the Fibonacci sequence and golden section
Because my undergraduate degree was civil engineering, I studied mathematics at university to second year. My great regret is that there was so little emphasis on theory of numbers and so very much time spent on calculus. The emphasis on calculus meant that I spent most of my time going through the motions, as it was never really my cup of tea (and I have never used any of it in practice). I knew about the Fibonacci Sequence, but never had the opportunity to play with it – until now. This chapter has been the most intriguing experience it is possible to imagine with paper, scissors and glue.
One comment I will make at this stage, before I get into my samples, is about scale. I worked at three different scales in this module: 1 unit to ½ inch, 1 unit to 1 centimetre, and 1 unit to ½ centimetre. I wouldn’t usually work in inches, but knew that I would be spending some time in Sydney, and the cutting mat I have there, for some reason, is marked in inches (it is probably American – I wish they’d learn to love SI units!). As I worked through, I realised that, working at 1 unit to ½ inch, I wouldn’t be able to fit a 21 cm square on an A4 sheet, whereas I could do so at 1 unit to 1 cm. When I returned from Sydney, I began working at this more conventional scale – until my samples began running off the page. Some of the images below have been photographed instead of scanned, because the samples were A3 or larger format. I did try stitching these together first in Micrografx Picture Publisher with varying degrees of success – after a while I gave up and took photographs instead. This was immeasurably easier than it has been in the past because my lovely sister gave me a tripod for my birthday. One of the larger samples I made again at a different scale. Both versions appear below. And so to the samples, as Samuel Pepys might have said (but almost certainly never did) .
Designing with the Fibonacci Sequence
I tried the first three exercises with two slightly different tonal columns, one with four and one with five gradations of tone. Image 1 shows the version with five different papers.
Image 2 shows this arrangement, with the pattern cut into strips with widths in the Fibonacci series. I wanted to make a version with five strips, so I made another arrangement with four gradations of tone in landscape format. This sample is shown in image 3.
This one reminds me of an abstract landscape. Image 4 shows the same series of papers arranged with every second strip reversed, and in mirror image.
Image 5 shows my plain and patterned papers with series of strips arranged alternately, working from either end.
Image 6 then shows this design cut at right angles into Fibonacci series strips, glued with gaps in the Fibonacci series of proportions.
Image 7 shows the same design as image 6, with no gaps, and alternate strips flipped.
The block in image 8 incorporates five strips in tonal order, with the block cut in half and joined ‘head-to-head’.
Image 9 shows this design cut into strips, and the alternate strips offset. This is one of my favourites.
The sample in image 10 is where the scale issue really began to bite. Using a scale of 1 unit to 1 cm, the strips ended up absurdly leggy. The sample is about 40 cm square.
I decided to make this again, with a scale of 1 unit to ½ cm and strips 2 cm wide instead of 1 cm. The resulting sample is shown in image 11. It’s far easier to get the visual impact of playing with Fibonacci in this second version, I think.
I also made two different samples with diagonal strips. The first one was made from the arrangement in the image 12 swatch. I made a 19 cm square, which I cut diagonally into 2 cm strips, then arranged the strips to form a 23 cm square by offsetting alternate strips along the diagonal. This looked interesting, but not interesting enough, so I then swapped alternate strips with their opposite counterparts from the far corner of the design (image 13). The square which appears in centre of the design is a 15 cm square.
The second diagonal sample is based on the image 14 swatch. I made this one by cutting a 21 cm square along the diagonal with a 1 cm strip in the centre, then 2 cm, 3 cm and 5 cm strips working outwards. I swapped alternate strips with their counterparts from the opposite side of the design, then offset alternate strips by the width of the narrowest band of the original arrangement (0.5 cm) to create a fractured design (image 15).
Designing with the Golden Section
Image 16 shows my drawing on squared paper of the golden section.
The ratio of each Fibonacci number to the previous number converges towards a number variously identified as the golden ratio, golden number, or f, which has an approximate value of 1.618. The golden spiral (shown in red on the drawing), or Fibonacci spiral, is a logarithmic spiral which expands the width of the spiral by f each 90 degrees. In addition to its frequent occurrence in nature, the golden ratio has been important in art (e.g. in its use by Salvador Dali) and architecture (e.g. in its use by Le Corbusier, in the United Nations building, for instance) because of its pleasing proportions.
My attempt at a tonal design using the formula of the Golden Section appears in image 17. As this is based upon a 1 unit to ½ inch scale, the largest square measures 13 units.
I made a second sample based upon the Golden Section, in which the squares are divided in half diagonally to reproduce the approximation of the spiral (image 18). I dithered over this for a while trying to work out where to place the lighter and darker toned triangles to produce a tonal spiral, and decided in the end to adopt the approach whereby if I were walking along the spiral to the centre of the design, the darker triangles would always be on my right.
I did make a couple of other variations based upon shapes other than parallel-sided strips. The design in image 19 is a tonal column of isosceles triangles, each with equal base lengths (10 cm) and with heights of successive numbers in the Fibonacci Sequence.
The other sample (image 20) is made from trapezia each with the same width but with heights at each end as successive numbers in the Fibonacci Sequence (so 0 and 1, 1 and 1, 1 and 2, 2 and 3 and so on). When stacked, these shapes form a curved design and the curvature may become more pronounced as more shapes are added.
I must say that I’ve tremendously enjoyed my creative foray into number theory. I’ll have this chapter in mind when I come to my major work for this module.