The mathematics of sea shells – UnicMinds

The sheer beauty of the seashell’s geometric shape arouses curiosity about the beautiful symmetry of the seashell. To put it simply – expansion, rotation and twisting are the three rules of shells in this world!
Because seashells are about curves, we need to understand the intuition of curves.
Curve intuition
When we look at y=x, this means that it is a straight line. However, y = x^2 starts to change as the value of x increases, and it looks like a curve. Likewise, y = log x is a curve. Essentially, if you have to represent a curve mathematically, you’ll end up using an exponential or logarithmic or parabolic function around that axis. So, what is the intuitive difference between exponential curve and logarithmic curve? – In an exponential curve, the change is slow at first and then the curve changes exponentially faster and faster, whereas in a logarithmic curve, the change is fast at first and then the curve changes more and more slowly. In a parabolic curve, the slope of the curve increases linearly, whereas a logarithmic curve has a slope that decreases more and more with time, and an exponential curve has a slope that increases more and more with time.
Sea shells are logarithmic spirals – fast at first, then slower
The basic pattern for making seashells depends on the logarithmic spiral. In order to understand logarithmic spirals, we first need to understand logarithms – logarithms convert multiplication into addition. Logarithms and exponentials are opposites.
A logarithmic spiral is a spiral like the one shown below and is extremely common in nature. This specific geometry has the evolutionary advantage of adapting to animals of all sizes and growing proportionally as the animal grows. You can also draw a logarithmic spiral within the swirl of a pine cone.
Logarithmic spirals have many interesting properties and show an interesting connection between φ and e. One way to make a logarithmic spiral is to take a golden rectangle and put another golden rectangle inside it, so that the width of the larger rectangle is the length of the smaller rectangle. Now add another rectangle within the smaller rectangle and continue adding rectangles as long as possible. If a spiral were inscribed in this series of rectangles, the spiral would be a logarithmic spiral. The general equation of a logarithmic spiral is r=eθ, and this golden spiral (composed of a golden rectangle) is its transformation.
Graph polar equations such as r=eθ We start with θ equal to zero. From the equation, if we substitute 0 into θ, we get a radius of 1. From this point, we draw a line by increasing the angle theta and letting it determine the distance r of the line we draw from the origin. The above equation is special because a given increase in θ always results in an increase in r in the same proportion.
The general equation for these spirals is r=eaθwhere a is a constant. If a is positive, the spiral will start at (0,1) and grow outward as θ increases. If a is a negative value, as θ increases, the spiral will become smaller and smaller and spiral back to the origin. If a=0, the exponent of e is always 0, so r is always 1 regardless of θ. This is the equation of the unit circle.
The curve of a sea shell looks like a logarithmic spiral of the following equation:
ρ = Asin(β) exp (θ cot(α)), where A, α and β are parameters of the model.
The fun of playing with various models in Matlab
Various mathematical models are needed to simulate the growth of the curve along various axes for different types of shells. The existence of spirals is not limited to shells in the living world. We can see logarithmic spirals in many places, from flowers to pine cones, from shells to fly wings.
It is important for children to gain an intuitive understanding and then use calculus as a tool of exploration to gain a deeper understanding of mathematical representations. MATLAB is an excellent tool for visually exploring various curve representations, and at UnicMinds we recommend using this tool to enhance understanding.
The mastery of nature in making these designs is an absolute marvel of mathematics and physics. We are still working on many applications to adapt new learning models across the animal and plant kingdoms to achieve a universal understanding of nature.
Hope this helps, thanks.
You might like to read: Teaching kids about good intentions, how nature inspires our computational thinking, and light diffusion explained!