Second image requires cutting paths. There's no method to create the outer partial circles unless you either mask them or cut them to delete portions.
What I posted above is as far as you can go before you start cutting circles.
I wouldn't use the appearance method on this one, since it's rather difficult to get an exact result. I would rather create circles with 6 anchors (this is possible with the script by s.h.). Those cricles can then scnap exactly. After that I would also be careful with the shapebuilder tool since it might produce unwanted result (especially closing of paths you don't want closed). Instead direct select the segments you dont need and just delete them.
Edit: to create the partly circles at the edges, first create them as full circles, then delete parts of them
Here is a quite different way, silly maybe, but completeæly accurate and very simple, especially if you just let it grow without deleting (much) underway. If you remove the outer circle, you may just click and delete top duplicates in the different stages where deletion is mentioned (Undo is your friend when you overdo it).
You may add the large circle as a copy (3 diameters) at any time if desired.
Because they have imposed a limit on how many answers can be marked as helpful I am refusing to mark any.
Yall will have to live with good ol fashion gratitude.
I really appreciate all of your ideas!
I had seen the Flower of Life as being made up of Reuleaux Triangles with Tricuspoids inside of them.
I had honestly never seen it as being made out of this single shape:
What do we call it?