rotation matrix for -pi/5 = -36 deg about the z axis R rotation matrix for -2pi/5 = -72 deg about the z axis | | | | |[3-[5]]/2[2] [5+[5]]]/2[2] 0| |[3+[5]]/2[2] [5-[5]]]/2[2] 0 | **2 | | | | | | | | R**2= |-[5+[5]]]/2[2] [3-[5]]/2[2] 0| = |-[5-[5]]]/2[2] [3+[5]]/2[2] 0 | | | | | | | | | | 0 0 1| | 0 0 1 | | | | | In the z=0 plane, the unit projections are: angle x=R sin(-angle) y = R cos(angle) 0 0 1 -72 [5+[5]]/(2[2]) [3-[5]]/(2[2]) -144 [5-[5]]/(2[2]) -[3+[5]]/(2[2]) 144 -[5-[5]]/(2[2]) -[3+[5]]/(2[2]) 72 -[5+[5]]/(2[2]) [3-[5]]/(2[2]) angle x y -36 [5-[5]]/(2[2]) [3+[5]]/(2[2]) -108 [5+[5]]/(2[2]) -[3-[5]]/(2[2]) 180 0 -1 108 -[5+[5]]/(2[2]) -[3-[5]]/(2[2]) 36 -[5-[5]]/(2[2]) [3+[5]]/(2[2]) The trig functions of pi/5 and 2pi/r can be derived from the half angle formula and law of cosines appled to a pentagon inscribed in a circle and the fact that an arc subtends an angle of one half the arc it the vertex is at the other side of the circle instead of the center, as shown at http://www.issi1.com/corwin/calculator/pentagon.txt . The dodecahedral and icosahedral angles are derived at http://www.issi1.com/corwin/calculator/proof.txt The rows of vertices and faces are at: top (0,0,1) vertices +36 (0,[10-2[5]]/[15],[5+2[5]]/[15]) faces (0,2/[5],1/[5]) vertices +36 (0,[10+2[5]]/[15],[5-2[5]]/[15]) vertices (0,[10+2[5]]/[15],-[5-2[5]]/[15]) faces +36 (0,2/[5],-1/[5]) vertices (0,[10-2[5]]/[15],-[5+2[5]]/[15]) bottom (0,0,-1) dodecahedron angle between face normals alpha arccosine 1/[5] angle to top row of vertices delta arccosine [5+2[5]]/[15] angle to second middle row of vertices epsilon arccosine [5-2[5]]/[15] check that delta dot epsilon = ([80]+[5])/15 = ([16]+1)[5]/15 = [5]/3 Unit vectors to faces top (0,0,1) A 0 (( 0 , 1 )2/[5],1/[5]) B -72 (( [5+[5]]/(2[2]), [3-[5]]/(2[2]))2/[5],1/[5]) D -144 (( [5-[5]]/(2[2]),-[3+[5]]/(2[2]))2/[5],1/[5]) 144 ((-[5-[5]]/(2[2]),-[3+[5]]/(2[2]))2/[5],1/[5]) 72 ((-[5+[5]]/(2[2]), [3-[5]]/(2[2]))2/[5],1/[5]) E -36 (( [5-[5]]/(2[2]), [3+[5]]/(2[2]))2/[5],-1/[5]) C -108 (( [5+[5]]/(2[2]),-[3-[5]]/(2[2]))2/[5],-1/[5]) 180 (( 0 , -1 )2/[5],-1/[5]) 108 ((-[5+[5]]/(2[2]),-[3-[5]]/(2[2]))2/[5],-1/[5]) 36 ((-[5-[5]]/(2[2]), [3+[5]]/(2[2]))2/[5],-1/[5]) bottom (0,0,-1) ---------------------------------------------------------------------- z axis A (0,0,1) top row B (0,[4/5],[1/5]) D ([4/5][5+[5]]]/(2[2]),[4/5][3-[5]]/(2[2]),[1/5]) =([5+[5]]]/[10]),[3-[5]]/[10],[1/5]) (+,- ,[1/5]) (-,- ,[1/5]) E (-,+ ,[1/5]) bottom row C ([5-[5]]]/[10],[3+[5]]/[10],-[1/5]) (+,- ,-[1/5]) (0,-[4/5],-[1/5]) (-,- ,-[1/5]) (-,+ ,-[1/5]) -z axis (0,0,-1) ---------------------------------------------------------------------- icosahedron angle between face normals arccosine [5]/3 angle to top row of faces delta arccosine [5+2[5]]/[15] angle to second middle row of faces epsilon arccosine [5-2[5]]/[15] angle to top row of vertices arccosine 1/[5] Unit vectors to faces: -36 (( [5-[5]]/(2[2]), [3+[5]]/(2[2]))[10-2[5]]/[15],[5+2[5]]/[15]) V -108 (( [5+[5]]/(2[2]),-[3-[5]]/(2[2]))[10-2[5]]/[15],[5+2[5]]/[15]) 180 (( 0 , -1 ))[10-2[5]]/[15],[5+2[5]]/[15]) 108 ((-[5+[5]]/(2[2]),-[3-[5]]/(2[2]))[10-2[5]]/[15],[5+2[5]]/[15]) 36 ((-[5-[5]]/(2[2]), [3+[5]]/(2[2]))[10-2[5]]/[15],[5+2[5]]/[15]) -36 (( [5-[5]]/(2[2]), [3+[5]]/(2[2]))[10+2[5]]/[15],[5-2[5]]/[15]) W -108 (( [5+[5]]/(2[2]),-[3-[5]]/(2[2]))[10+2[5]]/[15],[5-2[5]]/[15]) 180 (( 0 , -1 ))[10+2[5]]/[15],[5-2[5]]/[15]) 108 ((-[5+[5]]/(2[2]),-[3-[5]]/(2[2]))[10+2[5]]/[15],[5-2[5]]/[15]) 36 ((-[5-[5]]/(2[2]), [3+[5]]/(2[2]))[10+2[5]]/[15],[5-2[5]]/[15]) 0 (( 0 , 1 ))[10+2[5]]/[15],-[5-2[5]]/[15]) X -72 (( [5+[5]]/(2[2]), [3-[5]]/(2[2]))[10+2[5]]/[15],-[5-2[5]]/[15]) -144 (( [5-[5]]/(2[2]),-[3+[5]]/(2[2]))[10+2[5]]/[15],-[5-2[5]]/[15]) 144 ((-[5-[5]]/(2[2]),-[3+[5]]/(2[2]))[10+2[5]]/[15],-[5-2[5]]/[15]) 72 ((-[5+[5]]/(2[2]), [3-[5]]/(2[2]))[10+2[5]]/[15],-[5-2[5]]/[15]) 0 (( 0 , 1 ))[10-2[5]]/[15],-[5+2[5]]/[15]) -72 (( [5+[5]]/(2[2]), [3-[5]]/(2[2]))[10-2[5]]/[15],-[5+2[5]]/[15]) -144 (( [5-[5]]/(2[2]),-[3+[5]]/(2[2]))[10-2[5]]/[15],-[5+2[5]]/[15]) 144 ((-[5-[5]]/(2[2]),-[3+[5]]/(2[2]))[10-2[5]]/[15],-[5+2[5]]/[15]) 72 ((-[5+[5]]/(2[2]), [3-[5]]/(2[2]))[10-2[5]]/[15],-[5+2[5]]/[15]) ---------------------------------------------------------------------- The dot product of a vector to a vertex and that to the same face is the same for the dodecahedron and icosahedron and thus the radius of the inscribed sphere divided by the radius of the superscribed sphere is the same for both: A dot V = [5+2[5]]/[15] . The reciprocal, [3][5-2[5]], is the superscribed radius divided by the inscribed radius. The chord of a pentagonal face subtends an angle whose cosine is V dot X = -[25-20]/[15] + ([100 -20]/[15]) [3+5[5]]/2[2] = -1/[3] + [2][3+[5]]/[3] = ([2][3+[5]] - 1)/[3] = 4([3+[5]]/(2[2]) - 1/4)/[3] The length of a dodecahedron edge divided by the superscribed radius is 2 sin(arcsin(2/[5]) /2) = 2[1-1/[5]]/[2] = [2][[5]-1]/[5] and divided by the inscribed radius is [6/5][7[5]-15] . inscribed sphere [5+2[5]]/[15] 1 [15+7[5]]/[24] [5+[5]]/[10] superscribed 1 [3][5-2[5]] [5+5[5]]/(2[2]) [3/2][3-[5]] edge [2/5][[5]-1] [6/5][7[5]-15] 1 [12/5][[5]-2] edge center r [3+[5]]/[6] [5-[5]]/[2] [5][5+2[5]]/(2[3]) 1 The lenght of a icosahedron edge divided by the superscribed radius is 2 sin(arccos([5]/3) /2) = 2[1-[5]/3]/[2] = [2][3-[5]]/[3] and divided by the inscribed radius is [2][25-11[5]] . inscribed sphere [5+2[5]]/[15] 1 [25+11[5]]/[40] [3+[5]]/[6] superscribed 1 [3][5-2[5]] [3][3+[5]]/(2[2]) [5-[5]]/[2] edge [2/3][3-[5]] [2][25-11[5]] 1 [4/3][5-2[5]] edge center r [5+[5]]/[10] [3/2][3-[5]] [3][5+2[5]][20] 1 ---------------------------------------------------------------------- top row V ([[3-[5]]/[6],[5+5[5]]/[30],[5+2[5]]/[15]) upper middle W (([7+3[5]] - [3-5[5]]) /?,[5+2[5]]/[15],[5-2[5]]/[15]) lower middle bottom (0, , ) ---------------------------------------------------------------------- --------------------------------------------------------------------------- cos(delta) = [5+2[5]]/[15] cos(delta) = [2+[5]]/[15] ? cos(epsilon) = [5-2[5]]/[15] --------------------------------------------------------------------------- ? ([5+[5]]]/2[2] y ,[3-[5]]/2[2] y,-[1-yy]) (0,y,-[1-yy]) [3-[5]]/2[2] yy + 1-yy = [5]/3 ([3-[5]]/2[2] -1) yy = [5]/3 -1 need half angle formula to get out of this (1-cos(f)) = 2cos2(f/2) = 2 ([3+[5]]/2[2])**2 (3+[5])/4 yy = (3-[5])/3 yy = 4/3 (14 - 6[5]) / (9-5) = (14-6[5])/3 y = [2/3][7-3[5]] = sin(delta) cos(delta) = [1-yy] = [6[5]-11]/[3] ---------------------------------------------------------------------------- ? yy = (3-[5])2[2]/ ( 2[2] - [3-[5]] ) = (3-[5])2[2]/ ( (2[2][3-[5]] - (3-[5])) / [3-[5]] ) = aaa2[2]/ (2[2]a-aa) = aaa2[2] (2[2]a+aa) / (8aa-aaaa) = a2[2] (2[2]a+aa) / (8-aa) (8 - 3+[5]) = a2[2] (2[2]a+aa) (5-[5]) / (25-5) = [3-[5]]2[2] (2[2][3-[5]]+(3-[5])) (5-[5]) /20 ref http://www.issi1.com/corwin/calculator/icosahedron.jpg http://www.issi1.com/corwin/calculator/proof.txt http://www.issi1.com/corwin/calculator/summary.txt http://www.issi1.com/corwin/calculator/pentagon.txt http://www.issi1.com/corwin/calculator/unit_v.txt http://www.issi1.com/corwin/calculator/platonic.txt http://www.issi1.com/corwin/calculator/possible.txt