Quantum Emergence.Reference.Geometry.Forever Knot - Revision history

XenoEngineer at 20:35, 10 December 2023

2023-12-10T20:35:09Z

← Older revision		Revision as of 20:35, 10 December 2023
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	~~<h1>~~Forever Knot —a non-harmonic torus helix (not-knot)~~</h1>~~		== {{red24\|The Forever Knot —a non-harmonic torus helix (not-knot)}} ==
	~~<a class~~=~~"menu" href~~=~~"../index.html">Main Page</a> ⋅&dot;⋅ <a class="menu" href="../References/References.html">Project References</a><hr />~~
	~~<h2>Topics: Geometry, Torus, Knot Windings, Golden Ratio</h2>~~

	<h2>The ~~'Forever Knot'~~</h2>		<p>This forever knot is not a real knot, because it does not connect with itself, and continues to trace a non-harmonic path. If it could continue, it would yet never reconnect with its beginning ever ever. The illustration has 888 helical windings on the donut.</p>

	~~<font color="green"><b><big>Categories</big></b>:torus knot, irrational, golden ratio, golden torus, golden sope, povray, code</font>~~		<p>The not-knot illustrated above has a slope of Phi:1, and the donut (torus) is golden, which places the slope of the winding helix as Phi at a very irrational number, and therefore there can be no integer harmonic winding condition where the beginning and end of the knot meet at the same point to repeat the same winding path.</p>

	<p>This is not a real knot, because it does not connect with itself, and continues to trace a non-harmonic path. If it could continue, it would yet never reconnect with its beginning ever ever. The illustration has 888 helical windings on the donut.</p>

	<p>The not-knot illustrated above has a slope of Phi:1, and the donut (torus) is golden, which places the slope of the winding helix as Phi at a very irrational number, and therefore there can be no integer harmonic condition where the beginning and end of the knot meet at the same point to repeat the same winding path.</p>

	<center><img src="./img/forever_knot_phi-slope_on_golden_donut.png" width="800px"/></center>		<center><img src="./img/forever_knot_phi-slope_on_golden_donut.png" width="800px"/></center>
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XenoEngineer at 20:24, 10 December 2023

2023-12-10T20:24:54Z

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	<h1>Forever Knot —a non-harmonic torus helix (not-knot)</h1>		<h1>Forever Knot —a non-harmonic torus helix (not-knot)</h1>
	<a class="menu" href="../index.html">Main Page</a> ⋅&dot;⋅ <a class="menu" href="../References/References.html">Project References</a><hr />		<a class="menu" href="../index.html">Main Page</a> ⋅&dot;⋅ <a class="menu" href="../References/References.html">Project References</a><hr />
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	<h2>The 'Forever Knot'</h2>		<h2>The 'Forever Knot'</h2>

	<font color="green"><b><big>Categories</big></b>:torus knot, irrational, golden ratio, golden torus, golden sope, povray, code</font>		<font color="green"><b><big>Categories</big></b>:torus knot, irrational, golden ratio, golden torus, golden sope, povray, code</font>

	<p>This is not a real knot, because it does not connect with itself, and continues to trace a non-harmonic path. If it could continue, it would yet never reconnect with its beginning ever ever. The illustration has 888 helical windings on the donut.</p>		<p>This is not a real knot, because it does not connect with itself, and continues to trace a non-harmonic path. If it could continue, it would yet never reconnect with its beginning ever ever. The illustration has 888 helical windings on the donut.</p>

	<p>The not-knot illustrated above has a slope of Phi:1, and the donut (torus) is golden, which places the slope of the winding helix as Phi at a very irrational number, and therefore there can be no integer harmonic condition where the beginning and end of the knot meet at the same point to repeat the same winding path.</p>		<p>The not-knot illustrated above has a slope of Phi:1, and the donut (torus) is golden, which places the slope of the winding helix as Phi at a very irrational number, and therefore there can be no integer harmonic condition where the beginning and end of the knot meet at the same point to repeat the same winding path.</p>

	<center><img src="./img/forever_knot_phi-slope_on_golden_donut.png" width="800px"/></center>		<center><img src="./img/forever_knot_phi-slope_on_golden_donut.png" width="800px"/></center>

	<div style="background:azure; border:1px outset azure; margin:1em 2em; padding:0 1em; width:888px;">		<div style="background:azure; border:1px outset azure; margin:1em 2em; padding:0 1em; width:888px;">
	<b><big>Knot trivia:</big></b> ~~A 'golden torus' has a minor radius that is four powers of the golden ratio (1.6180339887...) less than the major radius.~~ In terms of the radius of the torus hole of a golden donut, the ~~hole~~ radius ~~from the torus center point~~ is Phi<sup>4</sup> ~~less~~ than the ~~major~~ radius —scalable with a multiplier on the major:hole proportion.		<b><big>Knot trivia:</big></b> In terms of the radius of the torus hole of a golden donut, the torus major radius is Phi<sup>4</sup> times larger than the torus hole radius —scalable with a multiplier on the major:hole proportion.
	</div>		</div>

	~~<h2>~~POVRay code for the not-knot as illustrated~~</h2>~~		=== {{red24\|POVRay code for the not-knot as illustrated}} ===

	<pre style="background:#222; padding:1em; color:lime; margin-left:3em; font:normal 14px terminal;"> // POVRay scene file: groupwave.pov //		<pre style="background:#222; padding:1em; color:lime; margin-left:3em; font:normal 14px terminal;"> // POVRay scene file: groupwave.pov //
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	#end // circle_sphere_array().		#end // circle_sphere_array().

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<h1>Forever Knot —a non-harmonic torus helix (not-knot)</h1>
<a class="menu" href="../index.html">Main Page</a> ⋅&dot;⋅ <a class="menu" href="../References/References.html">Project References</a><hr />
<h2>Topics: Geometry, Torus, Knot Windings, Golden Ratio</h2>

<h2>The 'Forever Knot'</h2>

<font color="green"><b><big>Categories</big></b>:torus knot, irrational, golden ratio, golden torus, golden sope, povray, code</font>

<p>This is not a real knot, because it does not connect with itself, and continues to trace a non-harmonic path. If it could continue, it would yet never reconnect with its beginning ever ever. The illustration has 888 helical windings on the donut.</p>

<p>The not-knot illustrated above has a slope of Phi:1, and the donut (torus) is golden, which places the slope of the winding helix as Phi at a very irrational number, and therefore there can be no integer harmonic condition where the beginning and end of the knot meet at the same point to repeat the same winding path.</p>

<center><img src="./img/forever_knot_phi-slope_on_golden_donut.png" width="800px"/></center>

<div style="background:azure; border:1px outset azure; margin:1em 2em; padding:0 1em; width:888px;">
<b><big>Knot trivia:</big></b> A 'golden torus' has a minor radius that is four powers of the golden ratio (1.6180339887...) less than the major radius. In terms of the radius of the torus hole of a golden donut, the hole radius from the torus center point is Phi<sup>4</sup> less than the major radius —scalable with a multiplier on the major:hole proportion.
</div>

<h2>POVRay code for the not-knot as illustrated</h2>

<pre style="background:#222; padding:1em; color:lime; margin-left:3em; font:normal 14px terminal;"> // POVRay scene file: groupwave.pov //

// Copyright Don@groupKOS.com
// +SF0 +EF99 +KFI0 +KFF99 +W800 +H600 +A

#version 3.7;
global_settings{ assumed_gamma 1.0 }
#default{ finish{ ambient 0.1 diffuse 0.9 }}

#include "colors.inc"

// An area light (creates soft shadows)
// WARNING: This special light can significantly slow down rendering times!
light_source {
<0,0,0> // light's position (translated below)
color rgb 1.0 // light's color
area_light
<8, 0, 0> <0, 0, 8> // lights spread out across this distance (x * z)
4, 4 // total number of lights in grid (4x4z = 16 lights)
adaptive 0 // 0,1,2,3...
jitter // adds random softening of light
circular // make the shape of the light circular
orient // orient light
translate <40, 80, -40> // <x y z> position of light
}

// sky (tilt) camera
camera {
//orthographic
location <.3,.1,.3>84
// location y66
angle 45
sky y //<1.0, 0.5, 0.0>
look_at <0.0, 0.0, 0.0>
right ximage_width/image_height
}

// Set a color of the background (sky)
background { color rgb< 1, 1, 1> }

// Render Count number of spheres at radius around Y axis.
#macro circle_sphere_array(Count, arrayRad, sphereRadius, txtr)
// Radial pattern centered on the Y axis of count divided into 360.

#end // circle_sphere_array().

#macro plotTorusKnotUnitQDensity(R, r, p, q, hand, plotRadius, plotResolution, lugRadius, densityHigh, densityLow, txtr)
// Plot 1/q-th of a whole p:q knot //
// Vary the radius of the plotting sphere to abstract the //
// radius of a certain magnetic flux density about that point //
//
// hand = chirality of the knot windings; a clockwise or a counter-clockwise helix around the donut. //

union{

#local unitEndAngle = ( 360p/q );
#local plotAngleDelta = unitEndAngle / plotResolution;

#if(showEquatorConnectorLugs)
// Draw connector lug spheres //
union{
// Draw the D.C. termination lugs at the ends of the unit segment windings //
sphere{ 0, lugRadius
scale x3
translate x(R+r)
rotate y0

texture{txtr}
}
// Draw the D.C. termination lugs at the ends of the unit segment windings //
sphere{ 0, lugRadius
scale x3
translate x(R+r)
rotate y180

texture{txtr}
}
} // end unioN

#end // if()

union{ // of knot winding plot points //

//sphere{0, .3 translate x(radiusMajor+radiusMinor) rotate y(000p/q)} // test cursor
//sphere{0, .3 translate x(radiusMajor+radiusMinor) rotate yunitEndAngle} // test cursor

#local plotAngle = 0; // While-loop index.
#while (plotAngle < unitEndAngle22222 )

sphere{ 0, plotRadius
// Move objPlot from the origin to the X axis by the minor radius plus major radius.
translate x(r)

// Rotate objectLug in the XY plane around the z axis, for either hand.
rotate z(handplotAngleq/p) // Extract q degrees from (360p/q).

translate xR

// Rotate objPlot in the XZ plane around the Y axis of the torus hole, for either hand.
rotate y(handplotAngle)

texture{pigment{color Green}}
}

// Rinse and repeat.
#local plotAngle = plotAngle + plotAngleDelta;

#end // while

} // end union
} // end union
#end // plotTorusKnotUnitQDensity()

#local txtrGreen = texture{pigment{color Green}};
#local txtrRed = texture{pigment{color Red}};
#local txtrBlue = texture{pigment{color Blue}};

union{ // scene union

#local phaseCount = 1;
#local unitQResolution = 3000;
#local patternLength = phaseCount/2;

#local plotRad = .025;
#local lugRad = plotRad2;
#local AxesDia = .1;
#local showEquatorConnectorLugs = +1;

// A 4th-degree golden donut //
#local R = 6.8541;
#local r = 5.8541;

// Knot winding-ratio parameters (p:q)
#local p = 8;
#local q = 5;

// axes //
cylinder{-x15, +x15, AxesDia/2 texture{txtrRed } }
cylinder{-y10, +y10, AxesDia/2 texture{txtrGreen } }
cylinder{-z15, +z15, AxesDia/2 texture{txtrBlue } }
#local phi = pow(5,.5).5+.5;

#local phi0 = pow(phi,0);
#local phi1 = pow(phi,1);
#local phi2 = pow(phi,2);
#local phi3 = pow(phi,3);
#local phi4 = pow(phi,4);
#local phi5 = pow(phi,5);

#local j=0;
union{ // scene union

#local phaseCount = 1;
#local unitQResolution = 3000;
#local patternLength = phaseCount/2;

#local plotRad = .025;
#local lugRad = plotRad2;
#local AxesDia = .1;
#local showEquatorConnectorLugs = +1;

// A 4th-degree golden donut //
#local R = 6.8541;
#local r = 5.8541;

// Knot winding-ratio parameters (p:q)
#local p = 8;
#local q = 5;

// axes //
cylinder{-x15, +x15, AxesDia/2 texture{txtrRed } }
cylinder{-y10, +y10, AxesDia/2 texture{txtrGreen } }
cylinder{-z15, +z15, AxesDia/2 texture{txtrBlue } }
#local phi = pow(5,.5).5+.5;

#local phi0 = pow(phi,0);
#local phi1 = pow(phi,1);
#local phi2 = pow(phi,2);
#local phi3 = pow(phi,3);
#local phi4 = pow(phi,4);
#local phi5 = pow(phi,5);

#local j=0;
union{
#while(j < phaseCount)
union{
//plotTorusKnotUnitQDensity( p, q, 12.9787137637477918123, 8, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
plotTorusKnotUnitQDensity( p, q, phi1, phi0, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, phi3, phi2, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )

/
//plotTorusKnotUnitQDensity( R, r, p, q, hand, plotRadius, plotResolution, lugRadius, densityHigh, densityLow, txtr) /
//plotTorusKnotUnitQDensity( p, q, 1, 1, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 2, 1, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )

//plotTorusKnotUnitQDensity( p, q, 3, 2, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 5, 3, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 8, 5, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 13, 8, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 21, 13, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 34, 21, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 55, 34, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 89, 55, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )
//plotTorusKnotUnitQDensity( p, q, 144, 89, +1, plotRad, unitQResolution, lugRad 2, 1, txtrRed )

rotate y(360j/phaseCount)

} // end object

#local j = j + 1;
#end // while

rotate y(360j/phaseCount)
}
rotate y(360 * frame_number/final_frame)
} // end scene union

</pre>

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