From the William Theiss estate, sold some years ago (not to me dammit!) Looks to be a weave thingy (technical term) over gold lame with silver border stitching. Just like the Starfleet emblem that can still be found in craft stores, this patch seems to be based on readily available and inexpensive items.
I looks like it originally had two pointed parts and they ended up folding the lower half underneath to truncate it making it more narrow.
The thing is as seen on a grainy CRT television it could look like anything depending on your imagination. The Starfleet insignia looks like a glittering stylized starburst or nova. It's only under good resolution that it looks like a flower.
I should also give credit where it's due: Thanks Star Trek Prop Authority! http://www.startrekpropauthority.com/2009/10/star-trek-tos-starfleet.html
To me it looked like a cursive capital I. If this symbol had been assigned to the Intrepid, people would have cried Racist, because it also looks like a pointed ear.
One of the things that facinates me about TOS is how they found cost effective solutions that (at least back then) looked so "uncheap." The TOS production team was truly superb; this is yet another example. I always figured the patches worn on the uniforms were some highly stylized, esoteric things.... and after all these years to find out they were low cost off the shelf items found in craft shops, etc. ... My admiration is even greater than before to think they could make a silk purse out of a sows ear so well!
When you watch old Doctor Who serials from the 1960's and then switch to the original Star Trek, the latter suddenly looks like a big-budget production.
Oh, I agree. TOS special effects were excellent for the 60's, probably the best sci-fi effects for their time. In fact, I've seen some Doctor Who episodes from the 70's and 80's that can't touch Trek's effects from the 60's.