Creating Photo-realistic Images and Animations

A Pentagon:

We'll construct our dodecahedron from twelve congruent pentagons. Suitable coordinates for the 20 vertices of a dodecahedron centered at the origin are as follows:

\((\pm 1, \pm 1, \pm 1)\)

\((0, \pm 1/\phi, \pm\phi)\)

\((\pm 1/\phi, \pm\phi, 0)\)

\((\pm\phi, 0, \pm 1/\phi)\)

where \(\phi = (1+\sqrt5)/2\) is the golden ratio [3].

The POV-Ray scene description language is “Turing Complete”, which means it is capable of expressing any algorithm expressible by the familiar programming languages like C++ or Java. It has branching and looping constructs and allows the description of functions. Functions are most often defined in POV-Ray using the macro construct. We use the macro construct below to define a function called “Pentagon” which takes the coordinates of five points in space as input then constructs a pentagon as output. The union construct allows primitive objects to be gathered into a single compound object. The spheres are added at the endpoints to make the joints smooth. Two constants, phi and c = 1/phi are declared and used to make the code more readable. The code below creates one of the pentagons that form the final scene. Save it as dodec-04.pov and render it (Figure 5). (Changes from dodec-03.pov above are indicated in boldface.)

// povray +P +I <strong>dodec-04.pov</strong> +W640 +H360 +A
 
#include "golds.inc"
 
background{ color rgb<0.2,0.2,0.45>}
 
camera {
        location <0, -8, 0>
        up <0, 1, 0>
        right <-1.78, 0, 0>
        look_at <0, 0, 0>
        angle 60
        rotate <0, 0, 0>
}
 
light_source {
        <100,-100,100>
        color rgb<1,1,1>*2.0
}
 
<strong>#declare phi = (1 + sqrt(5)) / 2;
#declare c = 1 / phi;
 
#macro Pentagon(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4, x5, y5, z5)
        union{
                cylinder {<x1, y1, z1>,   <x2, y2, z2>,  0.1}
                sphere {<x1, y1, z1>, 0.1}
                cylinder {<x2, y2, z2>,  <x3, y3, z3>, 0.1}
                sphere {<x2, y2, z2>, 0.1}
                cylinder { <x3, y3, z3>, <x4, y4, z4>, 0.1}    
                sphere {<x3, y3, z3>, 0.1}          
                cylinder { <x4, y4, z4>, <x5, y5, z5>, 0.1}
                sphere {<x4, y4, z4>, 0.1}        
                cylinder { <x5, y5, z5>, <x1, y1, z1>, 0.1}
                sphere {<x5, y5, z5>, 0.1}
        }            
#end
 
object {
        Pentagon( 0, -phi, c, 0, -phi, -c, 1, -1, -1, phi, -c, 0, 1, -1,  1)
        texture { T_Gold_5A }
        scale 1.25
}  <br /></strong>

Figure 5: A pentagon constructed from five cylinders

The Final Scene:

The only thing left is to add the remaining 11 pentagons. Note that we have declared a dodecahedron to be the union of twelve pentagons. The code is given below -- save it as dodec-05.pov and render it (Figure 6). (Changes from dodec-04.pov are indicated in boldface.)

// povray +P +I <strong>dodec-05.pov</strong> +W640 +H360 +A<br /><br />#include "golds.inc"<br /><br />background{ color rgb<0.2,0.2,0.45>}<br /><br />camera {<br />        location <0, -8, 0><br />        up <0, 1, 0><br />        right <-1.78, 0, 0><br />        look_at <0, 0, 0><br />        angle 60<br />        rotate <0, 0, 0><br />}<br /><br />light_source {<br />        <100,-100,100><br />        color rgb<1,1,1>*2.0<br />}<br /><br />#declare phi = (1 + sqrt(5)) / 2;<br />#declare c = 1 / phi;<br /><br />
#macro Pentagon(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4, x5, y5, z5)
        union{
                cylinder {<x1, y1, z1>,   <x2, y2, z2>,  0.1}
                sphere {<x1, y1, z1>, 0.1}
                cylinder {<x2, y2, z2>,  <x3, y3, z3>, 0.1}
                sphere {<x2, y2, z2>, 0.1}
                cylinder { <x3, y3, z3>, <x4, y4, z4>, 0.1}    
                sphere {<x3, y3, z3>, 0.1}          
                cylinder { <x4, y4, z4>, <x5, y5, z5>, 0.1}
                sphere {<x4, y4, z4>, 0.1}        
                cylinder { <x5, y5, z5>, <x1, y1, z1>, 0.1}
                sphere {<x5, y5, z5>, 0.1}
        }            
#end<br /><br /><strong>#declare dodecahedron = <br />union {<br />        Pentagon( c, 0,  phi, -c, 0, phi,  -1,  1,  1, 0,  phi,  c,  1,  1,  1)<br />        Pentagon(-c, 0,  phi,  c,  0, phi,  1, -1,  1, 0, -phi,  c, -1, -1,  1)<br />        Pentagon( c, 0, -phi, -c,  0, -phi,-1, -1, -1, 0, -phi, -c,  1, -1, -1)<br />        Pentagon(-c, 0, -phi,  c,  0, -phi, 1,  1, -1, 0,  phi, -c, -1,  1, -1)<br />        Pentagon( 0, phi, -c,  0,  phi,  c, 1,  1,  1, phi,  c,  0,  1,  1, -1)<br />        Pentagon( 0, phi,  c,  0,  phi, -c,-1,  1, -1,-phi,  c,  0, -1,  1,  1)<br />        Pentagon( 0,-phi, -c,  0, -phi,  c,-1, -1,  1,-phi, -c,  0, -1, -1, -1)<br />        Pentagon( 0,-phi,  c,  0, -phi, -c, 1, -1, -1, phi, -c,  0,  1, -1,  1)            <br />        Pentagon( phi, c,  0,  phi, -c,  0, 1, -1,  1, c,  0,   phi, 1,  1,  1)<br />        Pentagon( phi,-c,  0,  phi,  c,  0, 1,  1, -1, c,  0,  -phi, 1, -1, -1)<br />        Pentagon(-phi, c,  0, -phi, -c,  0,-1, -1, -1,-c,  0,  -phi,-1,  1, -1)<br />        Pentagon(-phi,-c,  0, -phi,  c,  0,-1,  1,  1,-c,  0,   phi,-1, -1,  1)<br />}<br /><br />object {<br />        dodecahedron<br />        texture { T_Gold_5A }       <br />        scale 1.25 <br />}<br /></strong>

Figure 6: The Final Scene