Golden Section SpiralsAuthor(s): John SharpSource: Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 8-12Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215323 .

Accessed: 07/04/2014 11:59

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions

GOLDEN SECTION

spirals

by John Sharp

The Golden Section is a fascinating topic which is continually throwing up new ideas. It has a mythical status which leads many people to assume its presence when it has no relation to a problem. Sometimes this blindness to other alternatives is due to mathematical ignorance and sometimes to the use of intuition rather than logic. The following discussion of Golden Section spirals not only contains some interesting mathematics and new ideas but is also useful for teaching that a simple understanding of mathematical properties can destroy old myths. The Golden Section has the value 1.6180399 ... and is given the symbol 4.

The Golden Section rectangle and the standard spiral The Golden Section rectangle is often used to illustrate spiral similarity. It is easy to draw using an approximation with Fibonacci numbers and provides a means to draw a quite accurate logarithmic spiral quite quickly. This spiral is the source of many myths since non-mathematicians, and indeed some notable mathematicians, have made the error of think- ing that just because a spiral is logarithmic it is related to the Golden Section. But more of that later. It is only one of a number of possible spirals that can be drawn using the Golden Section rectangle and I would like to introduce a new additional one.

To draw the standard approximation to the Golden Sec- tion spiral using circles, draw a Golden Section rectangle and create the so called 'whirling squares' as follows. [Note: This name was given by the art historian Jay Hambidge in Ham-

bidge (1967) whose books are a mixture of myth and fact about the Golden Section.]

Cut off a square from one end of the rectangle thus giving another Golden Section rectangle. Continue in the same way with the resulting rectangle to yield another Golden Section rectangle.

Draw many more squares iteratively to give the whirling spiral of squares as shown in Fig 1. The ratio of the lengths of the sides of successive squares are in the Golden Section.

Fig. 1 Golden Section rectangle with squares

You can now draw the spiral as a series of circle quadrants as shown in Fig 2. The centre of each are is the comer of a square and the radius is the side of a square.

8 Mathematics in School, November 1997

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions

Fig. 2 Golden Section rectangle with circles

Although the curve is made of arcs of circles, it is a smooth curve. The reason for this is that successive circles have common tangents. Many curves can be approximated by a series of arcs if the centres of the two arcs pass through their common point, so that they have the same line as tangent at this point. Figure 3 shows how arcs of two circles have been made into an apparent single curve.

Fig. 3 Two circles, common tangent line joining centres

Finding the true spiral The spiral is like a logarithmic spiral since each rotation of 9030214666 means the radius of the circle is multiplied by the Golden Section. The centre of rotation is found from the whirling squares diagram as shown in Fig 4 by intersecting diagonals of the Golden Section rectangles. These two diagonals are at right angles.

Fig. 4 Diagonals

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. Knowing this centre, to find the true spiral for which we have an approximation made up of arcs of circles, there are now two properties which can be used to find its equation and allow us to define it:

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.

Mathematics in School, November 1997

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

The logarithmic or equilateral spiral Whole books could, and have been, written on this spiral. The essential point we need to know is its polar equation:

r = aeecot(a)

The constant a defines the starting radius vector when the angle is zero. The constant cot(a) defines the angle between the radius vector (the line from the pole to a point on the curve) and the tangent as shown in Fig 5.

30214666a

Fig. 5 Arc of a spiral showing angle

If we know the radius at two positions we can determine the value of the angle a and thus define the equation.

Adding some more lines to Fig 4 to create the radial vectors to points D, P, F and G of the Golden Section spiral gives us Fig 6.

A E

B

D

O F

P G JC

Fig. 6 Labelled rectangle with diagonals

Triangles FOE and EOD are similar; they are right angle triangles with hypotenuses in the ratio 4, and so

EO ED FO EF

thus the radii after 9030214666 degree rotations are in the ratio of the Golden Section. This means that if we draw a logarithmic spiral through these pointsy then

r = aeocot(a) and gr = ae(30214666 +

x/2)cot(a)

which after division and taking logs, gives

cot(a) = 2 In(4)/t and hence a has the value 72.9676.

.... degrees.

9

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions

The spiral through the points

Using the equation of the spiral, with pole at the centre through the points where the squares divide the sides of the rectangle, gives the spiral shown in Figure 7.

Fig. 7 True spiral through points

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 one important differ- ence is that the logarithmic spiral goes outside the rectangle, although only slightly. This is sometimes, but rarely, men- tioned in descriptions of this diagram, but I have never seen either a calculation, or description, of how much. Since the value is very small (only 0.165% of the shortest side of the rectangle) it does not show up without high magnification. In Figure 7 there is a line sticking out of the top of the rectangle. This is the position where the spiral cuts the rectangle when it returns inside. So although the rectangle goes through the points, it does not fit neatly into the rectangle.

The spiral to touch the rectangle side If the spiral were to touch the sides of the rectangle, the line from the pole would need to make an angle of 72.9676a with the side of the rectangle. If we 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:

A T E BI

D

O

F

P GR 1C

Fig. 8 General radial lines

The right angle triangles POC, COB and BOA 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

10

formed by rotating the lines DF and GE, then triangles ROC, SOB and TOA and the triangle formed from the point where BP extended cuts AD produced, together with points O and U, are all similar so that sides OR, OS, OT and OU are successively in the Golden Section.

This means that the spiral which touches the four sides of the rectangle is the same one as the one in Figure 7, except that it is rotated slightly, so that it touches a little way along. The touching point (the point equivalent to point U in Figure 8) is 8.228% of the short side of the rectangle (that is, the ratio of DU to AD). The spiral then looks like this:

Fig. 9 Spiral correctly touching

The myth of the Nautilus shell Many books say categorically that the Nautilus shell cross section is a Golden Section spiral. One might expect non- mathematicians to make the mistake, seeing two spirals and not looking too closely. One does not expect the myth to be perpetrated in mathematics books and articles like Devlin (1995) and Stewart (1996). Comparing the following pair of spirals, the Golden Section spiral (on the left) visually with the Nautilus spiral (Fig 10) shows that the latter is much more tightly wound.

Fig. 10 Separated Golden Section spiral with Nautilus spiral on right

Measuring the Nautilus spiral gives a factor of about 3 for a rotation of 3600 (and so a tangent angle a of 80.080) which compares very unfavourably with the one for the Golden Section. There is a very good description of how to measure the values for a spiral in Land (1975) and there are pictures of cross sections or X-rayed Nautilus shells in Lockwood (1961) and Huntley (1970). Care has to be taken to use a diagram or photo with the angle of slice made perpendicu- larly, otherwise the value you obtain will not be correct. Sliced shells are also often not sliced through the centre of the shell and slices made in this way have different spirals depending on the curvature of the shell.

The Golden Section 'wobbly' spiral

It is easy to draw the approximate spirals on the computer, especially with CAD packages like AutoSketch, since it is

Mathematics in School, November 1997

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions

much easier to get accurate lines and arcs. When you draw an arc, you specify points such as the centre of the are and its ends. If you are drawing by hand, you know which way to draw the arc, but the computer sometimes makes a different guess. This leads to some unexpected results. In drawing the Golden Section spiral this way, I found the computer drew an are that was three quarters of a circle instead of the quarter I wanted. Interesting discoveries often come from such mis- takes, so I continued. From a single arc drawn in this way (Fig 11) the spiral on the Golden Section rectangle (Fig 12) develops into a completely different one. Separated from the rectangle (Fig 14 (left)) it looks like a spiral with a wobble.

Fig. 11 Golden Section rectangle with one 3/4 circle

Fig. 12 Golden Section rectangle with many 3/4 circles

Finding the equation of this spiral is an interesting exer- cise. Its centre is the same as the other spiral, at the intersec- tion of the diagonals (Fig 4). By using AutoSketch, I was able to create a series of radial vectors for equal angles over a 27030214666 range, and measure them (Table 1).

Table 1 45.9263, 44.1104, 42.9897, 42.2449, 41.5998, 41.7322, 42.4762, 43.3443, 44.5501, 45.9071, 48.0509, 50.6627, 53.2578, 56.7144, 60.1084, 63.8455, 67.6245, 71.1758, 74.4776, 77.4112, 79.1833, 80.3255, 81.7013, 81.8287, 80.9143, 79.3476, 77.0734, 74.3392

I could now plot these as a cartesian graph, that is, I unrolled the curve. The result was Fig 13a.

Mathematics in School, November 1997

Fig. 13a Plot of angle vs radii

Fig. 13b Standard exponential plot

Fig. 13c Subtracted plot

Fig. 13d Damped sine wave

This looked like an exponential curve, but with something added, so I tried plotting an exponential curve as well (Fig 13b), that is

y = aeek

I knew the angles, the initial value of a and its final value, so I could calculate k and obtain the full equation. Because the spiral goes through points of the golden section rectangle, and just like this could be thought of as the reverse of the one described above, the radius increases by a factor of # over an angle of 2700,

k = In(4)/(37t/2)

Then by subtracting the exponential curve, I obtained the curve shown in Fig 13c. This looks like a sine wave, but the amplitude is varying, that is, it is a damped sine wave. Also one wave is complete in a 2700 range, so its equation must be of the form:

y = F(0) sin(4 0/3) where F(0) is the damping function which must be an expo- nential function, most likely the one defining the spiral, so I tried the function as

y = qeek (1 + sin(4 0/3))

and tried varying the constant q until it matched the curved as closely as possible. This is an empirical approach, which adds a practical aspect to the problem. It has an analogy with the way some problems are pure mathematics and some are scientific; not quite pure and applied mathematics but nearly SO.

My guess was that the value of q was 2k which gave the best fit. Plotted with the wave in Figure 13c, this is the darker line in Figure 13d. Since the approximate curve using circles is somewhat artificial, the result relies on intuition as much as mathematics, as there will be no perfect fit. I could then recreate the wobbly spiral with the equation:

r = (1 + 2ksin(40/3)) aeek

which when plotted gave a curve (Figure 14 right) looking very similar to the arc approximation in Figure 12 shown at the left of Figure 14.

11

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions

Fig. 14 Circle version and plotted wobbly spiral

Spiral designs

The Golden Section is linked to the way plants grow with the Fibonacci series playing an important part in the description of phyllotaxis, for example in Huntley (1970). This could take up a whole book, never mind a few pages here. The Golden Section wobbly spiral may have a part to play here in that combining the spiral in a series of rotational repeats can generate flower and plant-like designs which are more com- plex than those from the conventional spirals seen in the centre of sunflowers. A few examples are shown in Figure 15 to conclude. M

Fig. 15 Flower effects

References Devlin, K. (1995) Mathematics, The Science of Patterns, W. H. Freeman Hambidge, J. (1967) The Elements of Dynamic Symmetry Dover Huntley, H. E. (1970) The Divine Proportion, Dover Land, F. Mathematics (1975) The Language of Mathematics, John Murray Lockwood, E. H. (1961) A book of curves, CUP Stewart, I. Mathematical Recreations, Scientific American, June 1996, p 92

(1996)

Author John Sharp, 20 The Glebe, Watford, Herts, WD2 6LR e-mail: 101503.432@compuserve.com

"Tumbling Blocks Sweater" pattern by Kaafe Fasset - September 1997. Photo: Kirsteen Hinds

12 Mathematics in School, November 1997

This content downloaded from 63.133.201.250 on Mon, 7 Apr 2014 11:59:53 AMAll use subject to JSTOR Terms and Conditions