The approximate methods, described in part 2, show the structural way of creating the spirals. This section looks at the mathematics of Golden Section spirals as it relates to the approximate methods using arcs of circles in part 2 and shows how to find the equations of the exact spirals.

MATHEMATICS OF THE TRUE GOLDEN SECTION SPIRALS
The spirals drawn using arcs of circles are approximations, but each one has a corresponding true spiral - in most cases a logarithmic spiral - that can be represented by an equation. Since they are all generated by using properties of the Golden Section, the question arises as to how many different spirals are there and indeed, if they are the same.

The spiral and the Golden Section rectangle
The spiral drawn using quarter circles in the set of whirling squares is a like a logarithmic spiral since each rotation of 90º means the radius of the circle is multiplied by the Golden Section. (Thus, the multiplication factor for a full rotation of 360º is f⁴.) It is not a true logarithmic spiral, however, because each quarter circle of the spiral has a different centre, whereas a logarithmic spiral rotates about a single pole. To analyse the mathematics, we have to go back to the whirling squares diagram and look at its spiral similarity.

Figure 30

As shown in Figure 30, the centre of rotation is found from the whirling squares diagram by intersecting diagonals of the Golden Section rectangles. These two diagonals are at right angles. That the centre is the intersection of successive diagonals of the Golden Section rectangles can be seen by the spiral symmetry of the squares and Golden Section rectangles. Although these diagonals have been drawn for the largest two rectangles, they are also the diagonals of the other Golden Section rectangles.

Knowing this centre, to find the true mathematical spiral for which we have an approximation made up of arcs of circles, there are now two properties that can be used to find its equation:

1. the spiral goes through the point where the arcs of circles meet, that is where the squares are cut off;
2. the spiral is tangent to the side of the rectangle.

These are in fact two problems yielding different results, although, as we shall see, the same spiral.

The essential mathematics of the logarithmic spiral is embodied in its polar equation, previously examined in part 1:

Recall that q is measured in radians, and the constant a defines the angle between the radius r (the line from the pole to a point on the curve) and the tangent, as shown in Figure 31.

Figure 31

If we know the value of the radius at two different values of q, we can determine the value of the tangent angle a and thus define the equation.

Adding some more lines to Figure 30 to create the radial vectors to points D, E, F and G of the Golden Section spiral gives us Figure 32:

Figure 32

Since OE and DF are perpendicular, triangles FOE and EOD are similar; they are right triangles with hypotenuses in the ratio f. Thus

so that the radii after 90º (i.e., p/2 radian) rotations are in the ratio of the Golden Section. This means that if we draw a logarithmic spiral through these points, then

which, after division and taking natural logs, gives

and hence a » 72.9676 degrees.

For future calculations, we note the that the formula for a - given the multiplication factor, M, for a full 360º (i.e., 2p radian) rotation - may be obtained analogously, yielding

in fact, the equation for the spiral, r = aeq cot a, can then be written

While this last equation is simpler, it does not explicitly show a.

Now that we have the equation and we know the pole, we can plot the true spiral through the points where the squares divide the sides of the rectangle, which gives the spiral shown in Figure 33.

Figure 33

This looks remarkably similar to the one from the arcs of circles, and if the two are included in the same diagram then it is difficult to see the lines apart unless the diagram is magnified and the lines are thin.

The short line sticking up at the top of Figure 33 is not a mistake. It shows an important property of the true logarithmic spiral through the points: it goes outside the rectangle, although only slightly. The short line is located at the position where the spiral cuts the rectangle when it returns inside. It is sometimes, but rarely, mentioned in descriptions of this diagram, but I have never seen either a calculation, or description, of how much. The amount it goes out is very small (only 0.165% of the shortest side of the rectangle) and so does not show up without high magnification. The area around the marked point at the top of Figure 33 shown in Figure 34. The arc of the spiral which is outside the rectangle is barely apparent, and not enough for a pencil drawing to show up.

Figure 34

The spiral to touch the rectangle side
We have seen that although the rectangle goes through the points, it does not fit neatly into the rectangle. If the spiral were to touch the sides of the rectangle, the line from the pole would need to make a tangent angle of 72.9676° with the side of the rectangle. So if we take the spiral that goes though the points and rotated it, would it touch all the sides in the same way? It would, because any four radii at right angles from the pole are successively in the ratio of the Golden Section. This may be seen from the following diagram.

Figure 35

(Note that this diagram is for illustrative purposes and is not an accurate representation of how the spiral is placed in Figure 33.)

The right angles triangles POC, COB, BOA and TOU are all similar, with hypotenuses in the ratio of the Golden Section, so the other corresponding sides are also in this ratio. If R, S, T and U are the points of intersection of any four other radii formed by rotating the lines DF and GE, then triangles ROC, SOB, TOA and UOZ (where Z is the intersection point of BP extended with AD extended) are all similar, so that the sides OR, OS, OT and OU are successively in the Golden ratio.

This means that the spiral that touches the four sides of the rectangle is the same one as the one in Figure 32, except that it is rotated slightly, so that it touches a little way along the side and not at the point where the vertex square sits. The touching point (the point equivalent to point U in Figure 35) is 8.228% of the short side of the rectangle (that is the ratio of DU to AB). The spiral in position then looks like this:

Figure 36

The calculation of the angle of rotation is too detailed to include here, but it is slightly over 3.75°.

The triangular and pentagonal spirals
Similar techniques can be used to find the equations of the spirals from the triangles (Figures 20 and 21) and the pentagonal one (Figure 28). Finding the centre is not as obvious and requires knowledge of the centre of spiral similarity of a triangle. This is a special point of a triangle known as the Brocard point; for full details see my explanation [Sharp 1999], available on the Association of Teachers of Mathematics website.

For the Golden Section spiral for the triangle LLS (Figure 20), the radial vector has a ratio of f after a rotation of 108° (and hence f10/3 after a full rotation, since 360° = (10/3) x 108°). This gives the tangent angle a » 75.6788°.

For the Golden Section spiral for the triangle SSL (Figure 21), the radial vector has a ratio of f after a rotation of 144°. This gives the tangent angle a » 79.1609°.

For the pentagon case (Figures 28 and 29), the spiral has a radial vector has a ratio of f 2after a rotation of 108°. This gives the tangent angle a » 62.9520°.

Comparing the tangent angle of 72.9676° for the Golden Section spiral described by the rectangle, this shows that these four spirals, while all defined by the Golden Section, are very different.

THE EQUATION OF THE WOBBLY SPIRAL
The approximate methods for drawing spirals in Part 2 gives rise to some complicated spirals. The only one whose equation is relatively easy to find is the wobbly spiral for the Golden rectangle. Full details for derivation of the equation are given in [Sharp 1997].

The equation is

where k = 2(ln f)/3p.

If you have software for plotting curves, I welcome you to try using this equation.