Spirals: Logarithmic Spiral

Base Logarithmic Spiral

The equations the corresponds with a logarithmic spiral is $$r = a*e^{k*\theta}+b$$ In this equation:
a = controls the distance from the origin along the x-Axis
b = changes the starting value of the spiral along the x-Axis
k = controls the tightness of the successive rotations

Similar to the Archimedean Spiral applet, press the draw button and watch how easy it is for a Logarithmic Spiral to form.
What do you notice about relation between the circle, point C and the segment?



Now that you have ad a chance to see how the spiral is constructed, move the a and k sliders below to see how they change the spiral.



In contrast to the Archimedean/Arithmetic Spiral and all of its variations, the Logarithmic Spiral forms a geometric progression. A unique attribute of this spiral is that the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity.

Mathematicians believed that as a result of this self-similarity property, the spiral was able to evolve in nature. An extension to this unique spiral, mentioned earlier, is the Golden Spiral or Fibonacci Spiral. In order to construct the spiral itself we need to create the sequence that corresponds to it.

Fibonacci Sequence

Sequence Definition: an ordered list of number or terms that often depend on a formula which dictates what the terms in the sequence are and how to find them

EX: the sequence {2,4,8,16,32. . .} can be described as the $$2^n$$
n = to what term in the sequence we are on. Thus when: n = 2 $$2^n = 2^2 = 4$$

The Fibonacci sequence is a sequence defined by its two previous terms. This requires that the first two terms of the sequence are given in order to find the next term.
In this case the first term is 0 and the second term is 1. The next number is found by adding up the two numbers before it. Hence $$0 + 1 = 1$$ is the third term in the sequence, $$1 + 1 = 2$$ is the fourth term and $$1 + 2 = 3$$ is the fifth term.
Using this pattern the sequence formed is as follows {0,1,1,2,3,5,8,13. . .} and can be written as a function such that $$x_{n-2} + x_{n-1} = x_{n}$$
Where:

$$x_{n}$$		is the term we are looking for
$$x_{n-1}$$		is the previous term
$$x_{n-2}$$		is the term before that

Fibonacci Sequence

In order to create the spiral, these numbers in the sequence are used to create squares. Each square corresponds to a single number in the sequence where the width and height of the square matches that of the number in the sequence. Thus the area of corresponding squares are ass follows:

n = 0		Area = 0
n = 1		$$1*1 = 1$$
n = 2		$$1*1 = 1$$
n = 3		$$2*2 = 4$$
n = 4		$$3*3 = 9$$
$$. . .$$		$$ . . .$$
n = n		$$n*n = n^2$$

Next take the squares and stack them in a counter clockwise order with all of the sides touching, matching the pattern of top, left, bottom, right.
Below is an applet that demonstrates this process



Now that we have these squares positioned together lets draw the actual spiral! Go to the first square and draw a quater cicle starting from the lower left corner and ending at the top right corner. In the second square draw another quarter circle starting where the previous ended and ending the quarter circle at the opposite corner. Continuing this process you should end up with the spiral similar to the one below.



But why is this called the Golden Spiral, what makes it so special?
In mathematics there is a ratio that is referred to as the golden ratio often called phi or denoted Φ. This ratio occurs when the ratio of two quantities is the same as the ratio of their sum to the larger quantity.
Imagine you have two numbers a and b where $$a < b$$ These two quantities have a golden ratio if $$\frac{b}{a} = \frac{(a+b)}{b}$$ In the Fibonacci Sequence when you compare two consecutive terms, their ratio comes close to Φ and the bigger terms in the sequence you take the closer to Φ you get!
In terms of the golden spiral, it was recognized that for each quarter turn that is made, the spiral get further from the origin, or wider, by a factor of Φ.