Theorizing the origins
Mathematical practice relating to intersecting circles can be seen from Babylonian clay tablets in 2100 – 1600 BC, although not presenting exactly the figure of the FOL. Eleanor Robson shows on her book the image of the four petal rosette or conclave square which ancient Babylonians called Apsamikkum that can be rendered as Cow’s Nose or Sound hole of the lyre. Ceremonial vessel from Indus Valley Harappa civilization 2600 – 2450 BC and others from Harappa and Mohenjo-daro indicates that this particular symbol was known in the Indus Valley already in 3000 BC. But because desing was known “from Halaf pottery of the sixth millennium BC to images of Neo-Assyrian textile” (The apsamikku in Neo-Babylonian Mathematics page 214) in the Syria, it is quite certain that Cow’s Nose symbol came somewhere from the ancient Fertile Crescent.
Flower of Life
The FOL symbol instead is basically produced by six intersecting circles around the seventh central circle forming a hexagon on the crossing points. It is even easier to draw with a plain drafting compass or dividers than Cow’s Nose since you continue forming the pattern only from the crossing points. In Cow’s Nose pattern you need to find straight lines and right angles as well. This can be verified by doing both figures manually. Below is a computer generated version of the Flower of Life pattern.
Outside of the scope of the essay, but intriguingly enough: if sound holes of the lyre or harp are called apsamikkum like Lawergren & Gurney in the Sound Holes And Geometrical Figures points, actual shapes of the holes resemble more like equilateral triangles of the FOL than concave squares or rectangles in some cases. It could rather be that apsamikkum refers to the hole, window and frame of the figure, not the exact shape of it. If the FOL symbol had any particular name for ancients, I leave it for future speculations.
Practice from clay tablets (IM 52916, Susu, MS 3051 on “A Remarkable Collection of Babylonian Mathematical Texts” by in Joran Friberg, page 207) shows equilateral triangle and hexagon calculations with the square root of 3 approximations. These arithmetic and geometric properties are fundamental in the FOL. In addition, sexagecimal place value system was already known in the 3rd millenium BC. Geometrical link between circle, hexagon and sexagecimal system is genuinely demonstrated by Jaime Vladimir. Yet the Sumerian brown stone cylinder seal with two six-petal rosettes originates from 3000 BC. Actually similar seals can repeatedly be found from the late Uruk and Jemdet Nasr period ie. 3100 – 2900 BC.
The pillar cone mosaic from the ancient Sumerian city, Uruk in 3400 – 3100 BC, shows a tessellation pattern in the temple pillars. 60 degrees zig-zag rhombus or lozenge pattern formed from two equilateral triangles is inside the FOL geometry. Those who want to go deeper to the triangle study should read “Mysteries of the equilateral triangle” by B.J. McCartin.
We can pretty safely conclude that already in the beginning of the 2nd millenium BC people on ancient Mesopotamia were playing with construction elements and pattern we find in the FOL. On the other hand, conclusion based strictly on artifacts is that four (or eight) pointed flowers and rosettes were more prominent and only later six pointed stars and petals came for decoration motifs and deeper mathematical interest. Still, Assyrian ornament [item 33] in the museum of Pergamo can stretch the origin of the FOL further, even to 3000 BC, if my request to the museum to give more information about the ornaments gets any attention and dating can be confirmed.