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.
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.
Why should a Ring-motif be transformed into the theme representing Walhalla?
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."
As regards the sources of the narrative part of the Periodoi, it is possible that the "recognition" motif was a literary commonplace.