Recursive wave

A recursive wave is a self-similar curve in three-dimensional space that is constructed by iteratively adding a helix around the previous curve.

Recursive waves of depths 1, 2 and 3.

Construction

A recursive wave of depth $n$ can be constructed as following:

$\psi _{0}(x)=x(ai+bi+ck)$

$\psi _{n}(x)=\psi _{n-1}(x)+R(A(n)g_{n}(x),\ \psi _{n-1}'(x),\ f(n)x+\alpha (n))$

where

$g_{n}(x)=|{\vec {w}}\times |\psi _{n-1}'(x)||$

and

$R({\vec {A}},{\vec {B}},\theta )=e^{{\vec {B}}\theta /2}{\vec {A}}e^{-{\vec {B}}\theta /2}$

Clarification

Each wave at non-zero depth $n$ is described by an amplitude $A(n)$, frequency $f(n)$ and phase offset $\alpha (n)$.

$g_{n}(x)$ represents a unit vector that is perpendicular to the previous curve at $x$. An arbitrary vector ${\vec {w}}$ is chosen to be the fixed "rag" vector.

$R$ is a function that rotates a vector ${\vec {A}}$ around an axis defined by a vector ${\vec {B}}$ by $\theta$ degrees. In this case it is expressed with quaternions.

See also

• Solenoid (DNA)