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Displacement

Let some particleΒ P be characterized by a orderedΒ  chain of events

\Psi^{\mathsf{P}} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2}, \, \mathsf{P}_{3} \, \ldots \, \mathsf{P}_{k} \, \ldots \, \right)

that is repetitive so that  \Psi may also be written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \,          \right)

where each orbital cycle  \mathsf{\Omega} is composed of  N sub-orbital events written as

\mathsf{\Omega} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2}, \,  \mathsf{P}_{3}  \, \ldots \, \mathsf{P}_{N} \right)

The orbital radius of P is noted by  R and  \lambda is its wavelength. A spatial orientation for P is specified by \delta_{\hat{m}}, \delta_{\hat{e}} and \delta_{z}. We use these characteristics to define the following numbers for describing sub-orbital events

    \begin{equation*} \begin{split} d \! x &\equiv \delta_{\hat{m}} \dfrac{R}{\, N \,} = \; \dfrac{R\cos{\! 2\theta}}{N} \rule{0px}{16px} \\  d \! y &\equiv \delta_{\hat{e}} \, \dfrac{R}{\, N \,} =-\dfrac{R\sin{\! 2\theta}}{N} \rule{0px}{16px} \\  d \! z &\equiv \delta_{z} \, \dfrac{\lambda}{\, N \,} \; = \; \dfrac{\lambda}{2\pi} \, d \! \theta \rule{0px}{16px}  \end{split} \end{equation*}

where  \theta is the phase angle of P. Then using the Cartesian unit vectors  \hat{x},  \hat{y} and  \hat{z}, a displacement vector for P is defined by

d \! \bar{r} \equiv d \! x \, \hat{x} + d \! y \, \hat{y} + d \! z \, \hat{z}

If we switch to implicitly using Cartesian basis vectors, we can express the displacement as an ordered set

d \! \bar{r} = ( d \! x, \, d \! y, \, d \! z )

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events may be large but not infinite. In principle  N is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that  N is large enough to make an approximation to spatial continuity, then allowing the use of calculus.

Displacement and spatial structure is suggested by the linear and circular patterns in this Maylasian ikat weaving.
Bidang (detail), Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 110 x 61 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.