Except for the square, I know no platonic solid has integer dihedral angles. Are there different regular divisions besides 360 of a circle that would allow integers for the dihedral angles of the other Platonic solids? Just wondering
No, because except for the cube, the other dihedral angles are irrational with respect to a complete circle.
By that definition, even the cube has irrational angles. I’m talking about dividing a circle into a numbers of degrees besides 360, that would allow the other solids to have integer angles. For instance, the tetrahedron almost divides a circles into 5 parts with its @70.53° angles, but not quite.
No, a cube’s dihedral angles are exactly 1/4 of a complete circle. None of the other platonic solids have a rational portion of a complete circle (I think. I’m not as well versed in the inverse trig functions as I would like.)
Not really, unless you divide a circle into an irrational number of degrees. Rational or irrational, there’s no single way to divide a circle into X degrees that will result in all Platonic solids having integral dihedral angles.
It makes more sense if you think of things in radian terms instead of degrees and look at how you can compute the dihedral angle, theta.
Say we denote a Platonic solid as {p,q} where p is the number of edges per face and q is the number of faces that meet at a vertex. So, a cube would be {4,3} while a tetrahedron would be {3,3}.
Then, the dihedral angle theta is given by:
sin(theta/2) = cos(pi/q) / sin(pi/p)
We need a combination of both p and q such that this cosine/sine ratio works out to something “nice” as a fraction of pi. A complete circle is 2pi. So, the key figure isn’t that we get an integer out but a rational fraction of pi.
Things work out for a cube because the dihedral angle works out to 1/4 of a circle or 2pi/4, i.e. theta = pi/2. So, as long as you divide a circle into a multiple of 4, the dihedral angle of a cube will be an integer.
Things don’t work out for the other Platonic solids.
For example, take the tetrahedron {3,3}. We have
sin(theta /2 ) = cos(pi/3) / sin(pi/3)
sin(theta / 2) = (1/2) / (sqrt(3)/2)
sin(theta / 2) = 1/sqrt(3)
This number doesn’t end up producing a nice fraction of pi. To produce an integer number of “degrees” for a tetrahedron, we would need to divide the circle into an irrational number of degrees.
The same thing holds for the other Platonic solids, cube excepted.
It just seems wrong that these solids would have an ultimately non-integer measurement about them. They are so simple and intuitive otherwise.