The geometry

In Section 2 we showed that we need to construct

$$\cos(72^\circ) = \frac{\sqrt{5} -1}{4}.$$

The only tools we have available are a straight edge with no marks some string and a piece of chalk. Thus given two points we can draw a straight line through them and we can draw a circle centered at one of the points that passes through the other. Furthermore using the compass (string and chalk) we can

i

erect a perpendicular to line at a point on the line;

ii

bisect a line segment; and

iii

copy the distance between two marked points on one line to another line.

The 7 steps to construct the pentagram are:

1.

Mark any two points A and B and draw a line $\ell$ through them. Draw a circle of radius |AB| centered at B. Let C be the the point other than A where the circle intersects $\ell$. Take AB to be our unit length, i.e. the length |AB|=1.

2.

Bisect AB at D and DB at E.

3.

Erect a line perpendicular to $\ell$ at E and mark off F on it such that |EF| = |AD|. Note that the Pythagorian formula shows that $\vert BF\vert = \sqrt{5}/4$ .

4.

Mark G on $\ell$ such that |EG|= |BF|

5.

Erect a line perpendicular to $\ell$ at G and Let Hbe the point where it intersects the circle.

The angle $\theta = HBG$ is $72^\circ$s. To see this observe that

$$\cos(\theta)=\frac{\vert BG\vert}{\vert BH\vert} =\frac{\vert... ...c{1}{4} =\frac{\sqrt{5}}{4}-\frac{1}{4} =\frac{\sqrt{5}-1}{4}.$$

6.

Using arc CH mark off the vertices of the pentagram.

7.

Draw the pentagram.