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Triangle Wreath

Regolo Bizzi had this wonderful triangle structure (http://regolo54.tumblr.com/post/134128840247/geometry) that got mewondering about which triangles have this property, how they connect and what would be a good way to do the outer rings. I messed around a bit with the color, but the Execute commands were causing weird behavior. If you can think of other features to add, let me know!