Saint Catherine was the daughter of the pagan governor of Alexandria. She declared that she would only marry someone who surpassed her in beauty, intelligence, wealth and status. It turned out that someone was Christ.
She went to the Roman Emperor to ask him to stop persecuting Christians. Although she failed, she did convert many of the philosophers she debated with, and also the Empress. Unamused, the Emperor had her condemned to be broken on the wheel. However, when she touched it, the wheel itself broke, so she was actually beheaded. An angel turned up, but rather than healing her for more torture as they seemed to have a habit of doing, it took her body to Mount Sinai.
Saint Catherine has a sword in the picture as she was one of the voices in Joan of Arc's head, and directed Joan to a shrine of Saint Catherine where she dug up the sword of Charles Martel. She gets a broken wheel for obvious reasons. An alternative would be to dress up "headless". Halo optional.
 This kind of thing never ends well.
 Probably our friend Diocletian or his co-emperor Maximian
 Hence Catherine Wheel.