A Universe Connected
In the year 1202 AD, a mathematician from Italy asked a very simple question: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” A question notably found in the third section of a publication called Liber Abaci which was written by Leonardo Pisano Fibonacci. The above question led to a sequence of numbers that the Western world knows as the “Fibonacci Sequence”. Although most accredit Fibonacci with the discovery of this numerical sequence, its existence had been observed long before Leonardo’s lifetime.
With each turn of a Rubik’s cube creates a permutation (rearrangement of color tiles)The study of permutations and combinations has long been a subject of research in areas of South Asia, particularly in India. Variations in numerical arrangements became popular in ancient Indian society and applied to astrology, medicine, music, and architecture. The first notable hint of this sequence’s existence came around 200 BC through the work of an ancient Indian mathematician named Pingala through his work with Chandaḥśāstra (metrical sciences). Following the work of Pingala, various other Indian mathematicians such as Virahanka, Gopala, and Hemachandra contributed work that gave evidence of the existence of this numerical anomaly. Not to discredit the brilliant work of Fibonacci but
his most recognized contribution to mathematics, not necessarily his most significant, led to the discovery of a ratio that has scientists in most practices mystified. This ratio, as observed numerically, appears to be ingrained within the fabric of universal order. Before going into too much detail about this ratio, it is essential first to understand how the Fibonacci sequence works and how through this sequence the ratio was discovered.
One of the most consistent and elusive numbers ever discovered is 1.161803398874989.
The Golden Ratio, as it is known to those seeking to discern the underlying architecture of the cosmos, is
a ratio apprised in quantum physics and in the overall construct of galaxies. This ratio has been applied to and observed in geometry, nature, and architecture.
The square labeled “2” has an area double of the 2 squares labeled “1”. At this point, the ratio between the sizes does not equal that of the Golden Ratio (Ratio 2:1 or 2), however, as the tiles grow larger by building on the base predicated by the previous two larger squares, the ratio approaches and will eventually equal that of the Golden Ratio. For instance: the area of the largest square of the image above is 21 units X 21 units or 21 units². The next largest squares have an area of 13 units² and 8 units² after that. Both the squares labeled 13 and 8 have a base that’s’ sum equals 21. Mathematicians have added additional geometry to the square Fibonacci tiling to create a spiral pattern. This is known as the Fibonacci spiral: illustrated below.
The Fibonacci spiral is incorporated into a tile pattern based on the sequence. As previously stated and observed in Table 1, ratios calculated from smaller integers of the Fibonacci sequence do not exhibit the Golden Ratio or Phi (φ)(Phi is the Greek letter denoting 1.61803…). As the numbers become larger throughout the calculation, the more a Fibonacci spiral resembles the Golden Spiral. A Golden spiral is based on geometrical square tiling similar to figure 1 except all correlating tiles are designed to have a ratio of φ (Phi) as seen below.
Sequence 1: a self-similar (mathematically identical continuous pattern) illustration of the Golden Spiral. Sequence 1 is a Golden spiral continuously winding through Golden-square tiles. This also resembles a Fib spiral at larger values as they approach affinity. Sequences, tiles, and spirals may all exhibit φ, but there are plenty of different ways to observe this ratio.
The golden ratio isn’t just a number or proportion; it is a reminder of the inherent connectivity that exists within our universe. The existence of the Fibonacci sequence and Golden ratio preceded mathematics as it can be observed in nature within human DNA and even within the anatomical structure of certain flowers. Now that some fundamentals have been discussed, let’s dive into a few real-world examples.
Although mathematics has deep-seeded roots in human history, DNA and mammalian procreation preceded human existence and therefore arithmetic. In 2005, a team of genealogists led by Luke Hutchison stumbled across some interesting results. In their research article entitled “Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships” Hutchinson et al. retroactively studied the number of ancestors that could potentially be the source of specific genes on a particular, heritable X chromosome (sex chromosome carried by both parents). This study used advanced genomic technology to discover what Leonardo Pisano Fibonacci had centuries ago, that in a controlled environment the Fibonacci sequence and Phi may be witnessed within aspects of mammalian lineage. This is one example of how the Fibonacci sequence appears to shape natural order, but another biological example is found in the anatomy of some flowers.
The disk florets of sunflowers have an arrangement that illustrates a spiral pattern that has Fibonacci undertones.
In sunflowers, the spirals that are created within the disk florets’ pattern (reproductive buds at the center of the flower) often have 34 spiral turning clockwise and 21 moving counter clockwise. Another Fibonacci anomaly may be found in the number of petals on certain flowers. For instance, Lillies have 3 petals, Roses have 8 interior plus 5 exterior petals, and Pyrethrum have 34 petals: all Fibonacci numbers. Even the way plants branch from their trunk follow a Fibonacci pattern. Although there is an exact correlation between these numbers and the Golden Ratio, because these numbers are too low to give the precise value of Phi, these patterns may be dictated by some underlying force exhibited by Phi. This force appears to manifest itself within the human psyche as evidenced by ancient architects use of dimensions inspired by the Fibonacci sequence Golden Ratio.
