Structural constraints often govern the patterns we find in poetry -- well-known in poetic history are rhythm-and-rhyme patterns including the sonnet and the villanelle and the limerick, and the syllable-counting pattern of some Haiku. Because many poems were shared orally, rather than in writing, patterns of counting and sound helped to ease the challenges of remembering.
For me a wonderful source for learning about new poetic forms is the blog of poet Marian Christie -- a writer and scholar, born in Zimbawe and now living in England , who has studied and taught both mathematics and poetry. In her very fine blog, Poetry and Mathematics, found here, Christie explores many of the influences that mathematics can have on poetry -- including, here in a recent posting, some effects transmitted by the SHAPE of a poem.