The included angles from poles
are designated as
is the distance between poles
. 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 . 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
sliders alter the "phase relation", which is related to the rotation of either or both poles
. 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.
 S. Wolfram, A New Kind of Science
, Champaign, IL: Wolfram Media, Inc., 2002.