Angles of an OctahedronDate: 07/20/98 at 18:33:51 From: Phani Subject: Octahedron I wanted to know the angle between two adjacent faces of an octahedron. Date: 07/22/98 at 11:27:54 From: Doctor Rick Subject: Re: Octahedron Hello, Phani. I assume you refer to a regular octahedron. It helps me to have a model of the regular octahedron in front of me. I have one that I made of index cards. Another can be seen at: http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html One way to compute the angle you want is to notice that the centers of the faces of a regular octahedron are the vertices of a cube, and that the diagonals of the cube are perpendicular to the faces of the octahedron. The angle between adjacent faces is the supplement of the angle between normals to the faces, which is the angle between diagonals of a cube. This angle can be computed most easily using vectors: make a cube with corners (+/-1, +/-1, +/-1). That's plus or minus 1 for each coordinate, giving 8 points. You want to find the angle between vectors A = (1, 1, 1) and B = (1, 1, -1), and you can do this using the rule that the dot product A.B = |A|*|B|*cos(angle). Then subtract this angle from 180 degrees. If you make a model of a tetrahedron with the same edge length as the octahedron, you will notice that the angle you are after is the supplement of the angle between faces of the tetrahedron.This angle in turn is the supplement of the "tetrahedral angle" that organic chemists care about - the angle at the center between two vertices of the tetrahedron. You can see this by the same argument I made above about faces and normals. And the upshot of this observation is that the angle you want is just the tetrahedral angle. If you don't want to do the calculation, just ask a chemist. Write back if you need more explanation. - Doctor Rick, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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