Mathematics and Origami

5. COROLLARY P

Enunciation:

• Let ́s assume that in a square whose side measures one unit, any of its four vertices is
  folded over any one of its opposite sides in such a way that the image of that vertex in-
  duces the distance x as shown in figs. 1 and 2 of Point 4.


• If x takes the form x =
  n

1

, n being a natural number or a rational number greater than 1, a

will have the value a =

1

2

n+

(see again fig. 2 in Point 4).

Before demonstrating the corollary, let ́s see two applications of it:

• n is a natural number. If we make, e.g., x =
  13

1

, a will be a =

7

1

14

2

=. This characteristic leads to

an exact procedure to divide a segment by folding, in any number of equal parts (see Point

9.16). We have to bear in mind (see Fig 2 of Point 4) that ratio a / x is biunivocal, i.e., it makes

no difference to fix x to get a, than viceversa.

• n is a rational number. If the square have a side of, e.g., 600 mm and we want x (^) = 273.5 mm, we
can figure out the value of a:
x in the form
n
1
ends up as: x =
2735
6000
1
273. 5
600
1
= , therefore a =
1
273. 5
600
2

• = 0.6262163
  600 x 0.6262163 = 375.72982 will be the mm measured by a.
  DEMONSTRATION: For this purpose we ́ll use the values of x, y, a, obtained in Point 4 as
  well as the simplification in the denominator of a. So we have:
  a =
  ()
  ()^2
  2 2

0. 251
   1
   x x
   x x

• 
  

  −
  −

  
  

  ()

  
  

0. 
   

   5 () 1 2
   1
   x
   x x
   −
   −

   
   

   ()x
   x

   
   


1. 
   

   51 +
   (1)
   If the corollary is true, it will be: a =
   1
   2
   n+

   
   

   
   
   
   
   
   

   
   

• n
  n
  1

0. 51
   1
   (2)
   Expressions (1) and (2) are identical for
   n
   x
   1
   = (corollary P ́s statement), therefore the cor-
   ollary is fulfilled.