Odds are good that, to an outsider, one of the most peculiar aspects of tabletop roleplaying is its way of generating random numbers. Whereas most board and wargames use "traditional" six-sided dice whose general shape and appearance hasn't changed for a few thousand years, RPGs - starting with Dungeons & Dragons in 1974 - have a three decades-long love affair with dice of different polyhedral shapes.
Precisely how, when, and by whom such dice were introduced into proto-roleplaying is a matter of some debate. David Wesely, whose contributions to the prehistory of roleplaying are now well established, claims to have "invented" - or at least "re-invented" - the use of such dice in gaming. In 2006, on the forums of the RPG collection website The Acaeum, he claimed: "I would like to lay claim to having 'invented' polyhedral dice. I was the first person to USE what were then being sold as 'Models of the five regular polyhedra' (for mathematics teachers to show to their students), AS DICE. I have since seen a book that claims that the Japanese were already using three D-20s, numbered 0-9 twice, to generate 3-digit decimal random numbers at some time before 1976. So it may be that they also invented this use for polyhedra, but I was unaware of them so I am at least an independant [sic] re-inventor. And it was my introducing the D4, D8, D12 and D20 to our gaming group in 1965 that led to them being used in RPGs and D&D."
Meanwhile, D&D co-creator, Dave Arneson, who had been a player in Wesely's free-form wargames campaigns, is sometimes credited with having first encountered polyhedral dice while visiting England and brought them back with him to use in his Blackmoor campaign. Similar stories are told about D&D's other co-creator, Gary Gygax. The truth of the matter may never be known with certainty and, like many aspects of early roleplaying history, it's quite possible that several people independently hit upon similar ideas.
What is certain is that Volume 1 of the original Dungeons & Dragons rules included among its "recommended equipment" four-sided, six-sided, eight-sided, twelve-sided, and twenty-sided dice -- all of them regular (or Platonic) solids. Interestingly, the rules at that time only made much explicit use of six and twenty-sided dice, with the others being relegated to very marginal uses until the release of the game's first supplement in 1975. Let's take a brief look at each of them.