MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

13.1.2 A CIRCUMFERENCE AS THE ENVELOPE OF ITS OWN TANGENTS

INSCRIBED WITHIN A SQUARE

Follow the process here after:

Its foundation is in the first figure: it consists basically in rotating a given square around

its center. Thus each turned side is equidistant to the center in the same value, i.e. the apothem

length. Then we can see that each side is a tangent to the circumference, and their midpoints are

the points of tangency.

The rest of figures show the process, whereas the last one displays the situation of cir-

cumference and given square after the three folds were performed. If we go on folding, we can

approximate as much as we wish the relation circumference / envelope.

CONCENTRIC WITH ANOTHER GIVEN ONE AND INTERIOR TO IT

The solution is to fold a stellate polygon inscribed in the given circumference. The re-

sulting ring shows in its inner side a convex polygon made by the folding lines: sought circum-

O

A

1

2

B

1

A

O

2

B

C

1 2