MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

and making the variable change

t= 2 cosω ;

2

cos

t

ω= (2)

(1) is transformed to:

t^3 +t^2 − 2 t− 1 = 0 (3)

Equation (3) was already “fold solved” in Point 7.14.5.

It is clear that the coefficients of (3) are:

1; p = 1; q = -2; r = -1

what leads to a discriminant

∆ =

3 2 2 3

3

3

27

1

27

2 9 27

4

1











 −

 +









 p − pq+ r q p

that is negative, i.e., to have three real solutions for (3). In consequence, there are three

different forms of simultaneous folding of point I (initial) and F (final) on axles X ́ Y ́ (fig 2).

According to (3), the vectors sequence in fig. 2 is:

I; 1; 1; -2; -1; F

Now we should explain how to configurate the direction of those vectors (fig. 2): if,

from one coefficient to the next there is no sign change, the advancing vector turns to the right;

if that change exists, it turns to the left.

Then (3) has three solutions in t, which correspond with other three solutions for cos ω

in (1); besides, t 1 t 2 t 3 are, respectively, the tangents of angles α, β, γ in fig.2:

t 1 = tg α → 2cos ω 1

t 2 = tg β → 2cos ω 2

t 3 = tg γ → 2cos ω 3

Angles ω 1 ω 2 ω 3 correspond to the heptagon vertices having distinct abscissas.

Relating (2) with fig. 2, it is

2 2 2 4

tg

2

cos 1 1

t AO AO

=

×

= = =

α

ω (4)

If the pair of heptagon upper vertices lie on AH, when we trace

7

360

ω 1 = , OR becomes

the radius of said heptagon, and it ́ll be:

OR

OA

cosω 1 = (5)

The consequence of equalising (4) and (5) is:

r = heptagon radius = OR = 4

Bringing r from O along OY ́ we obtain vertex V.

Now we can note this:

• OR measures 4 units; in fig. 2, first vector from I towards vertical Y, is taken as one
  unit.


• Angles α and ω 1 have a different though very close mesure. This is unimportant,

anyway: 51 , 428571

7

360

ω 1 = = and 51 , 272558

7

360

arctg 2 cos =







α= .

The above only means that radius OR is not parallel to its associated fold, although its con-

struction is perfect.