Important people generally had their likeness preserved in some way – either through sculptures, profiles on coins, or paintings their contemporaries made of them. In some cases, the art style was very realistic, so we have a very good Idea of what the person looked like (for example, ancient Rome/Greece, though busts became more idealized as time went on). You can get a pretty good idea of what important ancient Romans looked like since they loved a good statue. There are also quite a few surviving painted portraits of random people during this era.
For less important people, or people that don’t have any surviving portrayals, we have to rely on written accounts of what they looked like. If there aren’t any specific descriptions of what a person looked like, then you just don’t really know.
Also if you have their remains (i.e. skeleton, skull), then you can do additional facial reconstruction (though these use guesswork, but can give a pretty good idea of their face) and DNA testing to get an idea of things like eye color, hair color/texture, etc.
Sometimes we have contemporary accounts of their appearance that have survived in literature, but a lot is written decades or centuries after their death and based on rumours or biased accounts.
Socrates was known to be ugly, for instance, written in works by his students during his life:
> In Plato’s Theaetetus, the geometer Theodorus describes the young Theatetus to Socrates as “very like you, for he has a snub nose, and projecting eyes, although these features are not so marked in him as in you.”
> In Xenophon’s Symposium, Socrates himself says that he has protruding eyes, a snub nose, thick lips, and a paunch. He jokes that these features are to his advantage—his eyes, for instance, enabling him to “squint sideways and command the flanks.” Since Xenophon’s Symposium is set in 422 BCE, Socrates is describing himself at the age of around 48.
However, his ugliness may have been exaggerated deliberately:
> Socrates was condemned to death in part for “corrupting the youth,” and the story of Alcibiades trying and failing hard to seduce him may have been invented by Plato to help rehabilitate his reputation. In this context, making him as ugly as possible served to diminish any threat that he might have posed. It also created a golden opportunity to present wisdom, even in the ugliest of bodies, as infinitely more beautiful than the most beautiful of bodies.
From a [Psychology Today article](https://www.psychologytoday.com/gb/blog/hide-and-seek/202304/how-ugly-was-socrates)
Other people may have had epithets – given names – that describe their appearance. Plato was potentially given his epithet during his youth while wrestling, as it could have meant “broad-shouldered”.
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