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This is a condition known as syndactyly or webbed digits. It is a rather common congenital malformation that happens about once in every 2000-3000 births. It varies greatly: simple or complex, complete or incomplete. In the simple form, only skin connects the digits. In the complex form, either nails or bones are fused. The most severe (or complex) forms of this condition can include loss of digits and/or digits that are indistinguishable from one another. The severe forms are nearly always associated with some sort of syndrome, such as Apert syndrome. Operations to separate the digits are relatively common, though many parents or adults with the condition opt against it, especially in the simple form since it often does not interfere with normal function.
This is something I am quite familiar with. My father has complete simple syndactyly on one foot. Also, my significant other (pictured above) has incomplete simple syndactyly on both feet (though it is more extensive on the right foot). Neither have had an operation to separate them.
Famous people with this condition include Dan Aykroyd and Ashton Kutcher.
Though I have concentrated on toes, syndactyly can also just as easily occur on hands. Here is a baby with complete complex syndactyly where three fingers are fused:
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