Mathematics and Origami

8.2.3 EQUILATERAL TRIANGLE FROM A RECTANGLE (three solutions)
Solution 1

The rectangle is a paper strip, and the solution is very useful in hexaflexagons construction.

Justification:

a=b (simmetry r) ; g=d (simmetry s) ; a=d (vertical angles). Then:

a=b=g= 60

3

180

= ; e=f (simmetry r) ; w=a=e. So:

f=w=60º and ∆ABC, equilateral triangle.

It is impossible to develop a similar construction from a square because AB is greater than the

side of the supposed square.

Solution 2

We also begin with a paper strip. The equilateral ∆ABC is the solution. The reader may recall

that in point 8.2.2.2, Solution 1, the right angle of a square was trisected.

Solution 3

Former Solution 1 made use of the whole width of the paper strip to allow the construction of a

successive and indefinite quantity of equilateral triangles. On the contrary, present Solution 3

A

r B C

s

(^12)
3
4

=
30 º
30 º
A
B C
30 º