Turing Machines need not be constrained to a tape. In 1985, Christopher Langton considered a Turing machine that would move around on a grid. The squares of the grid could change color, just like the tape. The moving head has a state and position, as before. Also, the direction is stored. In the simplest two cases, the moving head always changes the color of the underlying cell, and the state never changes. Also, whenever the underlying cell is turned from white to black, the direction turns to the right. In Langton's Ant, when the color changes from white to black, the direction turns to the left. In the Binary Counter, which seems to be my own discovery, the direction doesn't change.