Solving for (X)-istential Patterns within Nature

Humans have sought to mathematically describe and characterize the structure, patterns, and evolution of natural organisms and processes for over two millennia. The ancient Greek philosopher Pythagoras (570-495 BCE), who famously developed the Pythagorean theorem, helped to pioneer and disseminate the idea that natural phenomena, or “the cosmos”, could be explained using numbers and numerical relationships. Whether it is the ridged pattern of a pinecone, the ripple effect of a stone dropped in a pond, or the branching patterns of a tree or a neuron…, natural structures, growth patterns, and events are woven into mathematical relationships that help us to clarify and predict elements within a complex world.

Imagine this: You are breeding a pair of rabbits, which are raised in ideal conditions that enable them to give birth to one male and one female offspring. This pair of offspring, in turn, becomes fertile in one month and also gives birth to one offspring of each sex. Consider that with every new pair of rabbits reaching fertility, the older pairs are still breeding. If this cycle continues, how many pairs of rabbits will you end up with in a year (see answer below)?

This was a thought experiment proposed by the mathematician Leonardo Bonacci in the 13^th century. The total number of breeding rabbits at the end of each month corresponds to what is now known as the Fibonacci sequence, a pattern by which each element is the sum of the two immediately preceding numbers. Starting with 1 and 1, as x₁ and x₂, respectively, it continues as 2, 3, 5, 8, 13…∞.

The mathematical equation is written as: x_n+2=x_n+1 + x_n

Numbers within the Fibonacci sequence are commonly observed in nature. One example is the number of petals in flowers: lilies and irises (3 petals), butter cups and wild roses (5 petals), and delphiniums (8 petals). Other plant structures, like sunflower seeds and pinecones, consist of spiral arrangements; the number of seeds in each spiral reflects a number in the Fibonacci sequence. While not a deliberate design, natural organisms may have “adopted” the Fibonacci sequence to maximize occupancy within a given space. For plants, this would mean arrangement of petals and leaves to maximize exposure to sunlight and other nutrients. Nonetheless, flower species exist that do not follow this pattern, such as poppies (4 petals). Clearly, other genetic, environmental, and evolutionary pressures lead to variations.

In addition to the Fibonacci sequence, other geometric shapes and patterns exist in the natural world. Fractal patterns, for one, are a potentially infinite repeating series in which an entire object is made up of smaller units resembling its likeness. Such mathematical patterns are ubiquitous in nature: the arteries of our circulatory system consist of large branching blood vessels, that extend into similar smaller branching units, called capillaries; similarly, ferns and broccoli consist of smaller units that mimic their overall structure. Tessellations, or tiling patterns, are also frequently observed in nature, with notable examples including the repeated hexagon structure of honeycombs, pineapple skin, turtle shell patterns, and fish scale arrangements. With regards to the honeycomb, its hexagonal arrangement permits maximal honey storage in its wells, while minimizing the wax required to coat the perimeter. Mathematicians have recently classified another 3-dimensional tiling pattern of “soft cells”, which lack corners; these tiling patterns are found in multiple biological structures, including nautilus shell chambers and muscle fibers. While the specific advantages of this shape are unclear, this ignites a conversation to understand how organismal adaptation and evolution favour the flexibility and resilience of certain structural patterns over others.

While the mathematical concepts above describe the structure of living things, math in nature extends beyond geometry and shapes. From predicting the transmission of minuscule viruses to planetary motion, math can be used to model natural phenomena. For instance, epidemiologists employ math to trace and predict patterns in infectious disease transmission. Vast quantities of clinical and surveillance data can now be collected from populations to allow for more precise estimation of biological, behavioural, and environmental traits that contribute to infectious disease spread. These data are interpreted using mathematical concepts like pathogen offspring distribution, which is the number of people infected by the original infected person, and generation time distribution, which is the length of time it takes for successive infections to occur from one individual to the next. These models help us to predict the severity of disease outbreaks and define key parameters, such as the number of infections that must occur before an outbreak can no longer be contained without public health intervention.

Beyond human health, scientists employ math to describe a variety of environmental occurrences. Tidal patterns are caused by the gravitational pull of the Sun and Moon on the Earth, in combination with other factors like the Earth’s rotation, the wind, and atmospheric pressure. An alignment of the Sun, relative to the Moon, can create exceptionally high or low tides (also known as “spring tides”). The gravitational pull exerted by either the Sun or the Moon, relative to the Earth, can be represented using an equation that depends on the gravitational constant, G, the mass of the two objects, and the distance between them. By incorporating the gravitational force and other parameters, geographers can use sophisticated mathematical modelling to more accurately predict tidal patterns. Understanding tidal patterns are important in activities such as navigation, fishing industries, and coastal engineering. Given the prevalence of math in the natural world, from infectious disease biology to oceanography, it would seem that Mother Nature was a formidable mathematician!

Many of the mathematical concepts discussed here provide a logical framework to understand how organisms could have evolved for survival. Inspired by nature’s wisdom, these mathematical concepts are used by artists and scientists alike in designing both art and man-made structures. Unravelling the parallels between art, science, and nature, through the incorporation of mathematical patterns, presents hidden beauty, order, and complexity in the universe. Continuing to view nature through a mathematical lens will help to enrich and foster in us a previously untapped appreciation and sense of wonder for the natural world.

Answer: 233 pairs of rabbits after 12 months