/* tetrahedron b4 = pi/2 - a4/2 = 35.26439 v1 = ( cos(b4), 0, sin(b4)) v2 = (-cos(b4), 0, sin(b4)) v3 = ( 0, cos(b4), -sin(b4)) v4 = ( 0,-cos(b4), -sin(b4)) cube faces v1 = 0, 0, 1 v2 = 1, 0, 0 v3 = 0, 1, 0 v4 = 0,-1, 0 v5 = -1, 0, 0 v6 = 0, 0,-1 octahedron faces edges v1 = ( 1, 1, 1)/3**0.5 e1 = ( 1, 0,1)/2**0.5 v2 = (-1, 1, 1)/3**0.5 e2 = ( 0, 1,1)/2**0.5 v3 = (-1,-1, 1)/3**0.5 e3 = (-1, 0,1)/2**0.5 v4 = ( 1,-1, 1)/3**0.5 e4 = ( 0,-1,1)/2**0.5 v5 = (-1, 1,-1)/3**0.5 v6 = ( 1, 1,-1)/3**0.5 v7 ( 1,-1,-1)/3**0.5 v8 = (-1,-1,-1)/3**0.5 */ /* dodecahedron 5.0*(cos(angle12))**2 = 1.0 angle12 = 63.43494882292 cos(pi/5) = 0.80901699 sin(pi/5) = 0.58778525 sin(0.4 pi) = 0.95105652 cos(0.4 pi) = 0.30901699 sin(0.8 pi) = 0.58778525 cos(0.8 pi) =-0.80901699 icosahedron vertices or dodecahedron faces unit vectors: v1 = 0,0,1 v2 = 0.89442719, 0, 0.44721360 v3 = 0.27639320, 0.85065081, 0.44721360 v4 = -0.72360680, 0.52573111, 0.44721360 v5 = -0.72360680,-0.52573111, 0.44721360 v6 = 0.27639320,-0.85065081, 0.44721360 v7 = -0.27639320, 0.85065081,-0.44721360 v8 = 0.72360680, 0.52573111,-0.44721360 v9 = 0.72360680,-0.52573111,-0.44721360 va = -0.27639320,-0.85065081,-0.44721360 vb = -0.89442719,-0.0,-0.44721360 vc = 0,0,-1 icosahedron faces or dodecahedron vertices w5 = (v6-v1) x (v2-v1) = 0.47022820, 0.34164078, 0.76084521 w1 = (v2-v1) x (v3-v1) = 0.47022820,-0.34164078, 0.76084521 unit(w5) = 0.49112347, 0.35682208, 0.79465447 unit(w1) = 0.49112347,-0.35682208, 0.79465447 angle20 = acos(unit(w1) dot unit(w5)) = acos(.74535597) angle20 = (5/9)**0.5 .729727656227 radians 41.81031489578 degrees w2 = (v3-v1) x (v4-v1) = -0.18759247, -0.57735027, 0.79465447 w3 = (v4-v1) x (v5-v1) = -0.60706200, 0.0, 0.79465447 w4 = (v5-v1) x (v6-v1) = -0.18759247, 0.57735027, 0.79465447 accurate to 7 decimal places: w6 = 2 9 8 0.982 24695 0.0 -0.187 59247 7 8 3 2 0.794 65448 0.577 35027 0.187 59247 8 3 8 7 0.303 53100 0.934 17236 -0.187 59247 9 7 4 3 -0.303 53100 0.934 17236 0.187 59247 10 4 7 b -0.794 65448 0.577 35027 -0.187 59247 w11 b 5 4 -0.982 24695 0.0 0.187 59247 12 5 b a -0.794 65448 -0.577 35027 -0.187 59247 13 a 6 5 -0.303 53100 -0.934 17236 0.187 59247 14 6 a 9 0.303 53100 -0.934 17236 -0.187 59247 15 9 2 6 0.794 65448 -0.577 35027 0.187 59247 w16= -0.49112347, -0.35682208, -0.79465447 w17= 0.18759247, -0.57735027,-0.79465447 16 = 0.60706200, 0.0, -0.79465447 18 = 0.18759247, 0.57735027,-0.79465447 20 = -0.49112347, 0.35682208,-0.79465447 icosohedron edges b20 = a12/2 = 31.71747441146 cos(b20) = sin(b20) = e1 = (cos(b20),0,sin(b20)) .850650808,0,.5257311121 e2 = (cos(b20)cos(0.4pi),cos(b20)sin(0.4pi),sin(b20)) e3 = (-cos(b20)cos(0.2pi),cos(b20)sin(0.2pi),sin(b20)) e4 = (-cos(b20)cos(0.2pi),-cos(b20)sin(0.2pi),sin(b20) e5 = (cos(b20)cos(0.4pi),-cos(b20)sin(0.4pi),sin(b20) -unit(e3+e4) dot e1 unit(cos(b20)cos(0.2pi),0,sin(b20)) dot (cos(b20),0,sin(b20)) .79465447,0,-.607061998 37.3773682 dodecahedron edges b12 = a20/2 = e1 = (cos(b12),0,sin(b12)) e2 = (-0.5*cos(b12),0.866*cos(b12),sin(b12)) e3 = (-0.5*cos(b12),-0.866*cos(b12),sin(b12)) */ /* http://www.dnaco.net/~michael/domes/books/mathematics.html geodesic@dnaco.net Michael Rader http://www.povray.org/rtn/rtnv4n1.html#art28 (0,+-p,+-1), (+-1,0,+-p), (+-p,+-1,0) (0,+-i,+-p), (+-p,0,+-i), (+-i,+-p,0), (+-1,+-1,+-1) p= (sqrt(5)+1)/2 i= (sqrt(5)-1)/2 = 1/9 = p-1 HSM Coxeter 1948 Regular Polytopes p 52-53 Methuen London http://www.faqs.org/faqs/graphics/algorithms-faq/ 6.07 http://cm.bell-labs.com/netlib/polyhedra/index.html http://netlib.bell-labs.com/netlib/polyhedra/index.html http://www.netlib.org/polyhedra/index.html http://www.cc.iastate.edu/old_answers/packages/graphics/algos.faq.html http://www2.hku.nl/~lennart1/docs/algos.txt http://www.chesworth.com/pv/graphics/graphics_algorithms_faq.htm http://www.scidiv.bcc.ctc.edu/Math/voroni.html */