To familiarize herself with a CNC milling machine, a friend wants to make some 12-sided gaming dice (regular dodecahedrons) out of solid aluminum. Her question: what is the dihedral angle between any two adjacent faces?

Three of the dodecahedron’s pentagonal faces meet at each vertex. Erect a small sphere around one vertex. Each of the three faces that meet there intersects the circle in a segment of great circle arc, forming a spherical triangle. The arc length of each side of this spherical triangle is the angle at the vertex of the regular pentagon, which is (5-2)*180/5 = 108°. The angles between the sides of the spherical triangle are equal to the dihedral angles between the faces of the dodecahedron, so we must merely solve for the angles of the spherical triangle. That can be accomplished through the law of cosines for spherical triangles (i.e., in spherical trigonometry):

cos a = cos b cos c + sin b sin c cos A, where lower case represents a side and upper case an angle, and side a is opposite angle A.

In this case all the sides are 108°, so cos 108 = (cos 108)^2 + ((sin 108)^2) cos x. Therefore cos x = (cos 108 – (cos 108)^2) / (sin 108)^2 = -.447213595499954, so x = 116.5650511770777 degrees, which is the answer, to the precision shown.

Posted by Charlie on 2003-03-31 09:10:40