# 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.

## References

- R. Honsberger,
*More Mathematical Morsels*, MAA, 1991

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