Shape 

Copyright David & Cynthia Thomas, 2009

Triangle Recipes--Directed Activity: Euclid’s First Triangle, Part III

Focus

Construct an equilateral triangle

Technologies

· Paper folding

References

· Euclid’s Elements Proposition 1

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html

· Euclid of Alexandria

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

Background

As with straightedge-and-compass constructions, paper folding is based on a set of fundamental postulates summarized in Table 1.2. 

Postulate	Illustration
1. Given two distinct points A and B, there is a unique fold containing A and B. [The paper is creased so that both points are on the fold.]
2. Given two distinct points A and B, there exists a unique fold that maps A onto B. [The paper is folded so that points A and B coincide.  This crease is the perpendicular bisector of segment AB.]
3. Given two creases l and m, there exists a unique fold that maps l onto m. [The paper is folded so that line l coincides with line m.  This forces the crease to pass through the vertex and bisect the angle.]
4. Given a point A and a crease l, there exists a unique fold through A perpendicular to l. [Line l is folded so that it coincides with itself and the crease passes through point A.  As a result the crease is perpendicular to line l.]
5. For given points A and B and crease l, there exists a fold that passes through A and maps B into l. [The paper is folded so that point B coincides with line l and the crease passes through point A.]
6. Given two points A and B and two creases l and m, there exists a unique fold that maps A into l and B into m. [The paper is folded so that point A coincides with line l and point B coincides with line m.]

Table 1.2: Postulates for Paper Folding

Tasks

1. Using paper folding, construct an equilateral triangle. 

2. Explain and justify the individual steps and final result of your construction.