Ballet shoes can be referred to as "flat shoes" because dancers wear them to dance en pointe. When a child starts ballet training for the first time, they will wear ballet shoes. As the dancer gets older, they will progress to wearing pointe shoes.
Raised heels are stated to have been a response to the problem of the rider's foot slipping forward in stirrups while riding. The "rider's heel," approximately 1-1/2" high, appeared around 1500. The leading edge was canted forward to help grip the stirrup, and the trailing edge was canted forward to prevent the elongated heel from catching on underbrush or rock while backing up, such as in on-foot combat.
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