+, -, *, /
Basic arithmetic operations.
a to the power of b.
Sine and cosine.
Logarithm with base e.
a mod b, a % b
a modulo b.
Mathematical constant pi.
Mathematical constant e.
Mathematical constant Gamma.
The absolut value of n.
The n-th Fibonacci Number.
Use OEIS sequence.
The n-th prime number.
The difference of the n-th prime numbers.
Use personal number squence.
Devisor fum function.
Devisor counting function.
Mathematics of Functional Walk Theory
A Walk is defined as a sequence of steps of connected graph vertices and graph edges.
Functional Walks are generated by using a main function f(n) and an additional step function g(n):
We are at some point w(n) = (x(n),y(n)). For n = 0 we define w(0) = (0,0).
We divide the 360 degrees or 2π in N equal directions.
We calculate the direction for the next step by
We move g(n) units in this direction and repeat the process.
This process can be summarized trough the recursion
The walk is generated by adding steps, which may be represented as vectors.
The recursion yields for the position (x,y) at a given moment n
The hole walk is therefore given by W(f,g,N).
Fascinating patterns emerge very quickly, e.g. by applying f(n) = n.
But also a variety of challenges and research opportunities arises when
explaining patterns, predicting walks, describing walks as curves or
investigating further properties. Also note the build-in OEIS tool.
Further Links: Wolfram: Walk
Walk Pattern Gallery