Recursive wave

Recursive waves of depths 1, 2 and 3.

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

Construction

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

${\psi }_0(x)=x(ai+bi+ck)$

${\psi }_n(x)={\psi }_{n-1}(x)+R(A(n)g_n(x),{\psi }_{n^{\prime }-1}(x),f(n)x+\alpha (n))$

where

$g_n(x)=|\overrightarrow{w}\times |{\psi }_{n^{\prime }-1}(x)||$

and

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

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 $\overrightarrow{w}$ is chosen to be the fixed "rag" vector.

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

See also

• Solenoid (DNA)