In order to verify my equation, I made use of technology.
According to my Graph 1, the points on the spiral show a
logarithmic behavior and its R 2 is 0.9827, which is close
to 1. Since, R 2 of the function indicated that my curve
fits to points well. My regression curve approximates the
real data values.
When I used excel to form my equation of spiral, the
equation appeared as;
y = 1.1798×e 0.1217x
MATHEMATICS
DEPARTMENT
Figure 1: Forming Spiral
Graph 1: Exponential Function of the Points
Figure 2: Labeling the Points
Therefore, I can say that my spiral shows a logarithmic
pattern and the equation that I found by hand is so
similar with the one that I found by using technology.
I chose to work with the 3 rd quadrant of the spiral;
therefore I measured my angles anticlockwise direction.
Every time when the spiral completed an entire
revolution and came to other branch of the spiral, I
added 2π to the angle. To say that my gastropod shell is
a logarithmic spiral, the radius of each point should have
increased with the same ratio. For the points (16.54,
8.04) and (10.19, 3.88) that I chose from my shell, the
equations 8.04=a. e b (16.54) and 3.88= a. e b (10.19) need to
be satisfied. I solved them simultaneously as below:
ln (3.88) – ln (a) = b (10.19)
ln (8.04) – ln (a) = b(16.54)
b= 0.115
a= 1.21
So, my equation is y = 1.21×e 0.115x
Bibliography:
Original Beauty. (2009, 6 27). Retrieved 10 9, 2016,
from Spirals in Nature: https://originalbeauty.
wordpress.com/2009/06/27/spirals-in-nature/
Eric Weisstein. (2002). Wolfram Math World.
Retrieved 9 4, 2016, from Logarithmic Spiral: http://
mathworld.wolfram.com/LogarithmicSpiral.html
Hazal YILDIRIM
12-A
THE CLAPPER 2016 - 2017 33