As I recall, it is not something particular about 60deg angles. But the impossibility of arbitrary angle trisection.
A 60 degree angle cannot be trisected by ruler and compass constructions.
https://twitter.com/AlgebraFact/status/1379531950239719441 On Twitter
Reply or react to this post via Webmentions or reply or like to the Mastodon, Twitter or Instagram post.
yeah, the key word being "arbitrary". you can trisect certain angles, if you know their degree, a 90° angle for example... since 60 degree is also a "nice" number, it might have been possible, but seems like that is not the case.