I think it's kind of dumb--if your chain brakes or slips, you have no way to stop--but to each their own, I guess. And yes, I HAVE seen a guy on a fixie break his chain and crash.
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1964 JRJ (Bob Jackson), 1973 Wes Mason, 1974 Raleigh Gran Sport, 1986 Schwinn High Sierra, 2000ish Colian (Colin Laing), 2011 Dick Chafe, 2013 Velo Orange Pass Hunter