There are trisectable angles that are not constructible.
As usual, a number is constructible if it is constructible with a compass and a straightedge. Not all numbers are constructible. While examples of non-constructible numbers abound, I have in mind a particular one (let me know if you come up with another suitable number). In the following I'll be talking of constructible angles.
The angle I have in mind is 2π/7. That the angle is not constructible has been shown in the 19th century by Gauss and Wantzel. If it were, one of the famous problems of antiquity - that of constructing a regular heptagon - would have a solution. As Gauss and Wantzel showed, it does not.
Since 2π/7 is not constructible, so is the angle 4π/7. For, if it were constructible, we could then, by angle bisection, obtain 2π/7, which is impossible.
Obviously, π is a constructible angle. It then follows that 3π/7 = π - 4π/7 is not constructible. For, if it were, so would be 4π/7.
Now examine the two angles: 4π/7 and 3π/7. Their difference is π/7 - one third of 3π/7. The situation becomes interesting. Assume the angle 3π/7 was constructed anyhow by non-traditional means. Starting with 3π/7, we can construct 4π/7 and, therefore, the difference - π/7.
In other words, if the angle 3π/7 was given in the first place, then it would be possible to trisect it with the traditional tools only. It appears that the angle well deserves to be called trisectable. For the sake of comparison, recollect that constructible π/3 = 60° is not trisectable.
π/2 = 90° is both constructible and trisectable. Since there are only countably many constructible angles, "most" of angles are not constructible. Most of these are probably not trisectable either. Is there a proven example?
Yes, Andrew Schultz found this one: π/21 is neither constructible, nor trisectable. The claim is based on the following identity:
1/21 + 2/7 = 1/3
As was already mentioned, π/3 is constructible but not trisectable. So, first of all, π/21 is not constructible. For, if it were,
On the other hand, if π/21 were trisectable, so would be
The argument is completely general. Let A be a property of angles, such as being constructible or being trisectable. Then, schematically,
A - A = A
which simply says that the difference of two angles with property A also possesses that property. (Which is of course true for A being either constructible or trisectable.) From here, it follows that
non-A + A = non-A
i.e., the sum of two angles - one with A, the other without, does not have that property.
As a corollary, the sum of a constructible but non-trisectable angle and a trisectable but non-constructible angle is neither constructible, nor trisectable.
Are there other examples?
Note: Ed Fisher has a valid objection to the above argument.
- R. Honsberger, More Mathematical Morsels, MAA, 1991