Origami Equilateral Triangle

Date: 04/26/2001 at 00:06:02
From: Oliver
Subject: Folding Construction: Equilateral Triangle

Hi,

I have spent quite a long time trying to figure out how to create an 
equilateral triangle by folding; not using the sides of a piece of 
paper and using nothing but your hands and, of course, your brain.

I have tried to make a hexagon from a piece of paper by folding, 
because that would give you six equilateral triangles, but I have had 
no luck in creating a hexagon from only folding.

I would appreciate your help and I am thankful you took the time to 
read this.

Oliver

Date: 04/26/2001 at 12:36:34
From: Doctor Peterson
Subject: Re: Folding Construction: Equilateral Triangle

Hi, Oliver.

I'm not quite sure what you mean by "not using the sides"; does that 
mean the edge of the paper can't form one of the sides of the 
triangle, or that you can't assume the sides are parallel, or what? I 
think what I've done probably is allowed, regardless of the 
interpretation.

Origami books teach a standard way to make a 60-degree angle by 
folding, which you can use to accomplish the task. If you have two 
parallel lines and a perpendicular to them, form a third parallel 
midway between the first two, and then fold the perpendicular AB at an 
angle so that point A lies on the midline (A'), while the fold passes 
through point B. Then the fold BC will be at the desired angle:

           B|
         ---*---------------------------
            |\ \
            | \    \
            |  \      \
            |   \         \
            |    \           \
         ---+-----\-------------+---------
            |      \           / A'
            |       \        /
            |        \      /
            |         \    /
            |          \ /
         ---+-----------X---------------
           A|            C

You should be able to prove easily that ABC is half of an equilateral 
triangle. I folded a triangle in the middle of a piece of paper using 
this method, with the parallel and perpendicular lines being 
previously formed folds.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/