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Some believe nature is unpredictable and arbitrary; natural structures and wildlife all grow without rhyme or reason. This, however, is not the case. Though some aspects of nature are dependent on randomized probability, many patterns in nature can be replicated and explained through mathematical concepts.

## Hexagons

Hexagons are seen throughout nature and are most well known for their usage in beehives. This shape is so common because of its stability and efficiency. Hexagons are able to fill a flat plane with no wasted space, maximizing the area for bees to store honey. Hexagons also have minimal surface area compared to their volume so bees can use the least amount of resources to build the most amount of space. Hexagons have a mostly equal distribution of surface tension, so they can withstand great pressure without collapsing.

## Fractals

Fractals are an even more common occurrence in nature than hexagons, seen in trees, rivers, and flowers. Fractals are one of the only mathematical concepts that create complex shapes which replicate natural structures; shapes like perfect circles and rectangles are rarely seen in nature. Fractals can show branching and randomness, which make them similar to the way many plants naturally grow. They are also able to describe the seemingly arbitrary structure of mountains and coastlines.

## Fibonacci Sequence and Golden Ratio

The Fibonacci Sequence is a series of numbers in which each number is equal to the sum of two previous numbers, and the first two numbers are both 1. The ratio of the numbers in the series approaches the Golden Ratio as the numbers increase. The Golden Ratio is also known as “φ” the Greek letter phi, which is approximately 1.618034. The Fibonacci Sequence and the Golden Ratio are so important because they are seen everywhere in nature, most notably in plant growth.

Fibonacci numbers and the Golden Ratio are used to make spirals seen in many petals, leaves and spiral shells. These spirals can be formed with squares with length equal to Fibonacci number or as a logarithmic spiral with a growth factor of the Golden Ratio. Furthermore, the Golden Angle, formed by two explementary angles whose ratio is the Golden Ratio, is the angle at which petals and leaves grow from one another.

Leaves grow at this specific angle because it maximizes the amount of sunlight that leaves are able to absorb. Alternatively, if leaves grew at an angle of 360 degrees from one another rather than the Golden Angle, this would minimize the amount of sunlight each leaf absorbs, because each leaf would be covered by the next.

There are many applications of math in the natural world and many mathematical explanations for natural growth. However, many natural patterns are still a mystery waiting to be solved. As always, stay curious!

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