I asked in the original review if any sf novelist has ever tried writing a society constructed along Plato's lines. This provoked a number of responses, with altariel proposing John Christopher's The Guardians, matgb suggesting Judge Dredd, and bellatrys putting forward Ayn Rand's Anthem. I have only a passing acquaintance with Dredd, and no knowledge of the other two at all, so can't comment further on them (though WikiPedia, for what that's worth, says that Silverberg's A Time of Changes is a response to Rand).
However, both saare_snowqueen and applez suggested that I consider a book that I have read, Aldous Huxley's Brave New World, as an attempt to write a society constructed along Plato's lines. Of course, I see the similarities: the rigid stratification into classes, the eugenics and abolition of the family, the social control. Both are works clearly in the utopian tradition of literature, if on different wings of it (Plato's account is not intended as fiction but as a manifesto; Huxley presents his fictional society as a dystopia rather than a utopia). And of course one must concede at the start that Huxley is writing in the shadow of Plato, Sir Thomas More, and H.G. Wells and maybe Fritz Lang (but not, at least according to Huxley, Zamyatin). But I don't think Huxley is trying to do Plato, or even to update Plato for the early twentieth century. There are some important differences:
The Arts: The question of aesthetics is crucial to both The Republic and Brave New World, and both authors agree that the arts are vital for a good society. But while Plato has good poetry and music at the heart of his educational programme (indeed, his strictures on dealing with bad poetry are pretty censorious), Huxley's future is utterly dumbed down, with one of the most tragic themes of the book being that Shakespeare has not only been removed from the scene but has become incomprehensible to society. Both authors have constructed their future societies around how they deal with the arts, but they take completely different tacks. I think this is by far the most important difference between the two books, and is enough on its own to demonstrate convincingly that Huxley was not constructing his Brave New World along the lines of The Republic in particular.
War: I have not seen many analyses of The Republic which even mention what for me was one of the most striking aspects of it: that his enlightened philosopher-rulers are also warriors. Indeed, we are led into the discussion of what makes a good defender of the state by a discussion of what makes a good soldier. (I wondered if Sherri S. Tepper's The Gate to Women's Country had drawn from this a little.) The threat of war is ever present in the background of The Republic; it is absent from Brave New World. Of course, this is partly because Plato sees his Republic as an enlightened citadel in the middle of Greek disorder surrounded by barbarian chaos, while Huxley's World encompasses most of the globe. But only partly; even a global government can face conflict (as autopope has pointed out), and Huxley's is unusual in that it doesn't.
Industry: At first sight a trivial difference, since obviously The Republic was written some time before the Industrial Revolution and Plato can therefore be forgiven for failing to foresee Henry Ford. And one could argue that industry shapes the background to in Huxley's book as war does in Plato's. But in fact Huxley is all about mass-production of people as well as machines, and about socialism as a political response to industrialisation. In Plato, the slave underclass is largely though not entirely invisible; the economic structure of Huxley's society is completely different.
The elite: Superficially a similarity between the two books, in fact there are very big differences between how the elite of the two societies are chosen and function. Plato's rulers are educated into the role, and although there is a rigid breeding programme, it is also accepted that you can be promoted into the ruling class (or demoted out of it) depending on ability; and once you are in the ruling class, all are effectively equal. But Huxley's Alphas are chosen before birth and hypnotised into their roles. Huxley is doing a bit of a riff on education as conditioning, but his heart isn't init: his social conditioning is about closing minds down, while Plato's is about opening them. (It should be added perhaps that Plato has only three classes, plus the invisible slaves, in his society while Huxley has five, each of which is subdivided.)
Drugs: The use of the recreational drug soma is central to Huxley's society. Plato's citizens drink wine, but not too often to excess (apart from the archetypal tyrant); apart from that they are clean-living and healthy warriors as well as philosophers. Again, this goes to the heart of the two projects. Huxley believes that his society is unsustainable without drugging and conditioning the entire population into compliance; Plato believes that enlightened thought will be enough to carry the day. If only it were!
Anyway, this was a very interesting question to consider!