGON

Begin with two paper strips long 10 to 15 times its width. Produce two knots like in

fig.1; one of them will be as shown; the other will be alike but set upside down.

Figs. 2, 3 and 4 show flattened folds just to ease the drawing; actually, the strips are

played loose as in fig 1. Only in fig. 5 the strips should be pulled tight and flattened (see pre-

cautions in Point 10.1.5).

B

F E

C

F

C

E

F

M

N

N M

F

M

F

N M

N

F

M

F

N

M

B

E

C

L