Feynmanian Visualisation of Kinetic Equivalence. A Visual Reconstruction of the Temporal Identity of Cusps through Variable Declination Circles

American theoretical physicist Richard P. Feynman (1918–1988) firmly believed that if you couldn’t visualise it, you didn’t understand it. Visualising a concept, in astronomical terms, is nothing more than drawing it in our own minds in three dimensions. Our work to illustrate the mathematical identity of a cusp, angular or otherwise, transforms an exceptionally difficult concept to elucidate (hour lines or cusps, or time of arrival) into a clear visual experience.

However, to fully grasp the mechanics of this, the reader must distinguish between the following:

• Oblique (Temporal/Seasonal) Hours: Time measured as a fraction of a specific diurnal arc.
• Declination: The angular separation of a celestial body from the plane of the equator.
• Circle of Declination: The apparent 24-hour trajectory of a zodiacal point at a given latitude.
• Diurnal Arc: The visible segment of that trajectory above the local horizon.
• Transit (Culmination): The moment a degree of the ecliptic intersects the local meridian.

Figure 1
Anatomy of an Organic Horizon. Invariance of the 1/6 ratio versus the variability of the ecliptic declination

Legend

• Yellow Trajectories: Diurnal arcs/circles of declination (i.e., points/degrees of the ecliptic/Zodiac).
• Dashed Grey Lines: Seasonal hour markers (consistently delineating twelve hours above and twelve below the local horizon).
• Red Columns: Temporal boundaries plotting the precise proportional divisions of every diurnal arc.
• Orange Nodes: Fractional indicators of duration; they denote the kinetic completion of every successive one-sixth of the corresponding diurnal arc.
• Outlined Nodes: Cusps (topocentric accuracy) of the house plotted pursuant to the corresponding fraction (e.g., 1/6, 2/6, 3/6) of the corresponding arc (degree of the sign).

Conceptual Explanation

1. Invariant Integrity of Primary Motion

As we move along different rings of declination, the “spatial” distance changes, but the “temporal” ratio (the 1/6th) remains an absolute constant. This is known as the true Ptolemaic/Placidian method (Worsdale, 1828; Gansten, 2009). Because every ecliptic degree intersects the horizon at a distinct coordinate (azimuth + altitude) dictated by its specific declination (dependent upon the day of the season), the absolute duration of its diurnal arc varies continuously, and therefore so does the number of minutes that each sixth represents.

However, by dividing these individual arcs into temporally proportional segments, each division strictly constitutes exactly one-sixth of its parent arc, or exactly two seasonal hours. Consequently, a single unit of celestial partitioning (a house) is mathematically defined by two temporal (or seasonal) hours, which guarantees a proportional (i.e, faithful to the corresponding diurnal arc or zodiacal degree on the local horizon) kinetic equivalence across the entire ecliptic-horizon relationship despite spatial inequalities (i.e., distance travelled by the corresponding point upon the ecliptic).

2. Anatomy of the Horizon

The illustration (Proportional Partitioning Across Varying Circles of Declination) depicts seven distinct diurnal arcs (circles of declination), ranging from the minimal temporal duration at the winter solstice (00º Capricorn) to the maximal duration at the summer solstice (00º Cancer). Despite their disparate spatial lengths, every arc is mathematically partitioned into six temporally equal segments, strictly corresponding to two seasonal (oblique) hours, otherwise known as temporal hours. Consequently, every topocentric house encapsulates exactly one-sixth (1/6) of its specific governing diurnal arc, showing that each individual cusp is geometrically bound to its unique circle of declination (of that zodiacal degree).

This makes it possible to visualise the necessary non-linear spatio-temporal calculation of each cusp (zodiacal degree/point on the ecliptic). While the spatial length of each arc is strictly dictated by its specific declination, precise topocentric construction requires kinetic equivalence. By segmenting every unique diurnal arc into exact one-sixth temporal intervals (two seasonal hours), this methodology guarantees that each cusp is calculated based upon the same phenomenon that produces the angular cusps, thus preserving the true mathematical identity of a cusp across the entire horizon.

Whether testing these coordinates (Asc, MC) under Placidian, Regiomontanian, Campanian, or Porphyrian houses in Kodiak, London, or the Caribbean, the physical reality remains invariant: these degrees fluctuate according to the angular motion specific to that latitude. Every system agrees that the Asc is the point where the ecliptic intersects the horizon (this degree completes its nocturnal arc) and that the MC is the point where the ecliptic intersects the local meridian (this degree culminates or completes half of its diurnal arc).[1]

This allows us to understand the reason for which the above eliminates the false contemporary necessity of resorting to reference frames foreign to declination, that is, of deriving all cusps from the equal segmentation of a foreign frame of reference (e.g., celestial equator, prime vertical), which assumes uniformity across varying ascensional times. This variability is due to the fact that each diurnal arc rises above the horizon under a different angular relationship (each diurnal arc constitutes a solar footprint). Unlike the celestial equator (Regiomontanus, 15th century) and the prime vertical (Campano of Novara, 13th century), they do not rise and set at the cardinal points East (azimuth 90º) and West (azimuth 270º).

Find more illustrations in our Visual Astronomy section, and stay tuned for the standalone illustration of this article.

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[1] If the Asc and MC represent the 6/6 and 3/6 proportional divisions of their own arcs, a consistent partitioning method must identify the ecliptic degrees that fulfil the 5/6, 4/6, 2/6, and 1/6 proportional intervals of their respective arcs. For absolute clarity, should a system identify an ecliptic degree as the 12th house cusp, but the Sun—upon occupying that degree—requires a temporal duration greater or less than 1/6th of its diurnal arc (or 1/3rd of its diurnal semi-arc) to travel from the horizon to that position, that system lacks topocentric fidelity. It would have attributed an arbitrary position and/or temporal coordinate to a physical ecliptic degree. Given that every degree possesses a unique, verifiable rate of ascension (oblique ascension), any cusp that deviates from this specific ‘speed’ represents a geometric incongruity rather than a physical reality.