Having been able to compare the different Golden Section spirals in Part 3, this section looks at the most enduring myth of Golden Section spirals: because a spiral is a logarithmic spiral it is a Golden Section one.

"A LITTLE KNOWLEDGE IS A DANGEROUS THING"
Using the mathematics of the Golden Section spirals in part 3, and the description of spirals in part 1, I now want to show that people with a slight knowledge of mathematics can make leaps of deduction, which can then be perpetuated as myths.

The simplest case of a mistake was one I heard on the radio recently by an artist who had been brought into one of a series of programmes on numbers. In the one on the Golden Section he went on to describe the spiral in the Golden Section rectangle. This is not an easy thing to do on the radio. It was made harder that his train of thought was:

1. The Greeks were the geometers who knew about and discovered the Golden Section.
2. Archimedes is a well-known Greek geometer who has a spiral named after him.
3. Therefore the spiral is a spiral of Archimedes.

This is quite rare. The most common misconception follows the logic:

1. Spirals of shells, particularly the nautilus are logarithmic spirals.
2. The Golden Section can be used to draw a logarithmic spiral.
3. Therefore shell spirals are related to the Golden Section and in particular, the nautilus shell spiral is a Golden Section spiral.

I will also describe how a spiral design extrapolated from the rose window of Chartres cathedral has been misattributed in the same way.

One of the amazing things about such misconceptions is that it is so widespread, even by mathematicians who should know better. It is a prime example of why geometry needs to be taught more widely and not only geometry, but the visual appreciation of shape and proportion. This is especially true for the education of artists, graphic designers and architects.

I can explain one way the errors about the Golden Section spiral are perpetuated. This anecdote affected me personally. I wrote an article on Golden Section spirals for a mathematics magazine a few years ago that included a section on why the nautilus shell was not a Golden Section spiral. The page proofs came back with a heading that was quite bland. I was horrified, however, when I received my sample copy. The designer had drawn a nautilus shell on the cover and put the same drawing at the heading of the article. The editorial in the next issue had an interesting apology and retraction. However, readers do not always see the next issue.

WHY THE NAUTILUS SHELL IS NOT A GOLDEN SECTION SPIRAL
The mathematics of the Golden Section spirals in part 3 allows the spirals to be quantified. We have seen that this has enabled us to say that each one is different. They are all logarithmic/equiangular spirals, but the tangent angle a is different.

The nautilus shell is a logarithmic spiral. Such a shape arises because a growing animal has the same proportions as it grows and the spiral fits the requirement to protect this shape as it gets larger.

When trying to measure a nautilus shell to determine the shape of the spiral, you can either work from a photograph or a sectioned specimen, which is essentially equivalent (though photos may add distortions, of course). In either case, there are numerous experimental difficulties. For example:

• the section may not be exactly in the right direction, that is it might not be in the right plane;
• the thickness of the shell means that there is a considerable error in deciding where to take the measurement;
• finding the centre (pole) of the spiral is not always easy.

The specimen in Figure 37 was measured as follows.

Figure 37

In order to approximate the multiplication factor for the nautilus logarithmic spiral, measurements were taken for four different 360° rotations of the spiral and the ratio of the radial vectors calculated for each rotation. The yellow marks show the measurement lines. Values obtained were 2.95, 3.02, 2.83, and 2.97, giving an average of 2.94. Although the errors are quite high it shows a value in the region of 3. Other measurements I have seen are also around 3. Since the shell is a living form, making statements other than this is as far as you can go. The creatures are no more uniformly shaped than you or I are.

A logarithmic spiral with a multiplication factor of 3 has a tangent angle of 80.08°. Compare this with the following table:

Note that a small change in tangent angle corresponds to a large change in the multiplication factor. So although the nautilus ratio is close to the SSL ratio, this is merely a coincidence. It is a very long way from the one for the Golden Section rectangle. In fact, comparing the Golden Section spiral (on the left in Figure 38 below) with the logarithmic spiral having a multiplication factor of 3 (on the right in Figure 38 below) and the nautilus in Figure 37, it is clear that the Golden Section rectangular spiral and the nautilus spiral simply do not match. There just are not enough turns with the Golden Section spiral.

Figure 38

Of course, one could specify any multiplication factor and use it to define a specific logarithmic spiral. In this way, one could define other Golden Section spirals, without appealing to any approximate spiral construction, simply by specifying the multiplication factor to be a chosen power of f (e.g., f, f2, f1/2, … - the possibilities are endless). With a suitable choice of the power, one could even produce a spiral very close to the nautilus spiral. This would be quite contrived, however, and it must be stressed that the spiral traditionally associated with the nautilus is the one corresponding to the Golden Section rectangle (i.e., multiplication factor f4), which is clearly far from a match.

There is a much longer and more detailed discussion on this subject in [Fonseca 1993]. He also finds the ratio for one turn of the nautilus as very close to 3. He describes how an artist makes errors in order to justify her assumption that the nautilus is a Golden Section spiral. It is a classic description of how the Golden Section is misused.

THE NORTH WINDOW OF CHARTRES
In his excellent Rose Windows [Cowen 1979], Painton Cowen superimposes a geometrical diagram over the north window of Chartres cathedral and in doing so makes the leap that since the Golden Section can be used to draw a logarithmic spiral, this is a Golden Section spiral.

Figure 39

The diagram Cowen uses is shown in Figure 39. Such a set of spirals bears a superficial resemblance to the sets of pentagon spirals shown in Figure 29. He has probably made the visual leap having seen a diagram of the spirals in a sunflower since it looks so close.

The overall spiral pattern has a twelve-fold symmetry that matches the symmetry of the window, which reflects the twelve apostles. He has drawn many auxiliary spirals like the one shown above, as well. The auxiliary spiral has a multiplication factor of (Ö2)8 = 16, which leads to a » 66.1895°, whereas the spirals of the overall pattern have multiplication factor (Ö2)24 = 4096, which leads to a » 37.0672°. There is no reason to call either of these spirals a Golden Section spiral.

REFERENCES
Cowen, Painton. 1979. Rose Windows. London: Thames and Hudson.

Fonseca, Rory. 1993. Shape and Order in Organic Nature: The Nautilus Pompilius. Leonardo 26: 201-204.

Hambidge, Jay. 1926. The Elements of Dynamic Symmetry. Rpt. New York: Dover, 1953.

Lawrence, J. Dennis. 1972. A Catalog of Special Plane Curves. New York: Dover.

Lockwood, E. H. 1967. A Book of Curves. Cambridge: Cambridge University Press.

Maor, Eli. 1994. e: The story of a number. Princeton: Princeton University Press.

Schattschneider, Doris. 1990. M C Escher: Visions of Symmetry. New York: W H Freeman.

Sharp, John. 1997. Golden section spirals. Mathematics in School 26, 5: 8-12.

Sharp, John. 1999. The Brocard point: A response to a Challenge. Micromath 15, 3. Republished on the Association of Teachers of Mathematics website (http://www.atm.org.uk/resources/articles/geometry/brocard/point.html)

Yates, Robert C. 1947. Curves and Their Properties. Rpt. Washington: National Council of Teachers of Mathematics, 1974.