The Dancing Bird Pinwheel is an extremely simple synthesis, and there are no major requirements for it.
You will automatically learn the recipe once you enter the Atelier.
Synthesizing the Dancing Bird Pinwheel
In order to complete this recipe, you will need four generic ingredients.
The Dancing Bird Pinwheel synthesis acts mostly as a filler for the main story. Use your ingredients with the lowest Quality and the lowest Elemental Level to save materials.
Start by adding a Lumber-category ingredient to the central Quality loop.
Insert a Plant-category ingredient in the Quality loop on the bottom-left corner.
Use a Flower-category ingredient in the Quality loop on the bottom-right.
Add a Lumber-category ingredient to the Quality loop on the top.
With the core loops filled out, you can now complete the synthesis.
Head back outside the Atelier. A long cutscene will play, during which you will learn Empel’s origins. Make sure to set some time aside to watch it.