Back in 2014 I watched a Numberphile video, by my friend Matt Parker, about Lucas Numbers. At the time I used the numbers to make a toilet roll cover which I referred to as Loo-case. I always intended to return to using these numbers but didn’t get round to it until 2018.

Most people are familiar with Fibonacci numbers but Lucas numbers are less well-known. Fibonacci lived about 600 years before Edouard Lucas but Fibonacci numbers did not have a name until Lucas named them. Lucas studied many patterns of numbers that follow the same rules and called them Lucas Sequences. Each term comes from adding the two that went before. Lucas numbers and Fibonacci numbers are both examples of a Lucas sequence.

Fibonacci Numbers
Lucas Numbers

0, 1, 1, 2, 3, 5, 8, 13 …
2, 1, 3, 4, 7, 11, 18, 29 …

For craft purposes Lucas numbers seem to be much more useful than Fibonacci numbers. Firstly, the Fibonacci numbers should really begin with zero though this is quite often omitted, in writing, and when the series is represented in other ways.

Also, beginning knitting with two single rows can be tricky. Starting with two, then adding one tends to be easier.

To be the Lucas series or the Fibonacci series the numbers must be used in the correct order. It is never correct to mix them up. I have seen very many projects labelled as Fibonacci where the numbers 1, 2, 3, 5, 8 … have been used in any order to create stripes making them nothing more than random numbers.

Once I had decided I wanted to make mitred squares using Lucas numbers I looked for a pleasing arrangement. Using the first six terms of the series gives a total of 28 ridges (2+1+3+4+7+11), which makes a square of approximately 15 cm (6”). I thought about using the numbers in reverse, from the outside of the square but decided I preferred the small numbers at the edges.

Using six terms and six colours meant that I could rotate the colours to create six different-looking squares. Six squares made a nice length for a baby blanket. I decided to use five squares across and rotated the order of the squares in each column. I looked at many different arrangements. This was the one I liked best.

One of the advantages of rotating the colours in the squares is that a complete column uses equal amounts of each colour.

I played with the idea of using bigger squares. The next size up would need 46 ridges, which would result in a square of approximately 23 cm (9”). To rotate the colours in the same way as for the baby blanket I would need to make seven different squares. The blanket would have been 160 cm (63”). This would be a reasonable length for an adult blanket. Larger squares would be unwieldy.

Each new number/colour that is added requires more than one and a half times as much yarn as has already been used. The relationship of the numbers in the series tends towards the Golden Ratio (which is written as the Greek letter φ or phi) so the lengths and areas in the Lucas number squares are closely related to the Golden Ratio. Many people believe that is why artists find such shapes appealing.