The dome is the leading idea or motif in Byzantine ecclesiastical architecture; the domes are placed over square, not circular apartments, and their bases are brought to a circle by means of pendentives.
Why should a Ring-motif be transformed into the theme representing Walhalla?
As regards the sources of the narrative part of the Periodoi, it is possible that the "recognition" motif was a literary commonplace.
The motif of the former is the poet's futile endeavour, in a dream, to ward off the arrows of Dame Beautee by Reason's "scheld of gold."
The fact that this theme is commonly called the " Ring-motif " is a glaring instance of what Wagner has had to endure from his friends.