THE MATHEMATICS BEHIND A GASTROPOD SHELL
In nature, there are some patterns that repeat itself.
Spirals are the one of the patterns that are found in
nature enormously. The spirals have common and
cosmic appeal to human beings because it is beautiful
and mysterious. They are found in the plants, animals,
earth and galaxies in other words all around us.
Mathematics can explain the logarithms, equations and
sequences of the patterns in the spirals (Original Beauty,
2009). Some famous mathematicians categorize the
type of spirals. The type that I will mention is logarithmic
spirals or known as equiangular spiral since any radius
vector makes the same angle with the curve.
If we state this equation in parametric form,
X= r cos θ= a r cos e bθ
Y= r sin θ= a r sin e bθ
y
r
0
x
Descartes was the first person who studied logarithmic
spirals in 1638 (Eric Weisstein, 2002). The logarithmic
spiral has a polar equation as r = ae bθ ; where r is the
distance from the origin, θ is the angle from the x-axis, a
and b are the constants.
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THE CLAPPER 2016 - 2017
To identify whether my shell is logarithmic or not, I first
placed the shell picture on a coordinate system in GSP.
Using coordinate system helped me to find coordinates
of the points on the shell. As seen in Figure 1, the shape
I got was a spiral but not sure what kind of spiral it
is. The next step that I followed was noting down the
coordinates of the points that I labelled, as in Figure 2.
Then, I calculated the distance from the origin because
it gave me the radii.
To find whether the spiral is logarithmic or not, I first
figured out the relationship between the angles and the
radii. I found the radii by the formula of distance from
the origin and I found the angles using the formula: X= r
cos θ and Y= r sin θ.