Hint for finding dodecahedral angle

Here is a short blurb giving my suggestions for finding the angle between faces of a dodecahedron.

First lie three "flaps" of the unfolded dodec flat in the xy-plane as shown below. The coordinates have been chosen to reduce computations. The first view is from directly above the origin O so you cannot see the z-axis; the second is from a viewpoint above the plane but away from the z-axis. A portion of the first quadrants of the xy-plane and yz-plane are marked.

Now we'll rotate two of the pentagonal flaps (by the same angle) until they meet. The angle through which this takes place is the supplement of the angle we want.

With the pentagon containing the points O, A, A held fixed in the xy-plane, rotate the pentagon containing O, A, B about the axis OA until the point B rises to lie in the yz-plane.  (We simultaneously, rotate the pentagon containing O, A, B about the axis OA until the point B rises to lie in the yz-plane, to see what is happening.  What we're actually requiring is that B and B meet, but because of the symmetric arrangement we've constructed, they meet in the yz-plane.)

The trick I suggest for finding this angle is to use the fact that the x-coordinate of B  will then be zero (when rotated to the final position) and this will ultimately give an equation we can solve.

So now what you have to do is remember how to rotate a point about a line! Of course, you'll need the coordinates of A and B in the original (flat) figure, but that should be easy, just some basic trig. You can assume any size for the pentagons, so I'd just let the edge have length 1 for convenience.

The method of rotating an arbitrary point about an arbitrary line should be in your notes, but you can also figure out how to do it in this case without resorting to a general method. Picture the path of the point B as it rotates about the axis OA. It is easy to calculate the distance of B from this line and its projection P onto the line. You can also use simple trig to do that, just by looking at the picture. Notice that |BP| is the radius of the circular arc on which B travels. As such, for any rotation angle θ, you can calculate the position of B using basic ideas (sines and cosines). I await your conclusions!

Created by Mathematica  (March 1, 2006)