Before going into the deeper mathematics of the Golden Section spirals, this section looks at spirals drawn as approximations using circles, together with variations on these approximations to create new spirals.

SPIRALS FROM THE GOLDEN RECTANGLE
We are now in a position to draw the first Golden Section spiral and develop new spirals using the same technique.

The spiral approximated with arcs of circles
Using the technique for spiral similarity shown in Figure 7, if you take a Golden Section rectangle, that is one with its sides in the ratio 1:f, and add a square to the shorter side, you end up with another golden rectangle. If you continue adding another square, then you produce another one and can do this indefinitely (Figure 8).

Figure 8

You can, of course, do the reverse, and subtract. Each time, if you keep the same orientation of the figure, you need to rotate the drawing. This brings home the rotation and magnification (or dilation if you are subtracting). This is the so-called 'whirling squares' so named by the art historian Jay Hambidge [1926].

Figure 8 was drawn with a true Golden Section rectangle. You can draw a similar figure if you use a rectangle with sides in the ratio of two successive terms of the Fibonacci sequence. The rectangles and squares then have integral sides that are Fibonacci numbers.

The most well-known Golden Section spiral is drawn from Figure 6, using arcs of circles. A quarter circle is drawn in each square so that the line joining the centres go though the touching point to give a smooth curve (Figure 9).

Figure 9

The pole for the spiral is found by drawing the diagonals of the Golden Section rectangles.

Figure 10

This allows us to unravel the mathematics of this spiral, but before doing that I would like to show that it is possible to use this technique for creating other spirals and designs using this set of whirling squares.

Doubling the spirals
Reflecting the spiral about the diagonal of the largest square yields a very satisfying design. The first stage with the construction squares is:

Figure 11

and then rotated by 45° on its own gives a smooth curve because of the way the arcs flow:

Figure 12

The anti-spiral from arcs of circles
The quarter arcs can be drawn in each square in different ways. Not all form a continuous curve. The following one forms a series of cusps.

Figure 13

Using the technique shown in Figures 12 and 13, the arcs no longer appear to be formed from quarter circles.

Figure 14

The "wobbly" spiral
I "discovered" this one by accident when I used a CAD program to draw the spiral of Figure 9. When you draw an arc with such a program, using the centre and two points on the arc, the program sometimes decides to go in the opposite direction from the one you want. Instead of drawing a quarter circle, I found it produced the corresponding three quarter one. This is the result on the second from largest square of the whirling squares.

Figure 15

The full drawing, with the three-quarter circles drawn for each square, then becomes:

Figure 16

I called this the "wobbly" spiral, since it appears to wobble back and forth. Without the construction squares it looks like this.

Figure 17

The "wobbly" anti-spiral
Using the technique described above (as in Figure 13), it is also possible to create an anti-spiral version of the wobbly spiral, although it does not quite have the wobble you would expect.

Figure 18

GOLDEN SECTION SPIRALS AND TRIANGLES
There are two Golden Section isosceles triangles formed by having two long or two short sides equal and the only right angled triangle with its sides in geometric progression is also a Golden Section one. It is also possible to produce Golden Section spirals using equilateral triangles.

The Golden Section triangles and their relationships
For ease of writing in the following, I will call the two parts of the Golden Section division Long and Short and use the first letter of these names to describe the triangles I am working with.

The two types of isosceles triangles having the Golden Section ratio of sides then become LLS and SSL. (An LLS triangle has angles of 72, 72, and 36 degrees; and an SSL triangle of 36, 36, and 108 degrees.) Dividing a side L in each of them in the Golden Section creates more isosceles triangles.

Figure 19

With the SSL triangle at the top of Figure 19, each division produces a LLS and another SSL which when subdivided in the same way creates the spiral of triangles. Similarly the LLS triangle below it when subdivided creates an SSL and another LLS and so on.

It is easier to draw the spiral using arcs of circles with the LLS triangle (Figure 20). They are not quarter circles as with the Golden Section rectangle but arcs subtending 108°. The centre and ends of the arcs are clearly defined, with the centre as the division point on one L side and the ends on the opposite L side.

Figure 20

The equivalent arc-spiral on the SSL triangle looks like this:

Figure 21

At first glance, the arcs seem to be drawn with centre on the point of division on the L side and ends at the endpoints of the S side. But the rule shown in figure 6 does not apply if this is the case because the line joining the centres does not pass though the joining point of the two arcs. If you try to draw a spiral this way, then the spiral looks odd because it is not smooth.

Figure 22

In order to draw the correct spiral, the arcs must be drawn with centre at the 'centroid' (centre of mass) of the corresponding triangle, which is the point of intersection of the angle bisectors of the three angles (actually, just two angle bisectors will suffice); the ends of the arc are at the endpoints of the L side.

Figure 23

The arcs in this case subtend an angle of 144°. Because it is a more complex diagram, it is well worth drawing and studying how the centres and ends of the arcs are related and how the bisectors come together.

Escher's Golden Section triangle
This triangle does not seem to appear in the Golden Section literature, and I have named it after Escher since it is the only case I can find of his using the Golden Section consciously. It is drawn in his notebooks as a tessellation (see [Schattschneider 1990: 83]). It is a right angled triangle with sides in the ratio 1 : Öf : f. If you write equations for a right-angled triangle with sides in geometric progression, you see that this is the only such triangle. (Specifically, this requires the sides of the triangle to satisfy the Pythagorean Theorem, a² + b² = c², as well as the geometric progression a/b = b/c. Since the scale of the triangle is inconsequential, we can take a=1 for convenience, and then solving simultaneously yields b = Öf and c= f.)

Figure 24

Although you can see a set of spiralling triangles, there is not the symmetry of the standard Golden Section triangles, and thus creating a spiral using circular arcs is not possible.

Double spirals from triangles
The double spiral of equilateral triangles in figure 25 below has been drawn using Fibonacci numbers, but the Golden Section can be used directly just as easily. If the drawing were created using the Golden Section, then the repeating parallelogram has sides that are in the Golden ratio. A long side of the parallelogram is divided in the ratio of the square of the Golden ratio.

Figure 25

A pair of spirals can be drawn using arcs of circles. The arcs used to create the spiral are drawn as follows. Consider the line on which an arc is drawn as the side of an equilateral triangle. Then the arc is part of the circumcircle of that triangle.

Figure 26

Because triangles tile the plane very easily, it can be adapted to use other triangles, for example with Golden Section isosceles triangles it looks like this:

Figure 27

SPIRALS AND THE PENTAGON
The pentagon can be treated in a similar way. For example, the centre of the largest arc in figure 28 is the point marked C.

Figure 28

There are a number of spirals that can be drawn in this way. The set of ten spirals is very much like natural forms composed of Golden Section spirals as you would see in the seeds of a sunflower, for example.

Figure 29