 It all started with a pair of slightly unbelievable baby rabbits, a baby boy rabbit and a baby girl rabbit. They were fully grown after one month and, being rabbits, they did what rabbits do best, so that the next month two more baby rabbits (again a boy and a girl) were born. The next month these babies were fully grown and the first pair had two more baby rabbits (again, handily a boy and a girl). Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature. This highly unrealistic situation (where rabbits never die and every month each adult pair produces a mixed pair of baby rabbits, who go on to mate the next month) was studied in the thirteenth century by Leonardo Pisano, better known as Fibonacci. He posed the problem “how many rabbits can a single pair produce after a year?” Fibonacci realised that the number of adult pairs in a given month was the total number of rabbits (both adults and babies) in the previous month

Adultsnth month = Rabbits(n-1)th month

and the number of baby pairs in a given month was the number of adult pairs in the previous month

Babiesnth month = Adults(n-1)th month = Rabbits(n-2)th month,

so the total number of pairs of rabbits in a particular month was the sum of the total pairs of rabbits in the previous two months:

Rabbitsnth month = Adultsnth month + Babiesnth month= Rabbits(n-1)th month + Rabbits(n-2)th month.

So following this through for a year we get the answer to Fibonacci’s question: 144 pairs of rabbits after a year.

1 pair in January
1 pair in February
2 pairs in March
3 pairs in April
5 pairs in May
8 pairs in June
13 pairs in July
21 pairs in August
34 pairs in September
55 pairs in October
89 pairs in November
144 pairs in December

The first nine of these – 1, 1, 2, 3, 5, 8, 13, 21, 34 – are the sequence of numbers you see on your cube. The sequence bares Fibonacci’s name after he published it in his great work, the Liber Abaci, in 1202. This book changed the face of western mathematics, not because of the Fibonacci numbers, but because it introduced the now familiar numerals 0,1,2,…,9 and demonstrated how easy they made calculations, compared with the cumbersome Roman numerals that were still being used then by mathematicians and merchants.

The sequence has many curious mathematical features. For example, every second Fibonacci number, starting with 5, is the long side of a right-angled triangle whose side lengths are whole numbers: 5 is the longest side of a triangle whose other sides are 3 and 4 (52=32+42) and 13 is the longest side of a triangle whose other sides are 5 and 12 (132=52+122) and so on.

The Fibonacci sequence appears in many places in nature, for example we can find its numbers in the turns of a shell. Start with two squares side by side, each with sides of length 1, for the first two numbers in the sequence. Then you can add a square with sides of length 2 and one with sides length 3 and so on. And if we inscribe quarter circles in each of these squares they build up into a spiral shape, not a perfect mathematical spiral but one very close to some of the spirals we see in nature, such as in nautilus shells, that are built up compartment by compartment.

Fibonacci numbers also appear in the spirals of seed heads on many plants. If you look closely at a sunflower you’ll notice the seeds appear in spirals, both clockwise and anticlockwise. And if you count the number of spirals in each direction, whether at the edge of the seedhead or closer in the middle, you are almost certain to find that they will be a pair of successive Fibonacci numbers.

This is because the ratio of successive Fibonacci numbers, where each is divided by the number before it (1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666…, 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538…), gradually approach a number known as the golden ratio. This number, approximately 1.618304, is an irrational number (it can’t be expressed as a simple fraction a/b where a and b are whole numbers), like pi and e on the other faces of your cube. Every irrational number can be better and better approximated by a sequence of fractions and mathematicians have found a way of measuring how well this approximation works. In this sense, the golden ratio is in fact the “worst approximable” number. This stems from the fact that you can represent it as the infinite continued fraction 1+1/(1+1/(1+1/(…) and is the reason behind the appearance of the golden ratio and Fibonacci numbers in plants. Plants produce their leaves and seeds from a growth tip that spirals around the plant or centre of the seed head as it goes. If it turned by something close to a simple fraction of a turn the seeds would soon line up creating loosely packed spirals of seeds. The most efficient way to pack seeds into the seed head is for the number of turns between seeds to be an irrational number that is not easily approximated by a fraction, such as the golden ratio, avoiding the widely spaced spiralling arms. Instead you get the closely packed Fibonacci spirals of seeds, with the number of clockwise and anticlockwise spirals at any point on the seed head a pair of successive Fibonacci numbers, as these approximate the golden ratio.