Golden torus Fibonacci knots
Jump to navigation
Jump to search
// Persistence of Vision Ray Tracer Scene Description File
// File: Groupkos.dev.geomagic_3-2_pattern_in_bifilar_13-8_torus_knots_800x600.pov
// POV-Ray Vers: 3.7 RC7 win64
// Desc: Polyphase 13:8 torus knot group w/ colored halves (bifilar).
// Auth: DonEMitchell -- a electrified knot group fanatic (XenoEngineer)
// Edited: March 9, 2020 Revisited and reorganized for posting at groupkos.com/dev/category:povray
// Edited: April 21, 2020 --added cylinder knot loop elements.
// Edited: April 22, 2020 for groupkos/dev propagation animation
// Edited: July 23, 2022 cleaned up for dev wiki posting.
// Date: November 29, 2023 --chiral grouping illustration
// -- HOW TO invoke an animation sequence --
// Put the control-code below into the POV-Ray text-box above the POV-Ray editor
// (without the comment slashes).
// Or from the menu: Render - Edit settings/Render.
// Then click Run (Alt G). Search POV-Ray help for 'Animation'.
// Or, right-click on the line below, and select "Copy [...] to the command line"
// +SF1 +EF299 +KFI1 +KFF299 +W800 +H600 +A
// The above POV-Ray control code causes POV-Ray to restart this file 299 times,
// creating 299 image frames.
// Each time the animation sequence runs, a variable named 'clock' is advanced
// from zero to one (0-1).
// The clock variable is used below to modulate parameters affecting the rendering.
// Note: POV-Ray does not generate an animated GIF, which is done with external utility.
#version 3.7;
#include "colors.inc"
global_settings {
max_trace_level 10 // Reflection levels before blackness.
assumed_gamma 1.2
charset utf8
// Used in global_settings, sets an overall brightness/ambient light level in the scene
ambient_light color rgb <1,1,1> * .5
}
#default{ finish{ ambient .5 diffuse 0.5 }}
sky_sphere{ pigment { gradient <-1,1,-1>
color_map { [0.00 rgb <0.6,0.7,0.80>*.2]
[0.35 rgb <0.0,0.1,0.8>*.2]
[0.65 rgb <0.0,0.1,0.8>*.2]
[1.00 rgb <0.6,0.7,1.0>*.2]
}
scale 2
} // end of pigment
}
//--------------------------- Macros ---------------------------------
// Axes macro: length and radii of axes, and boolean flags to include/exclude axes
#macro axes(length, radii, X, Y, Z)
#if (X) cylinder{ 0, < length, 0, 0>, radii texture { pigment { color rgb < .75,-.1,-.1 > }}} #end // X axis, red
#if (X) cylinder{ 0, <-length, 0, 0>, radii texture { pigment { color rgbt < 1, 0, 0, .6 > }}} #end // -X axis red translucent -adjust xmit
#if (Y) cylinder{ 0, < 0, length/2, 0>, radii texture { pigment { color rgb < -.1, .75,-.1 > }}} #end // Y axis, green
#if (Y) cylinder{ 0, < 0, -length/2, 0>, radii texture { pigment { color rgbt < 0, .8, 0, .6 > }}} #end // Y axis, green translucent
#if (Z) cylinder{ 0, < 0, 0, length>, radii texture { pigment { color rgb < 0, 0, 2 > }}} #end // Z axis, Blue
#if (Z) cylinder{ 0, < 0, 0, -length>, radii texture { pigment { color rgbt < .1, .1, .4, .6 > }}} #end // -Z axis blue translucent
#end //macro axes()
#macro torusKnotBifilar(objectBifilarA, objectBifilarB, objLugA, objLugB, radiusConductor, radiusConnector, major, minor, poloid, toroid, polyphase, delta)
union{
// Render objectBifilarA and B as connector lugs
object{objLugA translate x*(major+minor+radiusConnector*0)}
// Render a objectBifilarB as a connector lug
object{objLugB translate -x*(major+minor+radiusConnector*0)}
// Draw both bifilar objects in one loop from zero through 180 degrees.
#local n = 0; // n Is the while loop index.
// The clock variable modulates completion length of while loop to animate propagation of bifilar halves.
#while (n <= (180 * poloid * clock)) // Loop positive 180 degrees, and draw the objectBifilarB, below, looping the negative 180.
#local phaz = n * toroid/poloid; // Divide out poloid (multiplied already into n) to obtain toroidal rotation.
// 1st Bifilar half as objectBifilarA
#if( 01 )
object{ objectBifilarA
// Move the objectBifilar away from the origin on the X axis by the minor radius.
translate x*minor
// Toroidally rotate objectBifilar around Z axis in the XY plane.
rotate <0,0,phaz>
// Move the rotated objectBifilar the distance of the major radius in the X-axis.
translate x*major
// Rotate the object about the center Y axis of the torus.
rotate y*n
}
#end
#if( 01 )
object{ objectBifilarB
// Move the objectBifilar away from the origin on the X axis by the minor radius.
translate x*minor
// Toroidally rotate objectBifilar around Z axis in the XY plane.
rotate <0,0,phaz>
// Move the rotated objectBifilar the distance of the major radius in the X-axis.
translate x*major
// Rotate the object about the center Y axis of the torus.
rotate y*n
rotate y*180
}
#end
// Counter-toroid
#if( 0 )
object{ objectBifilarA
// Move the objectBifilar away from the origin on the X axis by the minor radius.
translate x*minor
// Toroidally rotate objectBifilar around Z axis in the XY plane.
rotate <0,0,-phaz>
// Move the rotated objectBifilar the distance of the major radius in the X-axis.
translate x*major
// Rotate the object about the center Y axis of the torus.
rotate y*n
}
#end
// 2nd Bifilar half as objectBifilarB (counter-toroid and counter-poloid rotations)
#if( 0 )
object{ objectBifilarB
// Move the objectBifilar away from the origin on the X-axis by the minor radius.
translate x*minor
// Toroidally Counter-rotate objectBifilarB around Z-axis in the XY-plane.
rotate <0,0,-phaz>
// Move the rotated objectBifilar the distance of the major radius in the X-axis.
translate x*major
// Counter-rotate the objectBifilarB about the center Y-axis of the torus.
rotate -y*n
rotate -y*(180)
}
#end
#if(showQuadrantConnectors)
sphere{
0,radiusConnector
translate x*(radiusMinor + radiusMajor)
rotate y*90
texture { pigment{rgbt<1,1,0,.0975>} }
}
sphere{
0,radiusConnector
translate x*(radiusMinor + radiusMajor)
rotate -y*90
texture { pigment{rgbt<1,1,0,.0975>} }
}
#end
#local n = n + delta;
#end // while
rotate y*polyphase // Rotate the completed knot to its phase-angle about the torus center Y-axis.
} // end union
#end // macro torusKnotBifilar()
// ----------------- Optics --------------------
camera {
//orthographic
//location <2*27-27*clock,8,0>
location z*28
look_at y*0
angle 58
sky y
right x*image_width/image_height
}
// general light definition
light_source {
<15, 10, 10> // position of the light source
color rgb 1.0*3 // color of the light
shadowless
}
// ---------------- Scene ------------------------
// 13/8 = 1.625
// Phi = 1.6180339887498948482045868343656
// Difference = 1.625 - Phi
// Difference = 0.00696601125010515179541316563436
// Torus knot parameters (loop/twist) //
#declare poloid = 3;
#declare toroid = 2;
//#declare poloid = 12 + 1*clock; // clock sequences from 0 to 1
//#declare toroid = 8;
#declare phaseCount = 1;
// Delta as increment between objects tracing torus loop in the torus knot macro.
// The knot is a string of overlapping spheres. Delta sets the separation.
#declare plotDelta = .05;
// Torus dimensions
// Torus parameters and knot ratio (winding number) construct a 'golden orthogonal (ish) Fibonacci torus knot' (GOFTK)
// FTK: Neighboring Fibonacci-sequence numbers (1,1,2,3,5,8,13,21...)
// Poloidal rotations = Fn+1
// Toroidal rotations = Fn
// Torus knot ratio: (Fn+1)/(Fn) = golden ratio (almost) --The knot ratio approximates Phi = 13/8 = 1.6025
#declare Phi = pow(5,.5) * .5 + .5; // Upper case Phi = 1.618...
#declare PhiIndex = 4; // Set the radix of Phi for the major radius.
// Calculate orthogonal major/minor radii.
#declare radiusMajor = pow(Phi, PhiIndex );
#declare radiusMinor = radiusMajor-pow(Phi, PhiIndex - 4 ); // Four degrees of Phi smaller.
#declare ySquash = 1;
#declare radiusConnector = .75;
#declare radiusConductor = .15;
#declare showConnectors = 0;
#declare showQuadrantConnectors = 0;
// Toggle parts of the scene.
#declare showBaseTorus = 10;
#declare showTorusKnots = 01;
#declare showAxes = 0;
#declare axesRadius = 0.1;
#declare colorA = Green*.5;
#declare colorB = Orange*.5;
// Center objects on the origin for passing to the macro
#declare objA = sphere{0, radiusConductor texture{pigment{color colorA}}};
#declare objB = sphere{0, radiusConductor texture{pigment{color colorB}}};
#declare objLugA = sphere{ 0, radiusConnector texture{pigment{color colorA }} }
#declare objLugB = sphere{ 0, radiusConnector texture{pigment{color colorB }} }
union{
// Bifilar torus knots.
#if(showTorusKnots)
#local phaseAngle = 360/phaseCount;
#local j = 0; // While-loop index
// Render multiple bifilar phases rotated apart by multiples of phaseAngle //
union{
#while(j < phaseCount)
// Call torus knot macro
// torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, major, minor, poloid, toroid, polyphase, delta)
torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, radiusMajor, radiusMinor, poloid, toroid, j*phaseAngle, plotDelta)
//object{torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, radiusMajor, radiusMinor, poloid, toroid, j*phaseAngle, plotDelta) rotate -y*90}
#declare j = j + 1; // Increment loop index
#end
scale y*ySquash
//clipped_by{plane{-x,0}}
}
#end // Moebius bifilar torus knots.
#if(showBaseTorus)
torus{ radiusMajor,radiusMinor *.9994
texture { pigment{rgbt<1,1,0,.985>} }
//texture { pigment{rgbt<1,1,1,.07> } }
//clipped_by{plane{z,0}}
scale y*ySquash
}
#end
#if(showAxes)
axes(radiusMajor+radiusMinor*1.0, axesRadius,1,1,1)
#end
} // end scene union{}