XenoEngineer: Created page with "{{menuAuricGeometry}} Auric: A term referring to 'golden' as used in the context of the geometry of a torus profile when major- and minor-radii are separated by four degrees of the golden ratio. == 4th Auric Torus Profile == ::'''Major_Radius = Phi^4th = ~6.854''' ::'''Minor_Radius = Major_Radius - Phi^0TH = ~6.854 - 1''' == Scaling the 4th Auric Torus Profile == ;The profile of a 4th auric tours is preserved when scaled..."

2024-09-23T18:13:45Z

Created page with "{{menuAuricGeometry}} Auric: A term referring to 'golden' as used in the context of the geometry of a torus profile when major- and minor-radii are separated by four degrees of the golden ratio. == 4th Auric Torus Profile == ::<big>'''Major_Radius = Phi4th = ~6.854'''</big> ::<big>'''Minor_Radius = Major_Radius - Phi0TH = ~6.854 - 1'''</big> == Scaling the 4th Auric Torus Profile == ;The profile of a 4th auric tours is preserved when scaled..."

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{{menuAuricGeometry}}

[[Auric]]: A term referring to 'golden' as used in the context of the geometry of a torus profile when major- and minor-radii are separated by four degrees of the golden ratio.

== 4th Auric Torus Profile ==
::<big>'''Major_Radius = Phi4th = ~6.854'''</big>
::<big>'''Minor_Radius = Major_Radius - Phi0TH = ~6.854 - 1'''</big>

== Scaling the 4th Auric Torus Profile ==
;The profile of a 4th auric tours is preserved when scaled by operating on the exponent in the radii term:
Given a scaling operator = <big>'''SO'''</big>
::<big>'''Major_Radius = PhiSO + 4'''</big>
::<big>'''Minor_Radius = Major_Radius - Phiso + 0'''</big>

Where,
::<big>'''Phi = 50.5 X 0.5 + 0.5 = 1.618033...'''</big>

== Geometric magic in the 4th auric torus profile ==
The slope of the helix of a torus knot wound smooth/tight on a 4th auric torus profile will have a slope of the helix of the torus knot equivalent to the knot ratio.

Which means, the <big>'''13:8'''</big> torus knot windings, smooth and tight, for a slope on the helix of <big>'''13/8 = 1.6250'''</big>

That means the 13:8 knot winding on a 4th auric torus profile is <big>'''0.0069'''</big>... units error from <big>'''Phi'''</big>.
 
The error from a helical slope of Phi is the error of the approximation of 13/8 is to Phi.
 
Successive adjacent-pairs in the Fibonacci number sequence (1,1,2,3,5,8,13,21,33, ...) if used as a knot winding ratio on a 4th auric torus, create helical slopes that error from golden on alternate sides of a slope of Phi.
 
A knot with a slope of Phi is not a knot by definition, as by definition, a knot is an integral harmony of a winding path, such that it self-connects, continuing on the same path. Integral path harmony.
 
As the golden ratio is very irrational, like Pi, a slope of Phi4:1 on a 4th auric torus would be an infinitely long line, never connectiing integrally with its beginning.
 
All of the Fibonacci adjacent pairs create a slope very near a golden slope.
 
 The '''''nearness''''' is exploited in the design of the bifurcated torus knot...
   ...because the bifurcated halves of a 13:8 | 3-group decompose to a 3:2 torus knot winding path —as a phase-sorted magnetic group-of-four-windings, which physical angle is a <big>'''negative err'''</big> from the golden helical slope — while the physical windings are at <big>'''positive err'''</big> to the golden slope.

Dev:4th auric geometry - Revision history