Tetralogy II (Vector Analytics)

Polar Coordinates

The polar coordinate is a locus on the circular array of values (set elements) arranged according to the radial coordinate (r) and the angular coordinate (φ, θ, t) [c.f. ISO Standard 31-11]. Polar coordinates assimilate in an algebraic curve - polar function form f((rφ), φ) representing circular (trigonometric) and orbital motions of the tetrad moments. The polar function form establishes the geodetic focus (diagnostic ‘fertilisation’) of the open orbit (hyperoctant space).

Spectral Determination

As identified in the Spectral Protocol, the t1/tx tetrad moment corresponds to a ‘Secondary Moment’ configured by the spectral field in phase-shift at t1, and at cross-section to the cosine function wave-shift at tx. Spectral modulations of the Secondary Moment are (a) Ontological, meaning that the physical phenomena are projectile and exhibit cross-grain formativity, and (b) Crystalline, involving spectral algorithms to identify epidemiological structuration that particularizes periodicity to tetrad moments (phase-shifts) of the tetralogue automaton state.

At a given locus in n-dimensional space, the hypercube emerges as a hyperoctant-space of dimension two. Spectral determination is configured by a one perpendicular unit-length timeshift (a line timeshifted to a square) which conforms to a unit hypercube of dimension two. Therefore, in the Secondary Moment, the spectral array manifests as a square (in hypercube-space) or quadrant (in hyperoctant-space).

Orthogonality

At dimension two-space, the Secondary Moment is configured by permutation representation (orthogonality) at axis of symmetry (or axial subset/subgroup of all isometries).

Quotient Orbital

The quotient orbital denotes the topological group action onto abstract objects in the topological space of the symmetry group (transformation group). In the Secondary Moment, the quotient orbital is given by the factorial moment generating function M_xt=Et^x manifesting as the Population Set (orbital frame), galvanised by an orthogonal rotation (functor). The resultant is the E-1 Psycho-Social Milestone vector space (function map) of the Population Set (in the codomain of the Universal Set).

Part B: Formal System (Cartesian Coordinates)

Computational methods for the formal sub-system of Cartesian coordinate space interfaces bijective functions of the polar operands on the hyperplanes of Euclidian space. Geometric interpolations of the formal system extend to methods of vector factorization and combinatorial computational geometry (algorithmic geometry). These methods of systematic vector calculus reference techniques of characteristic decomposition to factorial (polynomial) roots programmed to identify linear transformations of the partial integrals (set elements) that conform to the functor of the tetrad moment. Hence, in the syntax of Existemetrics, the Tertiary Moment is analogous to the ‘Germination’ stage of reproduction, delineating the granular structure as the groupoid catalyst of aetiological polytopes. The formal sub-system is thus a deductive apparatus of the MMS designative of homologic functions (characteristic energy) and coarse-graining endemic to the wave-shift tetralogue (T2) and aetiological frame of the tetrad (tertiary) moment.

Cartesian Coordinates

The Cartesian coordinate is a point projection in hyperoctant space (set elements) arranged according to three {x,y,z} mutually perpendicular hyperplanes. Mathematical operations of Cartesian coordinates are substantiated by the Cartesian Product determined by real enumeration R²=RxR of the Cartesian operands. Cartesian coordinates therefore, represent real distances of the tensor product in ambient space which is the Cartesian planar projection of the Tertiary Moment. The Cartesian tensor product form thus parameterizes and fixates the cyclo-symmetric geometries (diagnostic ‘germination’) of the open orbit (hyperoctant space).

Spectral Determination

As identified in the Spectral Protocol, the t2/ty tetrad moment corresponds to a Tertiary Moment configured by the spectral field in phase-shift at t2, and at cross-section to the sinusoid function wave-shift at ty. Spectral modulations of the Tertiary Moment are (a) Aetiological, whereby critical velocity of physical stimulants activate the tΩ Operator switch to originate sub-spectral aetiological polytopes (characteristic energy), and (b) Granular, symbolising spectral algorithms that prototype the homomorphology of the electro-dynamic image through phase-shifts of the tetrad (tertiary) moment.

At a given locus in n-dimensional space, the hypercube emerges as a hyperoctant-space of dimension three. Spectral determination is configured by a one perpendicular unit-length timeshift (a square timeshifted to a cube) which conforms to a unit hypercube of dimension three. Therefore, in the Tertiary Moment, the spectral array manifests as a cube (in hypercube-space) or octant (in hyperoctant-space).

Orthogonality

At dimension three-space, the Tertiary Moment is configured by permutation representation (orthogonality) at perpendicular planes of symmetry (or planar subset/subgroup of all isometries).

Quotient Orbital

The quotient orbital typifies the topological group action onto abstract objects in the topological space of the symmetry group (transformation group). In the Tertiary Moment, the quotient orbital is given by the Characteristic moment generating function φx(-it)=M_x(t) manifesting as the Existemetric E-1 Psycho-Social Milestone (orbital frame), mechanised by an orthogonal rotation (functor). The resultant is the E-2 Socio-Political Orientation vector space (function map) of the Population Set (in the codomain of the Universal Set).

How to cite this material:

MEF (2014) Tetralogy II (Vector Analytics). The Meta-Existential Framework. MEF e-Publishing.