“How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?” — Albert Einstein

Inthe garden, through a microscope, or deep in the depths of the ocean, patterns are all around us if we know how to look for them. Whether it's humans or nature, mathematics are used to contrive these patterns, orchestrating a fluent and alluring result. To look for these patterns you must have a basic knowledge of the fundamentals of mathematics, which is what this list will help you gain:

The Fibonacci Sequence

Mathematicians have been enthralled by the Fibonacci sequence for centuries. It’s ubiquity in both nature and man-made synthetic structures hints to its significance in both our lives and in the universe.

Here’s a quick refresher on what the Fibonacci sequence is:

The Fibonacci sequence is a series of numbers in which each subsequent number can be found by adding up the two numbers that precede it. It starts like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 24, 34, … and continues on forever.

The Fibonacci sequence was named after the man who came up with it, Leonardo Fibonacci, and the way he calculated it was quite peculiar. Over the course of a year, Fibonacci observed and calculated the ideal reproduction patterns of rabbits. The chart and diagram below provide an explanation and visual as to how the Fibonacci sequence was determined:

Although its discovery dates back to 1202, the Fibonacci sequence can be found from the microscale to the macroscale today. Not every object in the universe possesses it, but here are some prime examples:

Both spirals can be found in nature, and both follow the Fibonacci sequence. The logarithmic spiral, which is another name for the infamous spiral that the Fibonacci sequence emulates, can be seen in these natural spirals that theoretically continue to infinity. An interesting feature of spiral galaxies is that they defy Newtonian physics. According to Newtonian physics, these galaxies should begin to curve as a result of their rotation patterns, and their spiral arms should begin to wind until the galaxy disappears (the winding problem). Interestingly, neither of these phenomena occur: the spiral shape of the galaxy and the logarithmic spiral it mimics, are preserved.

Fractal Branching

Although fractal branching is not as well known as the Fibonacci sequence, it equates in its importance and presence in the universe. Fractal branching is a detailed pattern that is repeated over and over again, resulting in a shape that appears rather complex. These patterns look exactly as you would expect, and its most popular models are tree branches and snowflakes:

The basic process of fractal branching can be described as a beginning prototype that is copied and repeated. Using the example of a tree branch, one branch is split into other branches, that is split into more branches. It is as if two new, almost identical, smaller branches emerge from their origins. And so a tree with many branches can also be seen as an assembly of many smaller, similar-looking trees.

Voronoi Pattern

You may be unfamiliar with what a Voronoi pattern is, but I can assure you you’ve seen them everywhere. A Voronoi pattern is a pattern in which regions have points, and every point is the origin from where the region expands. The expansion of a region ends once it meets its neighbouring expanding region, thus, the radius from the end of each neighbouring region is equal in length to the point from which it expands.

Voronoi patterns can be found in giraffe skin, to cracked mud, to bubbles:

Voronoi patterns are synonymously tessellations in these examples, and appear as connecting pieces of a puzzle. Although they are not labelled, each region has a point from which they expand from, and connect to other regions with points of their own.

Patterns are truly found all around us in nature, all attributable to mathematics. One with a limited mathematical background may not think to look for them, but once a fundamental knowledge is gained, it's hard to not see that they are ubiquitous in the universe.

As Vincent van Gogh once said, “if you truly love nature, you will find beauty everywhere”. If you understand mathematical patterns, this beauty is not hard to find.