Biradial Matrix

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A biradial matrix consists of two sets of equally spaced radial lines separated by a distance . Use the "nodes" or "rays" buttons to change the display. The rays (nodes) can be indexed from zero starting at the segment, creating a biradial coordinate system with coordinate pairings at the nodes; check "intersection labels" to see their values. By adhering to specific connection algorithms, fundamental field structures are set with the "attract", "repel" and "hyperbola" buttons. The number of rays from each pole can be set using the sliders and , which changes the subsequent field structures. The angular phase relation between the poles (defined elsewhere) can be set with the sliders and . This geometric construct has many applications in physics. The "line thickness" and "node diameter" sliders vary the thickness of the lines and diameter of the nodes.

Contributed by: Russell Kramer (March 2017)
Open content licensed under CC BY-NC-SA



The included angles from poles and are designated as and . is the distance between poles and . There are numerous other variables that can be assigned to various segment lengths and other internal angles of the trapezoids defined by the intersecting rays. From this initial set of variables, an inverse-square equation based on the distance is derived, which corresponds to the field density as illustrated by the attraction, repulsion and hyperbola field lines. The nodes defined by the intersections of the rays can be designated with a biradial coordinate system allowing for a mathematical description of the field lines. These nodes and the resulting field lines appear related to the space networks described by Wolfram in [1]. The harmonic series is derived from the upper limit of rays from each pole in relation to the segment, indicating an underlying harmonic structure of the biradial matrix where the coordinate pairings also represent the ratios of the harmonic overtone series. The and sliders alter the "phase relation", which is related to the rotation of either or both poles and . As a conceptual model and harmonic coordinate system, the biradial matrix has many applications in physics. It is possible, for example, to define the gravitational equilibrium zone between two masses using the biradial matrix.


[1] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, Inc., 2002.

[2] R. Kramer. "Project #1: The Genesis Field and the Harmonic Structure of Space-Time." (Mar 9, 2017)

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