### Spirals Based on the Fibonacci Sequence

A polar coordinate system can be used to create spirals based on the Fibonacci sequence with Mathematica. The formulas used to determine the nth Fibonacci number yield complex results when n is a non-integer. The internal algorithm employed by the Mathematica command `Fibonacci[`n`]` drops the imaginary component of any complex results. As a result, y = `Fibonacci[`x`]` is a continuous function and generally increasing in the set of positive real numbers. Therefore, r = `Fibonacci[``]` yields a spiral. The next program uses a parametric coordinate system to generate such a spiral.

Another method of generating spirals from the Fibonacci sequence is based on the arrangement of adjacent squares that increase in size following the sequence of Fibonacci numbers. This method always yields logarithmic spirals. A line drawn from the origin to any point on such a spiral will form a constant angle with the tangent line at that point. It is unknown whether the spirals r = `Fibonacci[``]` or spirals produced in a similar manner possess any of the properties of the logarithmic spirals.

By manipulating the equations for the spirals in a three-dimensional parametric coordinate system, it is possible to produce graphic images that resemble shells.