One of the (many, many) things that irritate me about the world is the way that people who've never actually taken the trouble to read the great philosophers have no compunction in quoting their 'ideas' anyway. This means that these thinkers often get taken completely out of context and some of the most interesting ideas in human history are reduced, by popular usage, to mere platitudes. 
The classic example of this is the use of the word Platonic to mean an asexual relationship (the OED defines Platonic as 2. Applied to love or affection for one of the opposite sex, of a purely spiritual character, and free from sensual desire. Also of affection for one of the same sex.). All you have to do is actually read the Symposium to see how far from Plato's original discussion this usage is.
What Plato says
Just to set the record straight, here (in a grossly simplified version) is what Plato is actually saying (he puts the words in Socrates' mouth ; actually, he puts them in Socrates' mouth as being the words he heard from the oracle Diotima, but the argument is clearly his own):
If Love exists, Plato starts by arguing, it must be for something - that is to say it must have an object or end in mind. This end, Plato argues (more from empirics than anything else) is good and beauty - the purpose of which is to grant man happiness. Love, then, is the pursuit of everlasting good. But how is a mortal creature to achieve this end? If we are all doomed to die, surely this pursuit is bound to end in failure? It follows that he who loves must crave immortality, and must seek for ways to make himself immortal. The purpose of love, then, is generation - the extension of the self beyond the confines of one's own life.
How is this generation to be achieved, and what form must it take? That is, logically, a function of how the self is defined. Those who define the self as a physical entity, as bloodline or family, are "pregnant in the body" and must therefore achieve immortality through physical procreation - by having children in other words, and allowing these children to carry on their names, their memories, their bloodlines (Plato would have said genes, I'm sure, if he'd known about them).
But sexual reproduction is not the only way to ensure the immortality of the self. There is a nobler, higher way, one that is accessible to those who define their self in terms of ideas and interests, in terms of the soul; those who are, in Plato's terms "pregnant in the spirit". Such people will 'reproduce' through friendships / relationships with others - passing on their thoughts, their experience, their knowledge, ensuring that they live on in the minds of men rather than in their blood. The physical act of sex is irrelevant here - the true heirs of the spiritual lover are his works, and these are superior to mortal children precisely because they possess, in themselves, the power to last forever. Plato speaks of the 'children' of Homer and argues that these will be revered and respected long after all the mortal bloodlines of Homer's contemporaries have been lost. (Yeats writes: "Pardon, old fathers, if you still remain....Pardon that for a barren passion's sake / Although I have come close on forty-nine / I have no child, I have nothing but a book / Nothing but that to prove your blood and mine.")
These then, are the lesser mysteries of Love. But Plato does not stop here, he goes on to propose a heirarchy of love, a spiritual staircase of ever increasing vision. I quote: "the true order of going,or being led by another, to the things of love, is to begin from the beauties of earth and mount upwards for the sake of that other beauty, using these as steps only, and from one going on to two, and from two to all fair forms, and from fair forms to fair practices, and from fair practices to fair notions, until from fair notions he arrives at the notion of absolute beauty, and at last knows what the essenceof beauty is". There is thus, in Plato's conception of the world, an absolute beauty, a vision so dazzling that compared to it all lesser human loves fall away.
Why Platonic Love may not be so Platonic after all
A few things are important to notice here. First, Plato is not for a minute arguing against sexual pleasure / sensual gratification. His point is simply that these are not essential to the higher forms of love - they are not the reason for it. True Platonic lovers may sleep together, but the sexual tie is not what binds them to each other and therefore sexual fidelity is unimportant. Equally, reproduction is not the main purpose of this love (as it is with other forms of male-female relationships) so that gender becomes irrelevant as well - sexual relations with one of your own gender may actually make more sense, since it may be easier to pass on ideas / interests to someone of your own sex, which is the whole point of the relationship in the first place. The true Platonic lover finds beauty in everything, even sex, but sex is not more or less important than anything else and therefore is not what defines the relationship as it does with lower, more human loves.
In fact, the entire argument in the Symposium centres around (and glorifies) homosexual relationships between older men (lovers) and their younger proteges (beloveds). These are the relationships that the culture of the ancient Greeks praises and respects - Plato is explicitly critical of women and has nothing but contempt for relationships with them (political correctness is not a strong suit of the Greek philosophers - reading the Symposium can be a feminist's worst nightmare - but I think it's important to look beyond the bias of the times and see the truth in what Plato is saying without getting hung up on this - just because he's wrong about women doesn't mean he's wrong about anything else). It's ironic therefore, that the word Platonic should be used primarily in the sense of love or affection for the opposite sex, when that is certainly not the sense in which Plato meant it. One suspects that centuries of victorian morality that considered homosexuality absolutely taboo may have had something to do with this.
It should be obvious by now why the traditional use of the word Platonic is so wrong. What we normally consider critical for a 'Platonic' relationship - the absence of sex - isn't really important in Plato's view; on the other hand, what truly Platonic love requires - a deep, uplifting spiritual connection - may not exist in many relationships that we call 'Platonic' .
 I'm not even going to get into how much it irritates me that we live with a simplified (and misguided) image of these men. So Descartes is that "I think therefore I am" guy. Plato is the guy who said you can have love without sex (I mean, look, it's not even as though the Symposium is his major work, for crying out loud). And don't even get me started on what people keep doing to Nietszche.
 It's ironic that Plato uses Socrates as his mouthpiece to deliver a lengthy and fairly prescriptive discourse on Love. It's unclear that the real Socrates - who's entire credo was to question rather than to judge - would have been anything but suspicious of something stated this baldly as being 'the truth'
 The crummiest joke I ever heard about this was about this woman who gets caught sleeping with a man half her age and someone says to her "You said it was all Platonic!" to which she replies "It was. Play for him, tonic for me." (hey, I warned you it was a crummy joke)
 Knowing this can make conversations fairly complicated:
Friend: "So, this thing with you and her, is it Platonic or what?"
Yours Truly: "Errr...no, not really."
F: "So you're sleeping with her then."
YT: "No, no, nothing like that."
F: "So it's Platonic then"
YT: "No, it isn't. I mean we're not sleeping together but there's no major spiritual connection either. We're just friends. That is to say, not JUST friends, because you know friends are important too. More important than relationships, you know, they last longer and cause less stress. So friends. Not just friends. Just friends. I mean friends."
F: "So you want to sleep with her then."
YT: "No, no, I told you - it's not like that - it's totally platonic. I mean it's not really. I mean it's asexual. Not that we're actively avoiding sex or anything - it's just not part of the equation. So call it ultra-sexual. Or is it infra-sexual? Oh, damn it! Just forget about her okay. Let's just go watch the bloody movie"